On its takeoff roll, the airplane starts from restand accelerates according to a = a0

Problems 2 /1 through 2 /8 treat the motion of a particle which moves along the s-axis shown in the figure. The velocity of a particle is given by \(v=25 t^2-80 t- 200\) , where v is in feet per second and t is in seconds. Plot the velocity v and acceleration a versus time for the first 6 seconds of motion and evaluate the velocity when a is zero.

Problems 2 /1 through 2 /8 treat the motion of a particle which moves along the s-axis shown in the figure. The position of a particle is given by \(s=0.27 t^3- 0.65 t^2-2.35 t+4.4\), where s is in feet and the time t is in seconds. Plot the displacement, velocity, and acceleration as functions of time for the first 5 seconds of motion. Determine the positive time when the particle changes its direction.

Problems 2 /1 through 2 /8 treat the motion of a particle which moves along the s-axis shown in the figure. The velocity of a particle which moves along the s-axis is given by \(v=2-4 t+5 t^{3 / 2}\), where t is in seconds and v is in meters per second. Evaluate the position s, velocity v, and acceleration a when \(t=3 \mathrm{~s}\). The particle is at the position \(s_0=3 \mathrm{~m}\) when t=0.

Problems 2 /1 through 2 /8 treat the motion of a particle which moves along the s-axis shown in the figure. The displacement of a particle which moves along the s-axis is given by \(s=(-2+3 t) e^{-0.5 t}\), where s is in meters and t is in seconds. Plot the displacement, velocity, and acceleration versus time for the first 20 seconds of motion. Determine the time at which the acceleration is zero.

Problems 2 /1 through 2 /8 treat the motion of a particle which moves along the s-axis shown in the figure. The acceleration of a particle is given by a=2t-10, where a is in meters per second squared and t is in seconds. Determine the velocity and displacement as functions of time. The initial displacement at t=0 is \(s_0=-4 \mathrm{~m}\), and the initial velocity is \(v_0=3 \mathrm{~m} / \mathrm{s}\).

Problems 2 /1 through 2 /8 treat the motion of a particle which moves along the s-axis shown in the figure. The acceleration of a particle is given by \(a=-k t^2\), where a is in meters per second squared and the time t is in seconds. If the initial velocity of the particle at t=0 is \(v_0=12 \mathrm{~m} / \mathrm{s}\) and the particle takes 6 seconds to reverse direction, determine the magnitude and units of the constant k. What is the net displacement of the particle over the same 6 -second interval of motion?

Problems 2 /1 through 2 /8 treat the motion of a particle which moves along the s-axis shown in the figure. The acceleration of a particle is given by \(a=-k s^2\), where a is in meters per second squared, k is a constant, and s is in meters. Determine the velocity of the particle as a function of its position s. Evaluate your expression for \(s=5 \mathrm{~m}\) if \(k=0.1 \mathrm{~m}^{-1} \mathrm{~s}^{-2}\) and the initial conditions at time t = 0 are \(s_0=3 \mathrm{~m}\) and \(v_0=10 \mathrm{~m} / \mathrm{s}\).

Problems 2/1 through 2/8 treat the motion of a particle which moves along the s-axis shown in the figure. The acceleration of a particle is given by \(a=c_1+c_2 v\), where a is in millimeters per second squared, the velocity v is in millimeters per second, and \(c_1\) and \(c_2\) are constants. If the particle position and velocity at t = 0 are \(s_0\) and \(v_0\), respectively, determine expressions for the position s of the particle in terms of the velocity v and time t.

Calculate the constant acceleration a in g's which the catapult of an aircraft carrier must provide to produce a launch velocity of \(180 \mathrm{mi} / \mathrm{hr}\) in a distance of \(300 \mathrm{ft}\). Assume that the carrier is at anchor.

A particle in an experimental apparatus has a velocity given by \(v=k \sqrt{s}\), where v is in millimeters per second, the position s is millimeters, and the constant \(k=0.2 \mathrm{~mm}^{1 / 2} \mathrm{~s}^{-1}\). If the particle has a velocity \(v_0=3 \mathrm{~mm} / \mathrm{s}\) at t = 0, determine the particle position, velocity, and acceleration as functions of time, and compute the time, position, and acceleration of the particle when the velocity reaches \(15 \mathrm{~mm} / \mathrm{s}\).

Ball 1 is launched with an initial vertical velocity \(v_1=160 \mathrm{ft} / \mathrm{sec}\). Three seconds later, ball 2 is launched with an initial vertical velocity \(v_2\). Determine \(v_2\) if the balls are to collide at an altitude of \(300 \mathrm{ft}\). At the instant of collision, is ball 1 ascending or descending?

Experimental data for the motion of a particle along a straight line yield measured values of the velocity v for various position coordinates s. A smooth curve is drawn through the points as shown in the graph. Determine the acceleration of the particle when \(s=20 \mathrm{ft}\).

In the pinewood-derby event shown, the car is released from rest at the starting position A and then rolls down the incline and on to the finish line C. If the constant acceleration down the incline is \(2.75 \mathrm{~m} / \mathrm{s}^2\) and the speed from B to C is essentially constant, determine the time duration \(t_{A C}\) for the race. The effects of the small transition area at B can be neglected.

A ball is thrown vertically up with a velocity of \(30 \mathrm{~m} / \mathrm{s}\) at the edge of a \(60-\mathrm{m}\) cliff. Calculate the height h to which the ball rises and the total time t after release for the ball to reach the bottom of the cliff. Neglect air resistance and take the downward acceleration to be \(9.81 \mathrm{~m} / \mathrm{s}^2\).

A car comes to a complete stop from an initial speed of \(50 \mathrm{mi} / \mathrm{hr}\) in a distance of \(100 \mathrm{ft}\). With the same constant acceleration, what would be the stopping distance s from an initial speed of \(70 \mathrm{mi} / \mathrm{hr}\) ?

The pilot of a jet transport brings the engines to full takeoff power before releasing the brakes as the aircraft is standing on the runway. The jet thrust remains constant, and the aircraft has a near constant acceleration of \(0.4 \mathrm{~g}\). If the takeoff speed is \(200 \mathrm{~km} / \mathrm{h}\), calculate the distance s and time t from rest to takeoff.

A game requires that two children each throw a ball upward as high as possible from point O and then run horizontally in opposite directions away from O. The child who travels the greater distance before their thrown ball impacts the ground wins. If child A throws a ball upward with a speed of \(v_1=70 \mathrm{ft} / \mathrm{sec}\) and immediately runs leftward at a constant speed of \(v_A=16 \mathrm{ft} / \mathrm{sec}\) while child B throws the ball upward with a speed of \(v_2=64 \mathrm{ft} / \mathrm{sec}\) and immediately runs rightward with a constant speed of \(v_B=18 \mathrm{ft} / \mathrm{sec}\), which child will win the game?

During an 8-second interval, the velocity of a particle moving in a straight line varies with time as shown. Within reasonable limits of accuracy, determine the amount \(\Delta a\) by which the acceleration at \(t=4 \mathrm{~s}\) exceeds the average acceleration during the interval. What is the displacement \(\Delta s\) during the interval?

In the final stages of a moon landing, the lunar module descends under retrothrust of its descent engine to within \(h=5 \mathrm{~m}\) of the lunar surface where it has a downward velocity of \(2 \mathrm{~m} / \mathrm{s}\). If the descent engine is cut off abruptly at this point, compute the impact velocity of the landing gear with the moon. Lunar gravity is \(\frac{1}{6}\) of the earth's gravity.

A girl rolls a ball up an incline and allows it to return to her. For the angle \(\theta\) and ball involved, the acceleration of the ball along the incline is constant at \(0.25 \mathrm{~g}\), directed down the incline. If the ball is released with a speed of \(4 \mathrm{~m} / \mathrm{s}\), determine the distance s it moves up the incline before reversing its direction and the total time t required for the ball to return to the child's hand.

At a football tryout, a player runs a 40-yard dash in 4.25 seconds. If he reaches his maximum speed at the 16-yard mark with a constant acceleration and then maintains that speed for the remainder of the run, determine his acceleration over the first 16 yards, his maximum speed, and the time duration of the acceleration.

The main elevator A of the CN Tower in Toronto rises about \(350 \mathrm{~m}\) and for most of its run has a constant speed of \(22 \mathrm{~km} / \mathrm{h}\). Assume that both the acceleration and deceleration have a constant magnitude of \(\frac{1}{4}\)g and determine the time duration t of the elevator run.

A Scotch-yoke mechanism is used to convert rotary motion into reciprocating motion. As the disk rotates at the constant angular rate \(\omega\), a pin A slides in a vertical slot causing the slotted member to displace horizontally according to \(x=r \sin (\omega t)\) relative to the fixed disk center O. Determine the expressions for the velocity and acceleration of a point P on the output shaft of the mechanism as functions of time, and determine the maximum velocity and acceleration of point P during one cycle. Use the values \(r=75 \mathrm{~mm}\) and \(\omega=\pi \mathrm{rad} / \mathrm{s}\).

A train which is traveling at \(80 \mathrm{mi} / \mathrm{hr}\) applies its brakes as it reaches point A and slows down with a constant deceleration. Its decreased velocity is observed to be \(60 \mathrm{mi} / \mathrm{hr}\) as it passes a point \(1 / 2 \mathrm{mi}\) beyond A. A car moving at \(50 \mathrm{mi} / \mathrm{hr}\) passes point B at the same instant that the train reaches point A. In an unwise effort to beat the train to the crossing, the driver "steps on the gas." Calculate the constant acceleration a that the car must have in order to beat the train to the crossing by 4 seconds and find the velocity v of the car as it reaches the crossing.

Small steel balls fall from rest through the opening at A at the steady rate of two per second. Find the vertical separation h of two consecutive balls when the lower one has dropped 3 meters. Neglect air resistance.

Car A is traveling at a constant speed \(v_A=130 \mathrm{~km} / \mathrm{h}\) at a location where the speed limit is \(100 \mathrm{~km} / \mathrm{h}\). The police officer in car P observes this speed via radar. At the moment when A passes P, the police car begins to accelerate at the constant rate of \(6 \mathrm{~m} / \mathrm{s}^2\) until a speed of \(160 \mathrm{~km} / \mathrm{h}\) is achieved, and that speed is then maintained. Determine the distance required for the police officer to overtake car A. Neglect any non rectilinear motion of P.

A toy helicopter is flying in a straight line at a constant speed of \(4.5 \mathrm{~m} / \mathrm{s}\). If a projectile is launched vertically with an initial speed of \(v_0=28 \mathrm{~m} / \mathrm{s}\), what horizontal distance d should the helicopter be from the launch site S if the projectile is to be traveling downward when it strikes the helicopter? Assume that the projectile travels only in the vertical direction.

A particle moving along a straight line has an acceleration which varies according to position as shown. If the velocity of the particle at the position \(x=-5 \mathrm{ft}\) is \(v=-2 \mathrm{ft} / \mathrm{sec}\), determine the velocity when \(x= 9 \mathrm{ft}\).

A model rocket is launched from rest with a constant upward acceleration of \(3 \mathrm{~m} / \mathrm{s}^2\) under the action of a small thruster. The thruster shuts off after 8 seconds, and the rocket continues upward until it reaches its apex. At apex, a small chute opens which ensures that the rocket falls at a constant speed of \(0.85 \mathrm{~m} / \mathrm{s}\) until it impacts the ground. Determine the maximum height h attained by the rocket and the total flight time. Neglect aerodynamic drag during ascent, and assume that the mass of the rocket and the acceleration of gravity are both constant.

An electric car is subjected to acceleration tests along a straight and level test track. The resulting v - t data are closely modeled over the first 10 seconds by the function \(v=24 t-t^2+5 \sqrt{t}\), where t is the time in seconds and v is the velocity in feet per second. Determine the displacement s as a function of time over the interval \(0 \leq t \leq 10 \mathrm{sec}\) and specify its value at time \(t=10 \mathrm{sec}\).

A vacuum-propelled capsule for a high-speed tube transportation system of the future is being designed for operation between two stations A and B, which are \(10 \mathrm{~km}\) apart. If the acceleration and deceleration are to have a limiting magnitude of \(0.6 \mathrm{~g}\) and if velocities are to be limited to \(400 \mathrm{~km} / \mathrm{h}\), determine the minimum time t for the capsule to make the \(10-\mathrm{km}\) trip.

If the velocity v of a particle moving along a straight line decreases linearly with its displacement s from \(20 \mathrm{~m} / \mathrm{s}\) to a value approaching zero at \(s=30 \mathrm{~m}\), determine the acceleration a of the particle when \(s=15 \mathrm{~m}\) and show that the particle never reaches the 30 -m displacement.

A particle moves along the x-axis with the velocity history shown. If the particle is at the position x = -4 in. at time t = 0, plot the corresponding displacement history for the time interval \(0 \leq t \leq 10 \mathrm{sec}\). Additionally, find the net displacement and total distance traveled by the particle for this interval.

The 230,000 -lb space-shuttle orbiter touches down at about \(220 \mathrm{mi} / \mathrm{hr}\). At \(200 \mathrm{mi} / \mathrm{hr}\) its drag parachute deploys. At \(35 \mathrm{mi} / \mathrm{hr}\), the chute is jettisoned from the orbiter. If the deceleration in feet per second squared during the time that the chute is deployed is \(-0.0003 v^2\) (speed v in feet per second), determine the corresponding distance traveled by the orbiter. Assume no braking from its wheel brakes.

Reconsider the rollout of the space-shuttle orbiter of the previous problem. The drag chute is deployed at \(200 \mathrm{mi} / \mathrm{hr}\), the wheel brakes are applied at \(100 \mathrm{mi} / \mathrm{hr}\) until wheelstop, and the drag chute is jettisoned at \(35 \mathrm{mi} / \mathrm{hr}\). If the drag chute results in a deceleration of \(-0.0003 v^2\) (in feet per second squared when the speed v is in feet per second) and the wheel brakes cause a constant deceleration of \(5 \mathrm{ft} / \mathrm{sec}^2\), determine the distance traveled from \(200 \mathrm{mi} / \mathrm{hr}\) to wheelstop.

The cart impacts the safety barrier with speed \(v_0=3.25 \mathrm{~m} / \mathrm{s}\) and is brought to a stop by the nest of nonlinear springs which provide a deceleration \(a= -k_1 x-k_2 x^3\), where x is the amount of spring deflection from the undeformed position and \(k_1\) and \(k_2\) are positive constants. If the maximum spring deflection is \(475 \mathrm{~mm}\) and the velocity at half-maximum deflection is \(2.85 \mathrm{~m} / \mathrm{s}\), determine the values and corresponding units for the constants \(k_1\) and \(k_2\).

The graph shows the rectilinear acceleration of a particle as a function of time over a 12 -second interval. If the particle is at rest at the position \(s_0=0\) at time t = 0, determine the velocity of the particle when (a) \(t=4 \mathrm{~s}\), (b) \(t=8 \mathrm{~s}\), and (c) \(t=12 \mathrm{~s}\).

Compute the impact speed of a body released from rest at an altitude h = 650 miles above the surface of Mars. (a) First assume a constant gravitational acceleration \(g_{m_0}=12.3 \mathrm{ft} / \mathrm{sec}^2\) (equal to that at the surface) and (b) then account for the variation of g with altitude (refer to Art. 1/5). Neglect any effects of atmospheric drag.

The steel ball A of diameter D slides freely on the horizontal rod which leads to the pole face of the electromagnet. The force of attraction obeys an inverse-square law, and the resulting acceleration of the ball is \(a=K /(L-x)^2\), where K is a measure of the strength of the magnetic field. If the ball is released from rest at x = 0, determine the velocity v with which it strikes the pole face.

The falling object has a speed \(v_0\) when it strikes and subsequently deforms the foam arresting material until it comes to rest. The resistance of the foam material to deformation is a function of penetration depth y and object speed v so that the acceleration of the object is \(a=g-k_1 v-k_2 y\), where v is the particle speed in inches per second, y is the penetration depth in inches, and \(k_1\) and \(k_2\) are positive constants. Plot the penetration depth y and velocity v of the object as functions of time over the first five seconds for \(k_1=12 \mathrm{sec}^{-1}, k_2=24 \mathrm{sec}^{-2}\), and \(v_0= 25 \mathrm{in.} / \mathrm{sec}\). Determine the time when the penetration depth reaches 95% of its final value.

The electronic throttle control of a model train is programmed so that the train speed varies with position as shown in the plot. Determine the time t required for the train to complete one lap.

A projectile is fired downward with initial speed \(v_0\) in an experimental fluid and experiences an acceleration \(a=\sigma-\eta v^2\), where \(\sigma\) and \(\eta\) are positive constants and v is the projectile speed. Determine the distance traveled by the projectile when its speed has been reduced to one-half of the initial speed \(v_0\). Also, determine the terminal velocity of the projectile. Evaluate for \(\sigma=0.7 \mathrm{~m} / \mathrm{s}^2, \eta=0.2 \mathrm{~m}^{-1}\), and \(v_0=4 \mathrm{~m} / \mathrm{s}\).

The aerodynamic resistance to motion of a car is nearly proportional to the square of its velocity. Additional frictional resistance is constant, so that the acceleration of the car when coasting may be written \(a=-C_1-C_2 v^2\), where \(C_1\) and \(C_2\) are constants which depend on the mechanical configuration of the car. If the car has an initial velocity \(v_0\) when the engine is disengaged, derive an expression for the distance D required for the car to coast to a stop.

The driver of a car, which is initially at rest at the top A of the grade, releases the brakes and coasts down the grade with an acceleration in feet per second squared given by \(a=3.22-0.004 v^2\), where v is the velocity in feet per second. Determine the velocity \(v_B\) at the bottom B of the grade.

When the effect of aerodynamic drag is included, the y-acceleration of a baseball moving vertically upward is \(a_u=-g-k v^2\), while the acceleration when the ball is moving downward is \(a_d=-g+k v^2\), where k is a positive constant and v is the speed in meters per second. If the ball is thrown upward at \(30 \mathrm{~m} / \mathrm{s}\) from essentially ground level, compute its maximum height h and its speed \(v_f\) upon impact with the ground. Take k to be \(0.006 \mathrm{~m}^{-1}\) and assume that g is constant.

For the baseball of Prob. 2/45 thrown upward with an initial speed of \(30 \mathrm{~m} / \mathrm{s}\), determine the time \(t_u\) from ground to apex and the time \(t_d\) from apex to ground.

The stories of a tall building are uniformly 10 feet in height. A ball A is dropped from the rooftop position shown. Determine the times required for it to pass the 10 feet of the first, tenth, and one hundredth stories (counted from the top). Neglect aerodynamic drag.

Repeat Prob. 2/47, except now include the effects of aerodynamic drag. The drag force causes an acceleration component in \(\mathrm{ft} / \mathrm{sec}^2\) of \(0.005 v^2\) in the direction opposite the velocity vector, where v is in \(\mathrm{ft} / \mathrm{sec}\).

On its takeoff roll, the airplane starts from rest and accelerates according to \(a=a_0-k v^2\), where \(a_0\) is the constant acceleration resulting from the engine thrust and \(-k v^2\) is the acceleration due to aerodynamic drag. If \(a_0=2 \mathrm{~m} / \mathrm{s}^2, k=0.00004 \mathrm{~m}^{-1}\), and v is in meters per second, determine the design length of runway required for the airplane to reach the takeoff speed of \(250 \mathrm{~km} / \mathrm{h}\) if the drag term is (a) excluded and (b) included.

A test projectile is fired horizontally into a viscous liquid with a velocity \(v_0\). The retarding force is proportional to the square of the velocity, so that the acceleration becomes \(a=-k v^2\). Derive expressions for the distance D traveled in the liquid and the corresponding time t required to reduce the velocity to \(v_0 / 2\). Neglect any vertical motion.

A bumper, consisting of a nest of three springs, is used to arrest the horizontal motion of a large mass which is traveling at \(40 \mathrm{ft} / \mathrm{sec}\) as it contacts the bumper. The two outer springs cause a deceleration proportional to the spring deformation. The center spring increases the deceleration rate when the compression exceeds 6 in. as shown on the graph. Determine the maximum compression x of the outer springs.

Car A travels at a constant speed of \(65 \mathrm{mi} / \mathrm{hr}\). When in the position shown at time t = 0, car B has a speed of \(25 \mathrm{mi} / \mathrm{hr}\) and accelerates at a constant rate of \(0.1 \mathrm{~g}\) along its path until it reaches a speed of \(65 \mathrm{mi} / \mathrm{hr}\), after which it travels at that constant speed. What is the steady-state position of car A with respect to car B?

A block of mass m rests on a rough horizontal surface and is attached to a spring of stiffness k. The coefficients of both static and kinetic friction are \(\mu\). The block is displaced a distance \(x_0\) to the right of the unstretched position of the spring and released from rest. If the value of \(x_0\) is large enough, the spring force will overcome the maximum available static friction force and the block will slide toward the unstretched position of the spring with an acceleration \(a=\mu g-\frac{k}{m} x\), where x represents the amount of stretch (or compression) in the spring at any given location in the motion. Use the values \(m=5 \mathrm{~kg}, k=150 \mathrm{~N} / \mathrm{m}, \mu=0.40\), and \(x_0=200 \mathrm{~mm}\) and determine the final spring stretch (or compression) \(x_f\) when the block comes to a complete stop.

The situation of Prob. 2/53 is repeated here. This time, use the values \(m=5 \mathrm{~kg}, k=150 \mathrm{~N} / \mathrm{m}, \mu=0.40\), and \(x_0=500 \mathrm{~mm}\) and determine the final spring stretch (or compression) \(x_f\) when the block comes to a complete stop. (Note: The sign on the \(\mu \mathrm{g}\) term is dictated by the direction of motion for the block and always acts in the direction opposite velocity.)

The vertical acceleration of a certain solid-fuel rocket is given by \(a=k e^{-b t}-c v-g\), where k, b, and c are constants, v is the vertical velocity acquired, and g is the gravitational acceleration, essentially constant for atmospheric flight. The exponential term represents the effect of a decaying thrust as fuel is burned, and the term -cv approximates the retardation due to atmospheric resistance. Determine the expression for the vertical velocity of the rocket t seconds after firing.

The preliminary design for a rapid-transit system calls for the train velocity to vary with time as shown in the plot as the train runs the \(3.2 \mathrm{~km}\) between stations A and B. The slopes of the cubic transition curves (which are of form \(a+b t+c t^2+d t^3\) ) are zero at the end points. Determine the total run time t between the stations and the maximum acceleration.

A projectile is fired vertically from point A with an initial speed of \(255 \mathrm{ft} / \mathrm{sec}\). Relative to an observer located at B, at what times will the line of sight to the projectile make an angle of \(30^{\circ}\) with the horizontal? Compute the magnitude of the speed of the projectile at each time, and ignore the effect of aerodynamic drag on the projectile.

Repeat Prob. 2/57 for the case where aerodynamic drag is included. The magnitude of the drag deceleration is \(k v^2\), where \(k=3.5\left(10^{-3}\right) \mathrm{ft}^{-1}\) and v is the speed in feet per second. The direction of the drag is opposite the motion of the projectile throughout the flight (when the projectile is moving upward, the drag is directed downward, and when the projectile is moving downward, the drag is directed upward).

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) At time \(t=10 \mathrm{~s}\), the velocity of a particle moving in the x-y plane is \(\mathbf{v}=0.1 \mathbf{i}+2 \mathbf{j ~ m} / \mathrm{s}\). By time \(t=10.1 \mathrm{~s}\), its velocity has become \(-0.1 \mathbf{i}+1.8 \mathbf{j ~ m} / \mathrm{s}\). Determine the magnitude \(a_{\mathrm{av}}\) of its average acceleration during this interval and the angle \(\theta\) made by the average acceleration with the positive x-axis.

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) At time t = 0, the position vector of a particle moving in the x-y plane is \(\mathbf{r}=5 \mathbf{i} \mathrm{m}\). By time \(t=0.02 \mathrm{~s}\), its position vector has become \(5.1 \mathbf{i}+0.4 \mathbf{j} \mathrm{m}\). Determine the magnitude \(v_{\text {av }}\) of its average velocity during this interval and the angle \(\theta\) made by the average velocity with the positive x-axis.

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) At time t = 0, a particle is at rest in the x-y plane at the coordinates \(\left(x_0, y_0\right)=(6,0)\) in. If the particle is then subjected to the acceleration components \(a_x= 0.5-0.35 t \mathrm{in} . / \mathrm{sec}^2\) and \(a_y=0.15 t-0.02 t^2 \mathrm{in} . / \mathrm{sec}^2\), determine the coordinates of the particle position when \(t=6 \mathrm{sec}\). Plot the path of the particle during this time period.

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) The rectangular coordinates of a particle which moves with curvilinear motion are given by \(x= 10.25 t+1.75 t^2-0.45 t^3\) and \(y=6.32+14.65 t- 2.48 t^2\), where x and y are in millimeters and the time t is in seconds, beginning from t = 0. Determine the velocity v and acceleration a of the particle when \(t=5 \mathrm{~s}\). Also, determine the time when the velocity of the particle makes an angle of \(45^{\circ}\) with the x-axis.

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) For a certain interval of motion the pin A is forced to move in the fixed parabolic slot by the horizontal slotted arm which is elevated in the y-direction at the constant rate of \(3 \mathrm{in.} / \mathrm{sec}\). All measurements are in inches and seconds. Calculate the velocity v and acceleration a of pin A when \(x=6 \mathrm{in}\).

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) With what minimum horizontal velocity u can a boy throw a rock at A and have it just clear the obstruction at B?

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) Prove the well-known result that, for a given launch speed \(v_0\), the launch angle \(\theta=45^{\circ}\) yields the maximum horizontal range R. Determine the maximum range. (Note that this result does not hold when aerodynamic drag is included in the analysis.)

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) A placekicker is attempting to make a 64-yard field goal. If the launch angle of the football is \(40^{\circ}\), what is the minimum initial speed u which will allow the kicker to succeed?

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) In a basketball game, the point guard A intends to throw a pass to the shooting guard B, who is breaking toward the basket at a constant speed of \(12 \mathrm{ft} / \mathrm{sec}\). If the shooting guard is to catch the ball at a height of \(7 \mathrm{ft}\) at C while in full stride to execute a layup, determine the speed \(v_0\) and launch angle \(\theta\) with which the point guard should throw the ball.

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) A fireworks show is choreographed to have two shells cross paths at a height of 160 feet and explode at an apex of 200 feet under normal weather conditions. If the shells have a launch angle \(\theta=60^{\circ}\) above the horizontal, determine the common launch speed \(v_0\) for the shells, the separation distance d between the launch points A and B, and the time from launch at which the shells explode.

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) If a strong wind induces a constant rightward acceleration of \(16 \mathrm{ft} / \mathrm{sec}^2\) for the fireworks shells of Prob. 2/68, determine the horizontal shift of the crossing point of the shells. Refer to the printed answers for Prob. 2/68 as needed.

(In the following problems where motion as a projectile in air is involved, neglect air resistance unless otherwise stated and use \(g=9.81 \mathrm{~m} / \mathrm{s}^2\) or \(g=32.2 \mathrm{ft} / \mathrm{sec}^2\).) The center of mass G of a high jumper follows the trajectory shown. Determine the component \(v_0\), measured in the vertical plane of the figure, of his takeoff velocity and angle \(\theta\) if the apex of the trajectory just clears the bar at A. (In general, must the mass center G of the jumper clear the bar during a successful jump?)

Electrons are emitted at A with a velocity u at the angle \(\theta\) into the space between two charged plates. The electric field between the plates is in the direction E and repels the electrons approaching the upper plate. The field produces an acceleration of the electrons in the E-direction of eE/m, where is the electron charge and m is its mass. Determine the field strength E which will permit the electrons to cross one-half of the gap between the plates. Also find the distance s.

A boy tosses a ball onto the roof of a house. For the launch conditions shown, determine the slant distance s to the point of impact. Also, determine the angle \(\theta\) which the velocity of the ball makes with the roof at the moment of impact.

A small airplane flying horizontally with a speed of \(180 \mathrm{mi} / \mathrm{hr}\) at an altitude of \(400 \mathrm{ft}\) above a remote valley drops an emergency medical package at A. The package has a parachute which deploys at B and allows the package to descend vertically at the constant rate of \(6 \mathrm{ft} / \mathrm{sec}\). If the drop is designed so that the package is to reach the ground 37 seconds after release at A, determine the horizontal lead L so that the package hits the target. Neglect atmospheric resistance from A to B.

As part of a circus performance, a man is attempting to throw a dart into an apple which is dropped from an overhead platform. Upon release of the apple, the man has a reflex delay of 215 milliseconds before throwing the dart. If the dart is released with a speed \(v_0=14 \mathrm{~m} / \mathrm{s}\), at what distance d below the platform should the man aim if the dart is to strike the apple before it hits the ground?

A marksman fires a practice round from A toward a target B. If the target diameter is \(160 \mathrm{~mm}\) and the target center is at the same altitude as the end of the rifle barrel, determine the range of "shallow" launch angles \(\theta\) for which the round will strike the target. Neglect aerodynamic drag and assume that the round is directed along the vertical centerline of the target. (Note: The word "shallow" indicates a low-flying trajectory for the round.)

The pilot of an airplane carrying a package of mail to a remote outpost wishes to release the package at the right moment to hit the recovery location A. What angle \(\theta\) with the horizontal should the pilot's line of sight to the target make at the instant of release? The airplane is flying horizontally at an altitude of \(100 \mathrm{~m}\) with a velocity of \(200 \mathrm{~km} / \mathrm{h}\).

During a baseball practice session, the cutoff man A executes a throw to the third baseman B. If the initial speed of the baseball is \(v_0=130 \mathrm{ft} / \mathrm{sec}\), what launch angle \(\theta\) is best if the ball is to arrive at third base at essentially ground level?

A particle is launched from point A with a horizontal speed u and subsequently passes through a vertical opening of height b as shown. Determine the distance d which will allow the landing zone for the particle to also have a width b. Additionally, determine the range of u which will allow the projectile to pass through the vertical opening for this value of d.

If the tennis player serves the ball horizontally \((\theta=0)\), calculate its velocity v if the center of the ball clears the \(0.9-\mathrm{m}\) net by \(150 \mathrm{~mm}\). Also find the distance s from the net to the point where the ball hits the court surface. Neglect air resistance and the effect of ball spin.

A golfer is attempting to reach the elevated green by hitting his ball under a low-hanging branch in one tree A, but over the top of a second tree B. For \(v_0=115 \mathrm{mi} / \mathrm{hr}\) and \(\theta=18^{\circ}\), where does the golf ball land first?

If the launch speed of the golf ball of the previous problem remains \(v_0=115 \mathrm{mi} / \mathrm{hr}\), what launch angle \(\theta\) will put the first impact point of the ball closest to the pin? How far from the pin is this impact point?

An outfielder experiments with two different trajectories for throwing to home plate from the position shown: (a) \(v_0=42 \mathrm{~m} / \mathrm{s}\) with \(\theta=8^{\circ}\) and (b) \(v_0=36 \mathrm{~m} / \mathrm{s}\) with \(\theta=12^{\circ}\). For each set of initial conditions, determine the time t required for the baseball to reach home plate and the altitude h as the ball crosses the plate.

A ski jumper has the takeoff conditions shown. Determine the inclined distance d from the takeoff point A to the location where the skier first touches down in the landing zone, and the total time \(t_f\) during which the skier is in the air. For simplicity, assume that the landing zone BC is straight.

A projectile is launched with a speed \(v_0=25 \mathrm{~m} / \mathrm{s}\) from the floor of a 5-m-high tunnel as shown. Determine the maximum horizontal range R of the projectile and the corresponding launch angle \(\theta\).

A boy throws a ball upward with a speed \(v_0=12 \mathrm{~m} / \mathrm{s}\). The wind imparts a horizontal acceleration of \(0.4 \mathrm{~m} / \mathrm{s}^2\) to the left. At what angle \(\theta\) must the ball be thrown so that it returns to the point of release? Assume that the wind does not affect the vertical motion.

A projectile is launched from point O with the initial conditions shown. Determine the impact coordinates for the projectile if (a) \(v_0=60 \mathrm{ft} / \mathrm{sec}\) and \(\theta= 40^{\circ}\) and (b) \(v_0=85 \mathrm{ft} / \mathrm{sec}\) and \(\theta=15^{\circ}\).

A projectile is launched from point A with the initial conditions shown in the figure. Determine the slant distance s which locates the point B of impact. Calculate the time of flight t.

A team of engineering students is designing a catapult to launch a small ball at A so that it lands in the box. If it is known that the initial velocity vector makes a \(30^{\circ}\) angle with the horizontal, determine the range of launch speeds \(v_0\) for which the ball will land inside the box.

A snow blower travels forward at a constant speed \(v_s=1.4 \mathrm{ft} / \mathrm{sec}\) along the straight and level path shown. The snow is ejected with a speed \(v_r=30 \mathrm{ft} / \mathrm{sec}\) relative to the machine at the \(40^{\circ}\) angle indicated. Determine the distance d which locates the snowblower position from which an ejected snow particle ultimately strikes the slender pole P. At what height h above the ground does the snow strike the pole? Points O and Q are at ground level.

Determine the location h of the spot toward which the pitcher must throw if the ball is to hit the catcher's mitt. The ball is released with a speed of \(40 \mathrm{~m} / \mathrm{s}\).

A projectile is launched from point A with \(v_0=30 \mathrm{~m} / \mathrm{s}\) and \(\theta=35^{\circ}\). Determine the x- and y-coordinates of the point of impact.

A projectile is fired with a velocity u at right angles to the slope, which is inclined at an angle \(\theta\) with the horizontal. Derive an expression for the distance R to the point of impact.

A projectile is launched from point A with an initial speed \(v_0=100 \mathrm{ft} / \mathrm{sec}\). Determine the minimum value of the launch angle \(\alpha\) for which the projectile will land at point B.

A projectile is launched from point A and lands on the same level at D. Its maximum altitude is h. Determine and plot the fraction \(f_2\) of the total flight time that the projectile is above the level \(f_1 h\), where \(f_1\) is a fraction which can vary from zero to 1 . State the value of \(f_2\) for \(f_1=\frac{3}{4}\).

A projectile is launched with speed \(v_0\) from point A. Determine the launch angle \(\theta\) which results in the maximum range R up the incline of angle \(\alpha\) (where \(0 \leq \alpha \leq 90^{\circ}\) ). Evaluate your results for \(\alpha=0,30^{\circ}\), and \(45^{\circ}\).

A projectile is ejected into an experimental fluid at time t = 0. The initial speed is \(v_0\) and the angle to the horizontal is \(\theta\). The drag on the projectile results in an acceleration term \(\mathbf{a}_D=-k \mathbf{v}\), where k is a constant and v is the velocity of the projectile. Determine the x- and y-components of both the velocity and displacement as functions of time. What is the terminal velocity? Include the effects of gravitational acceleration.

A test car starts from rest on a horizontal circular track of \(80-\mathrm{m}\) radius and increases its speed at a uniform rate to reach \(100 \mathrm{~km} / \mathrm{h}\) in 10 seconds. Determine the magnitude a of the total acceleration of the car 8 seconds after the start.

If the compact disc is spinning at a constant angular rate \(\dot{\theta}=360 \mathrm{rev} / \mathrm{min}\), determine the magnitudes of the accelerations of points A and B at the instant shown.

The car moves on a horizontal surface without any slippage of its tires. For each of the eight horizontal acceleration vectors, describe in words the instantaneous motion of the car. The car velocity is directed to the left as shown for all cases.

Determine the maximum speed for each car if the normal acceleration is limited to \(0.88 \mathrm{g}\). The roadway is unbanked and level.

An accelerometer C is mounted to the side of the roller-coaster car and records a total acceleration of \(3.5 \mathrm{~g}\) as the empty car passes the bottommost position of the track as shown. If the speed of the car at this position is \(215 \mathrm{~km} / \mathrm{h}\) and is decreasing at the rate of \(8 \mathrm{~km} / \mathrm{h}\) every second, determine the radius of curvature \(\rho\) of the track at the position shown.

The driver of the truck has an acceleration of \(0.4 \mathrm{~g}\) as the truck passes over the top A of the hump in the road at constant speed. The radius of curvature of the road at the top of the hump is \(98 \mathrm{~m}\), and the center of mass G of the driver (considered a particle) is \(2 \mathrm{~m}\) above the road. Calculate the speed v of the truck.

A particle moves along the curved path shown. The particle has a speed \(v_A=12 \mathrm{ft} / \mathrm{sec}\) at time \(t_A\) and a speed \(v_B=14 \mathrm{ft} / \mathrm{sec}\) at time \(t_B\). Determine the average values of the normal and tangential accelerations of the particle between points A and B.

A ship which moves at a steady \(20-\mathrm{knot}\) speed ( \(1 \mathrm{knot}=1.852 \mathrm{~km} / \mathrm{h}\) ) executes a turn to port by changing its compass heading at a constant counterclockwise rate. If it requires 60 seconds to alter course \(90^{\circ}\), calculate the magnitude of the acceleration a of the ship during the turn.

A sprinter practicing for the \(200-\mathrm{m}\) dash accelerates uniformly from rest at A and reaches a top speed of \(40 \mathrm{~km} / \mathrm{h}\) at the \(60-\mathrm{m}\) mark. He then maintains this speed for the next 70 meters before uniformly slowing to a final speed of \(35 \mathrm{~km} / \mathrm{h}\) at the finish line. Determine the maximum horizontal acceleration which the sprinter experiences during the run. Where does this maximum acceleration value occur?

A train enters a curved horizontal section of track at a speed of \(100 \mathrm{~km} / \mathrm{h}\) and slows down with constant deceleration to \(50 \mathrm{~km} / \mathrm{h}\) in 12 seconds. An accelerometer mounted inside the train records a horizontal acceleration of \(2 \mathrm{~m} / \mathrm{s}^2\) when the train is 6 seconds into the curve. Calculate the radius of curvature \(\rho\) of the track for this instant.

A particle moves on a circular path of radius \(r= 0.8 \mathrm{~m}\) with a constant speed of \(2 \mathrm{~m} / \mathrm{s}\). The velocity undergoes a vector change \(\Delta \mathbf{v}\) from A to B. Express the magnitude of \(\Delta \mathbf{v}\) in terms of v and \(\Delta \theta\) and divide it by the time interval \(\Delta t\) between A and B to obtain the magnitude of the average acceleration of the particle for (a) \(\Delta \theta=30^{\circ}\), (b) \(\Delta \theta=15^{\circ}\), and (c) \(\Delta \theta=5^{\circ}\). In each case, determine the percentage difference from the instantaneous value of acceleration.

A particle moves along the curved path shown. If the particle has a speed of \(40 \mathrm{ft} / \mathrm{sec}\) at A at time \(t_A\) and a speed of \(44 \mathrm{ft} / \mathrm{sec}\) at B at time \(t_B\), determine the average values of the acceleration of the particle between A and B, both normal and tangent to the path.

An overhead view of part of a pinball game is shown. If the plunger imparts a speed of \(3 \mathrm{~m} / \mathrm{s}\) to the ball which travels in the smooth horizontal slot, determine the acceleration a of the ball (a) just before it exits the curve at C and (b) when it is halfway between points D and E Use the values \(r=150 \mathrm{~mm}\) and \(\theta=60^{\circ}\).

For the pinball game of Prob. 2/109, if the plunger imparts an initial speed of \(3 \mathrm{~m} / \mathrm{s}\) to the ball at time t = 0, determine the acceleration a of the ball (a) at time \(t=0.08 \mathrm{~s}\) and (b) at time \(t=0.20 \mathrm{~s}\). At point F, the speed of the pinball has decreased by 10% from the initial value, and this decrease may be assumed to occur uniformly over the total distance traveled by the pinball. Use the values \(r=150 \mathrm{~mm}\) and \(\theta=60^{\circ}\).

The speed of a car increases uniformly with time from \(50 \mathrm{~km} / \mathrm{h}\) at A to \(100 \mathrm{~km} / \mathrm{h}\) at B during 10 seconds. The radius of curvature of the hump at A is \(40 \mathrm{~m}\). If the magnitude of the total acceleration of the mass center of the car is the same at B as at A, compute the radius of curvature \(\rho_B\) of the dip in the road at B. The mass center of the car is \(0.6 \mathrm{~m}\) from the road.

A minivan starts from rest on the road whose constant radius of curvature is \(40 \mathrm{~m}\) and whose bank angle is \(10^{\circ}\). The motion occurs in a horizontal plane. If the constant forward acceleration of the minivan is \(1.8 \mathrm{~m} / \mathrm{s}^2\), determine the magnitude a of its total acceleration 5 seconds after starting.

Consider the polar axis of the earth to be fixed in space and compute the magnitudes of the velocity and acceleration of a point P on the earth's surface at latitude \(40^{\circ}\) north. The mean diameter of the earth is \(12742 \mathrm{~km}\) and its angular velocity is \(0.7292\left(10^{-4}\right) \mathrm{rad} / \mathrm{s}\).

The car C increases its speed at the constant rate of \(1.5 \mathrm{~m} / \mathrm{s}^2\) as it rounds the curve shown. If the magnitude of the total acceleration of the car is \(2.5 \mathrm{~m} / \mathrm{s}^2\) at point A where the radius of curvature is \(200 \mathrm{~m}\), compute the speed v of the car at this point.

At the bottom A of the vertical inside loop, the magnitude of the total acceleration of the airplane is 3g. If the airspeed is \(800 \mathrm{~km} / \mathrm{h}\) and is increasing at the rate of \(20 \mathrm{~km} / \mathrm{h}\) per second, calculate the radius of curvature \(\rho\) of the path at A.

A golf ball is launched with the initial conditions shown in the figure. Determine the radius of curvature of the trajectory and the time rate of change of the speed of the ball (a) just after launch and (b) at apex. Neglect aerodynamic drag.

If the golf ball of Prob. 2/116 is launched at time t = 0, determine the two times when the radius of curvature of the trajectory has a value of \(1800 \mathrm{ft}\).

The preliminary design for a "small" space station to orbit the earth in a circular path consists of a ring (torus) with a circular cross section as shown. The living space within the torus is shown in section A, where the "ground level" is \(20 \mathrm{ft}\) from the center of the section. Calculate the angular speed N in revolutions per minute required to simulate standard gravity at the surface of the earth \(\left(32.17 \mathrm{ft} / \mathrm{sec}^2\right)\). Recall that you would be unaware of a gravitational field if you were in a nonrotating spacecraft in a circular orbit around the earth.

A spacecraft S is orbiting Jupiter in a circular path \(1000 \mathrm{km}\) above the surface with a constant speed. Using the gravitational law, calculate the magnitude v of its orbital velocity with respect to Jupiter. Use Table D /2 of Appendix D as needed.

Two cars travel at constant speeds through a curved portion of highway. If the front ends of both cars cross line CC at the same instant, and each driver minimizes his or her time in the curve, determine the distance \(\delta\) which the second car has yet to go along its own path to reach line DD at the instant the first car reaches there. The maximum horizontal acceleration for car A is \(0.60 \mathrm{~g}\) and that for car B is \(0.76 \mathrm{~g}\). Which car crosses line DD first?

The figure shows a portion of a plate cam used in the design of a control mechanism. The motion of pin P in the fixed slot of the plate cam is controlled by the vertical guide A, which travels horizontally at a constant speed of \(6 \mathrm{in.} / \mathrm{sec}\) over the central sinusoidal portion of the slot. Determine the normal component of acceleration when the pin is at the position x = 2 in.

A football player releases a ball with the initial conditions shown in the figure. Determine the radius of curvature \(\rho\) of the path and the time rate of change \(\dot{v}\) of the speed at times \(t=1 \mathrm{sec}\) and \(t=2 \mathrm{sec}\), where t =0 is the time of release from the quarterback's hand.

During a short interval the slotted guides are designed to move according to \(x=16-12 t+4 t^2\) and \(y=2+15 t-3 t^2\), where x and y are in millimeters and t is in seconds. At the instant when \(t=2 \mathrm{~s}\), determine the radius of curvature \(\rho\) of the path of the constrained pin P.

The particle P starts from rest at point A at time t = 0 and changes its speed thereafter at a constant rate of 2g as it follows the horizontal path shown. Determine the magnitude and direction of its total acceleration (a) just before it passes point B, (b) just after it passes point B, and (c) as it passes point C. State your directions relative to the x-axis shown (CCW positive).

In the design of a timing mechanism, the motion of pin P in the fixed circular slot is controlled by the guide A, which is being elevated by its lead screw. Guide A starts from rest with pin P at the lowest point in the circular slot, and accelerates upward at a constant rate until it reaches a speed of \(175 \mathrm{mm} / \mathrm{s}\) at the halfway point of its vertical displacement. The guide then decelerates at a constant rate and comes to a stop with pin P at the uppermost point in the circular slot. Determine the n- and t-components of acceleration of pin P once the pin has traveled \(30^ {\circ}\) around the slot from the starting position.

An earth satellite which moves in the elliptical equatorial orbit shown has a velocity v in space of \(17970 \mathrm{~km} / \mathrm{h}\) when it passes the end of the semiminor axis at A. The earth has an absolute surface value of g of \(9.821 \mathrm{~m} / \mathrm{s}^2\) and has a radius of \(6371 \mathrm{~km}\). Determine the radius of curvature \(\rho\) of the orbit at A.

In the design of a control mechanism, the vertical slotted guide is moving with a constant velocity \(\dot{x}=15 \mathrm{in.} / \mathrm{sec}\) during the interval of motion from \(x=-8 \mathrm{in.}\) to \(x = +8 \mathrm{in.}\). For the instant when \(x=6 \mathrm{in.}\), calculate the n- and t-components of acceleration of the pin P, which is confined to move in the parabolic slot. From these results, determine the radius of curvature \(\rho\) of the path at this position. Verify your result by computing \(\rho\) from the expression cited in Appendix C/10.

In a handling test, a car is driven through the slalom course shown. It is assumed that the car path is sinusoidal and that the maximum lateral acceleration is 0.7g. If the testers wish to design a slalom through which the maximum speed is \(80 \mathrm{~km} / \mathrm{h}\), what cone spacing L should be used?

A particle which moves with curvilinear motion has coordinates in meters which vary with time t in seconds according to \(x=2 t^2+3 t-1\) and y = 5t - 2. Determine the coordinates of the center of curvature C at time \(t=1 \mathrm{~s}\).

A projectile is launched at time t = 0 with the initial conditions shown in the figure. If the wind imparts a constant leftward acceleration of \(5 \mathrm{~m} / \mathrm{s}^2\), plot the n- and t-components of acceleration and the radius of curvature \(\rho\) of the trajectory for the time the projectile is in the air. State the maximum magnitude of each acceleration component along with the time at which it occurs. Additionally, determine the minimum radius of curvature for the trajectory and its corresponding time.

A car P travels along a straight road with a constant speed \(v=65 \mathrm{mi} / \mathrm{hr}\). At the instant when the angle \(\theta=60^{\circ}\), determine the values of \(\dot{r}\) in \(\mathrm{ft} / \mathrm{sec}\) and \(\dot{\theta}\) in \(\mathrm{deg} / \mathrm{sec}\).

The sprinter begins from rest at position A and accelerates along the track. If the stationary tracking camera at O is rotating counterclockwise at the rate of \(12.5 \mathrm{deg} / \mathrm{s}\) when the sprinter passes the 60-m mark, determine the speed v of the sprinter and the value of \(\dot{r}\).

A drone flies over an observer O with constant speed in a straight line as shown. Determine the signs (plus, minus, or zero) for \(r, \dot{r}, \ddot{r}, \theta, \dot{\theta}\), and \(\ddot{\theta}\) for each of the positions A, B, and C.

Motion of the sliding block P in the rotating radial slot is controlled by the power screw as shown. For the instant represented, \(\dot{\theta}=0.1 \mathrm{rad} / \mathrm{s}\), \(\ddot{\theta}=-0.04 \mathrm{rad} / \mathrm{s}^2\), and \(r=300 \mathrm{~mm}\). Also, the screw turns at a constant speed giving \(\dot{r}=40 \mathrm{~mm} / \mathrm{s}\). For this instant, determine the magnitudes of the velocity v and acceleration a of P. Sketch v and a if \(\theta=120^{\circ}\).

Rotation of bar OA is controlled by the lead screw which imparts a horizontal velocity v to collar C and causes pin P to travel along the smooth slot. Determine the values of \(\dot{r}\) and \(\dot{\theta}\), where \(r=\overline{O P}\), if \(h=160 \mathrm{~mm}, x=120 \mathrm{~mm}\), and \(v=25 \mathrm{~mm} / \mathrm{s}\) at the instant represented.

For the bar of Prob. 2/135, determine the values of \(\ddot{r}\) and \(\ddot{\theta}\) if the velocity of collar C is decreasing at a rate of \(5 \mathrm{~mm} / \mathrm{s}^2\) at the instant in question. Refer to the printed answers for Prob. 2/135 as needed.

The boom OAB pivots about point O, while section AB simultaneously extends from within section OA. Determine the velocity and acceleration of the center B of the pulley for the following conditions: \(\theta=20^{\circ}, \dot{\theta}=5 \mathrm{deg} / \mathrm{sec}, \ddot{\theta}=2 \mathrm{deg} / \mathrm{sec}^2, l=7 \mathrm{ft}\), \(\dot{l}=1.5 \mathrm{ft} / \mathrm{sec}, \ddot{i}=-4 \mathrm{ft} / \mathrm{sec}^2\). The quantities \(\dot{l}\) and \(\ddot{l}\) are the first and second time derivatives, respectively, of the length l of section AB.

A particle moving along a plane curve has a position vector r, a velocity v, and an acceleration a. Unit vectors in the r - and \(\theta\)-directions are \(\mathbf{e}_r\) and \(\mathbf{e}_\theta\), respectively, and both r and \(\theta\) are changing with time. Explain why each of the following statements is correctly marked as an inequality. \(\begin{array}{ccl} \dot{\mathbf{r}} \neq v & \ddot{\mathbf{r}} \neq a & \dot{\mathbf{r}} \neq \dot{r} \mathbf{e}_r \\ \dot{r} \neq v & \ddot{r} \neq a & \ddot{\mathbf{r}} \neq \ddot{r} \mathbf{e}_r \\ \dot{r} \neq \mathbf{v} & \ddot{\boldsymbol{r}} \neq \mathbf{a} & \dot{\mathbf{r}} \neq r \dot{\theta} \mathbf{e}_\theta \end{array}\)

Consider the portion of an excavator shown. At the instant under consideration, the hydraulic cylinder is extending at a rate of \(6 \mathrm{in.} / \mathrm{sec}\), which is decreasing at the rate of \(2 \mathrm{in.} / \mathrm{sec}\) every second. Simultaneously, the cylinder is rotating about a horizontal axis through O at a constant rate of \(10 \mathrm{deg} / \mathrm{sec}\). Determine the velocity v and acceleration a of the clevis attachment at B.

The nozzle shown rotates with constant angular speed \(\Omega\) about a fixed horizontal axis through point O. Because of the change in diameter by a factor of 2 , the water speed relative to the nozzle at A is v, while that at B is 4v. The water speeds at both A and B are constant. Determine the velocity and acceleration of a water particle as it passes ( a ) point A and (b) point B.

The radial position of a fluid particle P in a certain centrifugal pump with radial vanes is approximated by \(r=r_0 \cosh K t\), where t is time and \(K=\dot{\theta} is the constant angular rate at which the impeller turns. Determine the expression for the magnitude of the total acceleration of the particle just prior to leaving the vane in terms of \(r_0\), R, and K.

A helicopter starts from rest at point A and travels along the straight-line path with a constant acceleration a. If the speed \(v=28 \mathrm{~m} / \mathrm{s}\) when the altitude of the helicopter is \(h=40 \mathrm{~m}\), determine the values of \(\dot{r}, \ddot{r}, \dot{\theta}\), and \(\ddot{\theta}\) as measured by the tracking device at O. At this instant, \(\theta=40^{\circ}\), and the distance \(d=160 \mathrm{~m}\). Neglect the small height of the tracking device above the ground.

The slider P can be moved inward by means of the string S, while the slotted arm rotates about point O. The angular position of the arm is given by \(\theta=0.8 t-\frac{t^2}{20}\), where \(\theta\) is in radians and t is in seconds. The slider is at \(r=1.6 \mathrm{~m}\) when t = 0 and thereafter is drawn inward at the constant rate of \(0.2 \mathrm{~m} / \mathrm{s}\). Determine the magnitude and direction (expressed by the angle \(\alpha\) relative to the x-axis) of the velocity and acceleration of the slider when \(t=4 \mathrm{~s}\).

Cars A and B are both moving with constant speed v on the straight and level highway. They are side by-side in adjacent lanes as shown. If the radar unit attached to the stationary police car P measures "line-of-sight" velocity, what speed v ' will be observed for each car? Use the values \(v=70 \mathrm{mi} / \mathrm{hr}\), \(L=200 \mathrm{ft}\), and \(D=22 \mathrm{ft}\).

A fireworks shell P is launched upward from point A and explodes at its apex at an altitude of \(275 \mathrm{ft}\). Relative to an observer at O, determine the values of \(\dot{r}\) and \(\dot{\theta}\) when the shell reaches an altitude \(y= 175 \mathrm{ft}\). Neglect aerodynamic drag.

For the fireworks shell of Prob. 2/145, determine the values of \(\ddot{r}\) and \(\ddot{\theta}\) when the shell reaches an altitude \(y=175 \mathrm{ft}\). Refer to the printed answers for Prob. 2/145 as needed.

The rocket is fired vertically and tracked by the radar station shown. When \(\theta\) reaches \(60^{\circ}\), other corresponding measurements give the values \(r= 9 \mathrm{~km}, \ddot{r}=21 \mathrm{~m} / \mathrm{s}^2\), and \(\dot{\theta}=0.02 \mathrm{rad} / \mathrm{s}\). Calculate the magnitudes of the velocity and acceleration of the rocket at this position.

As it passes the position shown, the particle P has a constant speed \(v=100 \mathrm{~m} / \mathrm{s}\) along the straight line shown. Determine the corresponding values of \(\dot{r}, \dot{\theta}, \ddot{r}\), and \(\ddot{\theta}\).

Repeat Prob. 2/148, but now the speed of the particle P is decreasing at the rate of \(20 \mathrm{~m} / \mathrm{s}^2\) as it moves along the indicated straight path.

The diver leaves the platform with an initial upward speed of \(2.5 \mathrm{~m} / \mathrm{s}\). A stationary camera on the ground is programmed to track the diver throughout the dive by rotating the lens to keep the diver centered in the captured image. Plot \(\dot{\theta}\) and \(\ddot{\theta}\) as functions of time for the camera over the entire dive and state the values of \(\dot{\theta}\) and \(\ddot{\theta}\) at the instant the diver enters the water. Treat the diver as a particle which has only vertical motion. Additionally, state the maximum magnitudes of \(\dot{\theta}\) and \(\ddot{\theta}\) during the dive and the times at which they occur.

Instruments located at O are part of the ground traffic control system for a major airport. At a certain instant during the takeoff roll of the aircraft P, the sensors indicate the angle \(\theta=50^{\circ}\) and the range rate \(\dot{r}=140 \mathrm{ft} / \mathrm{sec}\). Determine the corresponding speed v of the aircraft and the value of \(\dot{\theta}\).

In addition to the information supplied in the previous problem, the sensors at O indicate that \(\ddot{r}=14 \mathrm{ft} / \mathrm{sec}^2\). Determine the corresponding acceleration a of the aircraft and the value of \(\ddot{\theta}\).

At the bottom of a loop in the vertical \((r-\theta)\) plane at an altitude of \(400 \mathrm{~m}\), the airplane P has a horizontal velocity of \(600 \mathrm{~km} / \mathrm{h}\) and no horizontal acceleration. The radius of curvature of the loop is \(1200 \mathrm{~m}\). For the radar tracking at O, determine the recorded values of \(\ddot{r}\) and \(\ddot{\theta}\) for this instant.

The member OA of the industrial robot telescopes and pivots about the fixed axis at point O. At the instant shown, \(\theta=60^{\circ}, \dot{\theta}=1.2 \mathrm{rad} / \mathrm{s}, \ddot{\theta}=0.8 \mathrm{rad} / \mathrm{s}^2, \quad \overline{O A}=0.9 \mathrm{~m}, \quad \dot{O A}=0.5 \mathrm{~m} / \mathrm{s}\), and \(\dot{O A}= -6 \mathrm{~m} / \mathrm{s}^2\). Determine the magnitudes of the velocity and acceleration of joint A of the robot. Also, sketch the velocity and acceleration of A and determine the angles which these vectors make with the positive x-axis. The base of the robot does not revolve about a vertical axis.

At the instant depicted in the figure, the radar station at O measures the range rate of the space shuttle P to be \(\dot{r}=-12,272 \mathrm{ft} / \mathrm{sec}\), with O considered fixed. If it is known that the shuttle is in a circular orbit at an altitude \(h=150 \mathrm{mi}\), determine the orbital speed of the shuttle from this information.

A locomotive is traveling on the straight and level track with a speed \(v=90 \mathrm{~km} / \mathrm{h}\) and a deceleration \(a=0.5 \mathrm{~m} / \mathrm{s}^2\) as shown. Relative to the fixed observer at O, determine the quantities \(\dot{r}, \ddot{r}, \dot{\theta}\), and \(\ddot{\theta}\) at the instant when \(theta=60^{\circ}\) and \(r=400 \mathrm{~m}\).

The small block P starts from rest at time t = 0 at point A and moves up the incline with constant acceleration a. Determine \(\dot{r}\) as a function of time.

For the conditions of Prob. 2 /157, determine \(\dot{\theta}\) as a function of time.

An earth satellite traveling in the elliptical orbit shown has a velocity \(v=12,149 \mathrm{mi} / \mathrm{hr}\) as it passes the end of the semiminor axis at A. The acceleration of the satellite at A is due to gravitational attraction and is \(32.23[3959 / 8400]^2=7.159 \mathrm{ft} / \mathrm{sec}^2\) directed from A to O. For position A calculate the values of \(\dot{r}, \ddot{r}, \dot{\theta}\), and \(\ddot{\theta}\).

A meteor P is tracked by a radar observatory on the earth at O. When the meteor is directly overhead \(\left(\theta=90^{\circ}\right)\), the following observations are recorded: \(r=80 \mathrm{~km}, \dot{r}=-20 \mathrm{~km} / \mathrm{s}\), and \(\dot{\theta}=0.4 \mathrm{rad} / \mathrm{s}\). (a) Determine the speed v of the meteor and the angle \(\beta\) which its velocity vector makes with the horizontal. Neglect any effects due to the earth's rotation. (b) Repeat with all given quantities remaining the same, except that \(\theta=75^{\circ}\).

The low-flying aircraft P is traveling at a constant speed of \(360 \mathrm{~km} / \mathrm{h}\) in the holding circle of radius \(3 \mathrm{~km}\). For the instant shown, determine the quantities r, \(\dot{r}, \ddot{r}, \theta, \dot{\theta}\), and \(\ddot{\theta}\) relative to the fixed x-y coordinate system, which has its origin on a mountaintop at O. Treat the system as two-dimensional.

At time t = 0, the baseball player releases a ball with the initial conditions shown in the figure. Determine the quantities r, \(\dot{r}, \ddot{r}, \theta, \dot{\theta}\), and \(\ddot{\theta}\), all relative to the x-y coordinate system shown, at time \(t=0.5 \mathrm{sec}\).

The racing airplane is beginning an inside loop in the vertical plane. The tracking station at O records the following data for a particular instant: \(r=90 \mathrm{~m}, \dot{r}=15.5 \mathrm{~m} / \mathrm{s}, \ddot{r}=74.5 \mathrm{~m} / \mathrm{s}^2, \theta=30^{\circ}\), \(\dot{\theta}=0.53 \mathrm{rad} / \mathrm{s}\), and \(\ddot{\theta}=-0.29 \mathrm{rad} / \mathrm{s}^2\). Determine the values of v, \(\dot{v}, \rho\), and \(\beta\) at this instant.

A golf ball is driven with the initial conditions shown in the figure. If the wind imparts a constant horizontal deceleration of \(4 \mathrm{ft} / \mathrm{sec}^2\), determine the values of r, \(\dot{r}, \ddot{r}, \theta, \dot{\theta}\), and \(\ddot{\theta}\) when \(t=1.05 \mathrm{sec}\). Take the r-coordinate to be measured from the origin.

The rectangular coordinates of a particle are given in millimeters as functions of time t in seconds by x = 30 cos 2t, y = 40 sin 2t, and \(z=20 t+3 t^2\). Determine the angle \(\theta_1\) between the position vector r and the velocity v and the angle \(\theta_2\) between the position vector r and the acceleration a, both at time \(t=2 \mathrm{~s}\).

A projectile is launched from point O with an initial velocity of magnitude \(v_0=600 \mathrm{ft} / \mathrm{sec}\), directed as shown in the figure. Compute the x-, y- and z-components of position, velocity, and acceleration 20 seconds after launch. Neglect aerodynamic drag.

A projectile is launched from point O at a speed \(v_0=80 \mathrm{~m} / \mathrm{s}\) with the goal of hitting the target A. At the launch instant, a strong horizontal wind begins blowing and imparts a constant acceleration of \(1.25 \mathrm{~m} / \mathrm{s}^2\) to the projectile in the same direction as the wind. If the launch conditions are chosen so that the particle would impact the target in the absence of wind effects, determine the impact coordinates of the projectile. (Note: There is more than one solution to this problem, so choose only answers which feature positive angles less than \(90^{\circ}\). Select the impact location which is closest to the target.)

If the launch speed of the projectile in Prob. 2/167 remains unchanged, what values of \(\theta\) and \(\phi\) (positive and less than \(90^{\circ}\) ) will ensure that the projectile impacts the target at A if the wind conditions are considered. Please list both possible combinations of the angles.

An amusement ride called the "corkscrew" takes the passengers through the upside-down curve of a horizontal cylindrical helix. The velocity of the cars as they pass position A is \(15 \mathrm{~m} / \mathrm{s}\), and the component of their acceleration measured along the tangent to the path is \(g \cos \gamma\) at this point. The effective radius of the cylindrical helix is \(5 \mathrm{~m}\), and the helix angle is \(\gamma=40^{\circ}\). Compute the magnitude of the acceleration of the passengers as they pass position A.

The radar antenna at P tracks the jet aircraft A, which is flying horizontally at a speed u and an altitude h above the level of P. Determine the expressions for the components of the velocity in the spherical coordinates of the antenna motion.

The rotating element in a mixing chamber is given a periodic axial movement \(z=z_0 \sin 2 \pi n t\) while it is rotating at the constant angular velocity \(\dot{\theta}=\omega\). Determine the expression for the maximum magnitude of the acceleration of a point A on the rim of radius r. The frequency n of vertical oscillation is constant.

A helicopter starts from rest at point A and travels along the indicated path with a constant acceleration a. If the helicopter has a speed of \(60 \mathrm{~m} / \mathrm{s}\) when it reaches B, determine the values of \(\dot{R}, \dot{\theta}\), and \(\dot{\phi}\) as measured by the radar tracking device at O at the instant when \(h=100 \mathrm{~m}\).

For the helicopter of Prob. 2/172, find the values of \(\ddot{R}, \ddot{\theta}\), and \(\ddot{\phi}\) for the radar tracking device at O at the instant when \(h=100 \mathrm{~m}\). Refer to the printed answers for Prob. 2/172 as needed.

The vertical shaft of the industrial robot rotates at the constant rate \(\omega\). The length h of the vertical shaft has a known time history, and this is true of its time derivatives \(\dot{h}\) and \(\ddot{h}\) as well. Likewise, the values of l, \(\dot{l}\), and \(\ddot{l}\) are known. Determine the magnitudes of the velocity and acceleration of point P. The lengths \(h_0\) and \(l_0\) are fixed.

An industrial robot is being used to position a small part P. Calculate the magnitude of the acceleration a of P for the instant when \(\beta=30^{\circ}\) if \(\dot{\beta}=10 \mathrm{deg} / \mathrm{s}\) and \(\ddot{\beta}=20 \mathrm{deg} / \mathrm{s}^2\) at this same instant. The base of the robot is revolving at the constant rate \(\omega=40 \mathrm{deg} / \mathrm{s}\). During the motion arms AO and AP remain perpendicular.

The car A is ascending a parking-garage ramp in the form of a cylindrical helix of \(24- \mathrm{ft}\) radius rising \(10 \mathrm{ft}\) for each half turn. At the position shown the car has a speed of \(15 \mathrm{mi} / \mathrm{hr}\), which is decreasing at the rate of \(2 \mathrm{mi} / \mathrm{hr}\) per second. Determine the r-, \(\theta-\), and z-components of the acceleration of the car.

The base structure of the firetruck ladder rotates about a vertical axis through O with a constant angular velocity \(\Omega=10 \mathrm{deg} / \mathrm{s}\). At the same time, the ladder unit OB elevates at a constant rate \(\dot{\phi}=7 \mathrm{deg} / \mathrm{s}\), and section AB of the ladder extends from within section OA at the constant rate of \(0.5 \mathrm{~m} / \mathrm{s}\). At the instant under consideration, \(\phi=30^{\circ}, \overline{O A}=9 \mathrm{~m}\), and \(\overline{A B}=6 \mathrm{~m}\). Determine the magnitudes of the velocity and acceleration of the end B of the ladder.

The rod OA is held at the constant angle \(\beta=30^{\circ}\) while it rotates about the vertical with a constant angular rate \(\dot{\theta}=120 \mathrm{rev} / \mathrm{min}\). Simultaneously, the sliding ball P oscillates along the rod with its distance in millimeters from the fixed pivot O given by \(R=200+50 \sin 2 \pi n t\), where the frequency n of oscillation along the rod is a constant 2 cycles per second and where t is the time in seconds. Calculate the magnitude of the acceleration of P for an instant when its velocity along the rod from O toward A is a maximum.

Beginning with Eq. 2/18, the expression for particle velocity in spherical coordinates, derive the acceleration components in Eq. 2/19. (Note: Start by writing the unit vectors for the R-, \(\theta-\), and \(\phi\)-coordinates in terms of the fixed unit vectors i, j, and k.)

In the design of an amusement-park ride, the cars are attached to arms of length R which are hinged to a central rotating collar which drives the assembly about the vertical axis with a constant angular rate \(\omega=\dot{\theta}\). The cars rise and fall with the track according to the relation \(z=(h / 2)(1-\cos 2 \theta)\). Find the R-, \(\theta-\), and \(\phi\)-components of the velocity v of each car as it passes the position \(\theta=\pi / 4 \mathrm{rad}\).

The particle P moves down the spiral path which is wrapped around the surface of a right circular cone of base radius b and altitude h. The angle \(\gamma\) between the tangent to the curve at any point and a horizontal tangent to the cone at this point is constant. Also the motion of the particle is controlled so that \(\dot{\theta}\) is constant. Determine the expression for the radial acceleration \(a_r\) of the particle for any value of \(\theta\).

The disk A rotates about the vertical z-axis with a constant speed \(\omega=\dot{\theta}=\pi / 3 \mathrm{rad} / \mathrm{s}\). Simultaneously, the hinged arm OB is elevated at the constant rate \(\dot{\phi}=2 \pi / 3 \mathrm{rad} / \mathrm{s}\). At time t = 0, both \(\theta=0\) and \(\phi=0\). The angle \(\theta\) is measured from the fixed reference x-axis. The small sphere P slides out along the rod according to \(R=50+200 t^2\), where R is in millimeters and t is in seconds. Determine the magnitude of the total acceleration a of P when \(t=\frac{1}{2} \mathrm{~s}\).

Rapid-transit trains A and B travel on parallel tracks. Train A has a speed of \(80 \mathrm{~km} / \mathrm{h}\) and is slowing at the rate of \(2 \mathrm{~m} / \mathrm{s}^2\), while train B has a constant speed of \(40 \mathrm{~km} / \mathrm{h}\). Determine the velocity and acceleration of train B relative to train A.

Train A is traveling at a constant speed \(v_A= 35 \mathrm{mi} / \mathrm{hr}\) while car B travels in a straight line along the road as shown at a constant speed \(v_B\). A conductor C in the train begins to walk to the rear of the train car at a constant speed of \(4 \mathrm{ft} / \mathrm{sec}\) relative to the train. If the conductor perceives car B to move directly westward at \(16 \mathrm{ft} / \mathrm{sec}\), how fast is the car traveling?

The jet transport B is flying north with a velocity \(v_B=600 \mathrm{~km} / \mathrm{h}\) when a smaller aircraft A passes underneath the transport headed in the \(60^{\circ}\) direction shown. To passengers in B, however, A appears to be flying sideways and moving east. Determine the actual velocity of A and the velocity which A appears to have relative to B.

A helicopter approaches a rescue scene. A victim P is drifting along with the river current of speed \(v_C=2 \mathrm{~m} / \mathrm{s}\). The wind is blowing at a speed \(v_W= 3 \mathrm{~m} / \mathrm{s}\) as indicated. Determine the velocity relative to the wind which the helicopter must acquire so that it maintains a steady overhead position relative to the victim.

A ship capable of making a speed of 16 knots through still water is to maintain a true course due west while encountering a 3-knot current running from north to south. What should be the heading of the ship (measured clockwise from the north to the nearest degree)? How long does it take the ship to proceed 24 nautical miles due west?

Train A travels with a constant speed \(v_A=120 \mathrm{km} / \mathrm{h}\) along the straight and level track. The driver of car B, anticipating the railway grade crossing C, decreases the car speed of \(90 \mathrm{~km} / \mathrm{h}\) at the rate of \(3 \mathrm{~m} / \mathrm{s}^2\). Determine the velocity and acceleration of the train relative to the car.

The car A has a forward speed of \(18 \mathrm{~km} / \mathrm{h}\) and is accelerating at \(3 \mathrm{~m} / \mathrm{s}^2\). Determine the velocity and acceleration of the car relative to observer B, who rides in a nonrotating chair on the Ferris wheel. The angular rate \(\Omega=3 \mathrm{rev} / \mathrm{min}\) of the Ferris wheel is constant.

For the instant represented, car A has an acceleration in the direction of its motion, and car B has a speed of \(45 \mathrm{mi} / \mathrm{hr}\) which is increasing. If the acceleration of B as observed from A is zero for this instant, determine the acceleration of A and the rate at which the speed of B is changing.

A drop of water falls with no initial speed from point A of a highway overpass. After dropping \(6 \mathrm{~m}\), it strikes the windshield at point B of a car which is traveling at a speed of \(100 \mathrm{~km} / \mathrm{h}\) on the horizontal road. If the windshield is inclined \(50^{\circ}\) from the vertical as shown, determine the angle \(\theta\) relative to the normal n to the windshield at which the water drop strikes.

Plane A travels along the indicated path with a constant speed \(v_A=285 \mathrm{~km} / \mathrm{h}\). Relative to the pilot in plane B, which is flying at a constant speed \(v_B=350 \mathrm{~km} / \mathrm{h}\), what are the velocities which plane A appears to have when it is at positions C and E? Both planes are flying horizontally.

For the planes of Prob. 2/192, beginning at the position shown, plane A increases its speed at a constant rate and acquires a speed of \(415 \mathrm{~km} / \mathrm{h}\) by the time it reaches position E, while plane B experiences a steady deceleration of \(1.5 \mathrm{~m} / \mathrm{s}^2\). Relative to the pilot in plane B, what are the velocities and accelerations which plane A appears to have when it is at positions C and E?

A sailboat moving in the direction shown is tacking to windward against a north wind. The log registers a hull speed of 6.5 knots. A "telltale" (light string tied to the rigging) indicates that the direction of the apparent wind is \(35^{\circ}\) from the centerline of the boat. What is the true wind velocity \(v_w\) ?

At the instant illustrated, car B has a speed of \(30 \mathrm{~km} / \mathrm{h}\) and car A has a speed of \(40 \mathrm{~km} / \mathrm{h}\). Determine the values of \(\dot{r}\) and \(\dot{\theta}\) for this instant where r and \(\theta\) are measured relative to a longitudinal axis fixed to car B as indicated in the figure.

For the cars of Prob. 2/195, determine the instantaneous values of \(\ddot{r}\) and \(\ddot{\theta}\) if car A is slowing down at a rate of \(1.25 \mathrm{~m} / \mathrm{s}^2\) and car B is speeding up at a rate of \(2.5 \mathrm{~m} / \mathrm{s}^2\). Refer to the printed answers for Prob. 2/195 as needed.

Car A is traveling at \(25 \mathrm{mi} / \mathrm{hr}\) and applies the brakes at the position shown so as to arrive at the intersection C at a complete stop with a constant deceleration. Car B has a speed of \(40 \mathrm{mi} / \mathrm{hr}\) at the instant represented and is capable of a maximum deceleration of \(18 \mathrm{ft} / \mathrm{sec}^2\). If the driver of car B is distracted, and does not apply his brakes until 1.30 seconds after car A begins to brake, the result being a collision with car A, with what relative speed will car B strike car A ? Treat both cars as particles.

As part of an unmanned-autonomous-vehicle (UAV) demonstration, an unmanned vehicle B launches a projectile A from the position shown while traveling at a constant speed of \(30 \mathrm{~km} / \mathrm{h}\). The projectile is launched with a speed of \(70 \mathrm{~m} / \mathrm{s}\) relative to the vehicle. At what launch angle \(\alpha\) should the projectile be fired to ensure that it strikes a target at C ? Compare your answer with that for the case where the vehicle is stationary.

The shuttle orbiter A is in a circular orbit of altitude \(200 \mathrm{mi}\), while spacecraft B is in a geosynchronous circular orbit of altitude 22,300 mi. Determine the acceleration of B relative to a nonrotating observer in shuttle A. Use \(g_0=32.23 \mathrm{ft} / \mathrm{sec}^2\) for the surface-level gravitational acceleration and \(R=3959 \mathrm{mi}\) for the radius of the earth.

After starting from the position marked with the "x", a football receiver B runs the slant-in pattern shown, making a cut at P and thereafter running with a constant speed \(v_B=7 \mathrm{yd} / \mathrm{sec}\) in the direction shown. The quarterback releases the ball with a horizontal velocity of \(100 \mathrm{ft} / \mathrm{sec}\) at the instant the receiver passes point P. Determine the angle \(\alpha\) at which the quarterback must throw the ball, and the velocity of the ball relative to the receiver when the ball is caught. Neglect any vertical motion of the ball.

Car A is traveling at the constant speed of \(60 \mathrm{~km} / \mathrm{h}\) as it rounds the circular curve of \(300-\mathrm{m}\) radius and at the instant represented is at the position \(\theta=45^{\circ}\). Car B is traveling at the constant speed of \(80 \mathrm{~km} / \mathrm{h}\) and passes the center of the circle at this same instant. Car A is located with respect to car B by polar coordinates r and \(\theta\) with the pole moving with B. For this instant determine \(v_{A / B}\) and the values of \(\dot{r}\) and \(\dot{\theta}\) as measured by an observer in car B.

For the conditions of Prob. 2/201, determine the values of \(\ddot{r}\) and \(\ddot{\theta}\) as measured by an observer in car B at the instant represented. Use the results for \(\dot{r}\) and \(\dot{\theta}\) cited in the answers for that problem.

A batter hits the baseball A with an initial velocity of \(v_0=100 \mathrm{ft} / \mathrm{sec}\) directly toward fielder B at an angle of \(30^{\circ}\) to the horizontal; the initial position of the ball is \(3 \mathrm{ft}\) above ground level. Fielder B requires \(\frac{1}{4}\) sec to judge where the ball should be caught and begins moving to that position with constant speed. Because of great experience, fielder B chooses his running speed so that he arrives at the "catch position" simultaneously with the baseball. The catch position is the field location at which the ball altitude is \(7 \mathrm{ft}\). Determine the velocity of the ball relative to the fielder at the instant the catch is made.

A place kicker A executes a "pooch" kick, which is designed to eliminate a potential return by the receiving team. The "pooch" kick features a high trajectory and short range, thereby preventing the deep kick returner B from reaching his maximum speed before encountering coverage. To offset this, player B hesitates for \(1.15 \mathrm{sec}\) after the ball is kicked, and then accelerates at a uniform rate, reaching his maximum speed at the instant he catches the ball. Determine the velocity of the football relative to the receiver at the instant the catch is made if the football is caught when it is \(4.5 \mathrm{ft}\) above the playing surface.

The aircraft A with radar detection equipment is flying horizontally at an altitude of \(12 \mathrm{~km}\) and is increasing its speed at the rate of \(1.2 \mathrm{~m} / \mathrm{s}\) each second. Its radar locks onto an aircraft B flying in the same direction and in the same vertical plane at an altitude of \(18 \mathrm{~km}\). If A has a speed of \(1000 \mathrm{~km} / \mathrm{h}\) at the instant when \(\theta=30^{\circ}\), determine the values of \(\ddot{r}\) and \(\ddot{\theta}\) at this same instant if B has a constant speed of \(1500 \mathrm{~km} / \mathrm{h}\).

At a certain instant after jumping from the airplane A, a skydiver B is in the position shown and has reached a terminal (constant) speed \(v_B=50 \mathrm{~m} / \mathrm{s}\). The airplane has the same constant speed \(v_A=50 \mathrm{~m} / \mathrm{s}\), and after a period of level flight is just beginning to follow the circular path shown of radius \(\rho_A=2000 \mathrm{~m}\). (a) Determine the velocity and acceleration of the airplane relative to the skydiver. (b) Determine the time rate of change of the speed \(v_r\) of the airplane and the radius of curvature \(\rho_r\) of its path, both as observed by the nonrotating skydiver.

If the velocity \(\dot{x}\) of block A up the incline is increasing at the rate of \(0.044 \mathrm{~m} / \mathrm{s}\) each second, determine the acceleration of B.

At the instant represented, \(v}_{B / A}=3.5 \mathbf{j} \mathrm{m} / \mathrm{s}\) Determine the velocity of each body at this instant. Assume that the upper surface of A remains horizontal.

At a certain instant, the velocity of cylinder B is \(1.2 \mathrm{~m} / \mathrm{s}\) down and its acceleration is \(2 \mathrm{~m} / \mathrm{s}^2\) up. Determine the corresponding velocity and acceleration of block A.

Determine the velocity of cart A if cylinder B has a downward velocity of \(2 \mathrm{ft} / \mathrm{sec}\) at the instant illustrated. The two pulleys at C are pivoted independently.

An electric motor M is used to reel in cable and hoist a bicycle into the ceiling space of a garage. Pulleys are fastened to the bicycle frame with hooks at locations A and B, and the motor can reel in cable at a steady rate of \(12 \mathrm{in} . / \mathrm{sec}\). At this rate, how long will it take to hoist the bicycle 5 feet into the air? Assume that the bicycle remains level.

Determine the relation which governs the accelerations of A, B, and C, all measured positive down. Identify the number of degrees of freedom.

Determine an expression for the velocity \(v_A\) of the cart A down the incline in terms of the upward velocity \(v_B\) of cylinder B.

Neglect the diameters of the small pulleys and establish the relationship between the velocity of A and the velocity of B for a given value of y.

Under the action of force P, the constant acceleration of block B is \(6 \mathrm{ft} / \mathrm{sec}^2\) up the incline. For the instant when the velocity of B is \(3 \mathrm{ft} / \mathrm{sec}\) up the incline, determine the velocity of B relative to A, the acceleration of B relative to A, and the absolute velocity of point C of the cable.

Determine the relationship which governs the velocities of the four cylinders. Express all velocities as positive down. How many degrees of freedom are there?

Collars A and B slide along the fixed right-angle rods and are connected by a cord of length L. Determine the acceleration \(a_x\) of collar B as a function of y if collar A is given a constant upward velocity \(v_A\).

The small sliders A and B are connected by the rigid slender rod. If the velocity of slider B is \(2 \mathrm{m} / \mathrm{s}\) to the right and is constant over a certain interval of time, determine the speed of slider A when the system is in the position shown.

For a given value of y, determine the upward velocity of A in terms of the downward velocity of B. Neglect the diameters of the pulleys.

Cart A has a leftward velocity \(v_A\) and acceleration \(a_A\) at the instant represented. Determine the expressions for the velocity and acceleration of cart B in terms of the position \(x_A\) of cart A. Neglect the diameters of the pulleys and assume that there is no mechanical interference. The two pulleys at C are pivoted independently.

Determine the vertical rise h of the load W during 10 seconds if the hoisting drum draws in cable at the constant rate of \(180 \mathrm{~mm} / \mathrm{s}\).

The hoisting system shown is used to easily raise kayaks for overhead storage. Determine expressions for the upward velocity and acceleration of the kayak at any height y if the winch M reels in cable at a constant rate \(\dot{l}\). Assume that the kayak remains level.

Develop an expression for the upward velocity of cylinder B in terms of the downward velocity of cylinder A. The cylinders are connected by a series of n cables and pulleys in a repeating fashion as shown.

If load B has a downward velocity \(v_B\), determine the upward component \(\left(v_A\right)_y\) of the velocity of A in terms of b, the boom length l, and the angle \(\theta\). Assume that the cable supporting A remains vertical.

The rod of the fixed hydraulic cylinder is moving to the left with a constant speed \(v_A=25 \mathrm{~mm} / \mathrm{s}\). Determine the corresponding velocity of slider B when \(s_A=425 \mathrm{~mm}\). The length of the cord is \(1050 \mathrm{mm}\), and the effects of the radius of the small pulley A may be neglected.

With all conditions of Prob. 2/225 remaining the same, determine the acceleration of slider B at the instant when \(s_A=425 \mathrm{~mm}\).

The two sliders are connected by the light rigid bar and move in the smooth vertical-plane guide. At the instant illustrated, the speed of slider A is \(25 \mathrm{~mm} / \mathrm{s}, \theta=45^{\circ}\), and \(\phi=15^{\circ}\). Determine the speed of slider B for this instant if \(r=175 \mathrm{~mm}\).

For the two sliders of Prob. 2 /227, determine the time rate of change of speed for slider B at the location shown if the speed of slider A is constant over a short interval which includes the position shown.

In a test of vertical leaping ability, a basketball player (a) crouches just before jumping, (b) has given his mass center G a vertical velocity \(v_0\) at the instant his feet leave the surface, and (c) reaches the maximum height. If the player can raise his mass center 3 feet as shown, estimate the initial velocity \(v_0\) of his mass center in position (b).

The position s of a particle along a straight line is given by \(s=8 e^{-0.4 t}-6 t+t^2\), where s is in meters and t is the time in seconds. Determine the velocity v when the acceleration is \(3 \mathrm{~m} / \mathrm{s}^2\).

A particle moving in the x-y plane has a velocity \(v=7.25 \mathbf{i}+3.48 \mathbf{j} \mathrm{m} / \mathrm{s}\) at a certain instant. If the particle then encounters a constant acceleration \(\mathbf{a}=0.85 \mathbf{j ~ m} / \mathrm{s}^2\), determine the amount of time which must pass before the direction of the tangent to the trajectory of the particle has been altered by \(30^{\circ}\).

An inexperienced designer of a roadbed for a new high-speed train proposes to join a straight section of track to a circular section of 1000 -ft radius as shown. For a train that would travel at a constant speed of \(90 \mathrm{mi} / \mathrm{hr}\), plot the magnitude of its acceleration as a function of distance along the track between points A and C and explain why this design is unacceptable.

While scrambling directly toward the sideline, the football quarterback Q throws a pass toward the stationary receiver R. At what speed \(v_Q\) should the quarterback run if the direction of the velocity of the ball relative to the quarterback is to be directly down the field as indicated? The speed of the ball relative to the quarterback is \(60 \mathrm{ft} / \mathrm{sec}\). What is the absolute speed of the ball? Treat the problem as two-dimensional.

Two airplanes are performing at an air show. Plane A travels along the path shown and, for the instant under consideration, has a speed of \(265 \mathrm{mi} / \mathrm{hr}\) that is increasing at a rate of \(4 \mathrm{mi} / \mathrm{hr}\) every second. Meanwhile, plane B executes a vertical loop at a constant speed of \(150 \mathrm{mi} / \mathrm{hr}\). Determine the velocity and acceleration which plane B appears to have to the pilot in plane A at the instant represented.

At time t = 0 a small ball is projected from point A with a velocity of \(200 \mathrm{ft} / \mathrm{sec}\) at the \(60^{\circ}\) angle. Neglect atmospheric resistance and determine the two times \(t_1\) and \(t_2\) when the velocity of the ball makes an angle of \(45^{\circ}\) with the horizontal x-axis.

A bicyclist rides along the hard-packed sand beach with a speed \(v_B=16 \mathrm{mi} / \mathrm{hr}\) as indicated. The wind speed is \(v_W=20 \mathrm{mi} / \mathrm{hr}\). (a) Determine the velocity of the wind relative to the bicyclist. (b) At what speed \(v_B\) would the bicyclist feel the wind coming directly from her left (perpendicular to her path)? What would be this relative speed?

Rotation of the arm OP is controlled by the horizontal motion of the vertical slotted link. If \(\dot{x}= 4 \mathrm{ft} / \mathrm{sec}\) and \(\ddot{x}=30 \mathrm{ft} / \mathrm{sec}^2\) when x = 2 in., determine \(\dot{\theta}\) and \(\ddot{\theta}\) for this instant.

Body A is released from rest in the position shown and moves downward causing body B to lift off the support at C. If motion is controlled such that the magnitude \(a_{B / A}=2.4 \mathrm{~m} / \mathrm{s}^2\) is held constant, determine the amount of time it takes for body B to travel \(5 \mathrm{~m}\) up the incline and the corresponding speed of body A at the end of that time period. The angle \(\theta=55^{\circ}\).

The launching catapult of the aircraft carrier gives the jet fighter a constant acceleration of \(50 \mathrm{~m} / \mathrm{s}^2\) from rest relative to the flight deck and launches the aircraft in a distance of \(100 \mathrm{~m}\) measured along the angled takeoff ramp. If the carrier is moving at a steady \(30 \mathrm{knots}\) ( \(1 \mathrm{knot} =1.852 \mathrm{~km} / \mathrm{h}\) ), determine the magnitude v of the actual velocity of the fighter when it is launched.

For the instant represented the particle P has a velocity \(v=6 \mathrm{ft} / \mathrm{sec}\) in the direction shown and has acceleration components \(a_x=15 \mathrm{ft} / \mathrm{sec}^2\) and \(a_\theta=-15 \mathrm{ft} / \mathrm{sec}^2\). Determine \(a_r, a_y, a_t, a_n\), and the radius of curvature \(\rho\) of the path for this position. (Hint: Draw the related acceleration components of the total acceleration of the particle and take advantage of the simplified geometry for your calculations.)

The coordinates of a particle which moves with curvilinear motion are given by \(x=10.25 t+ 1.75 t^2-0.45 t^3\) and \(y=6.32+14.65 t-2.48 t^2\), where x and y are in millimeters and the time t is in seconds. Determine the values of \(v, \mathbf{v}, a, \mathbf{a}, \mathbf{e}_t, \mathbf{e}_n\), \(a_t, \mathbf{a}_t, a_n, \mathbf{a}_n, \rho\), and \(\dot{\beta}\) (the angular velocity of the normal to the path) when \(t=3.25 \mathrm{~s}\). Express all vectors in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\).

The coordinates of a particle which moves with curvilinear motion are given by \(x=10.25 t+1.75 t^2- 0.45 t^3\) and \(y=6.32+14.65 t-2.48 t^2\), where x and y are in millimeters and the time t is in seconds. Determine the values of \(v, \mathbf{v}, a, \mathbf{a}, \mathbf{e}_r, \mathbf{e}_\theta, \mathbf{v}_r, \mathbf{v}_r, v_\theta\), \(\mathbf{v}_\theta, a_r, \mathbf{a}_r, a_\theta, \mathbf{a}_\theta, r, \dot{r}, \ddot{r}, \theta, \dot{\theta}\), and \(\ddot{\theta}\) when \(t=3.25 \mathrm{~s}\). Express all vectors in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Take the r-coordinate to proceed from the origin, and take \(\theta\) to be measured positive counterclockwise from the positive x-axis.

As part of a training exercise, the pilot of aircraft A adjusts her airspeed (speed relative to the wind) to \(220 \mathrm{~km} / \mathrm{h}\) while in the level portion of the approach path and thereafter holds her absolute speed constant as she negotiates the \(10^{\circ}\) glide path. The absolute speed of the aircraft carrier is \(30 \mathrm{~km} / \mathrm{h}\) and that of the wind is \(48 \mathrm{~km} / \mathrm{h}\). What will be the angle \(\beta\) of the glide path with respect to the horizontal as seen by an observer on the ship?

A small aircraft is moving in a horizontal circle with a constant speed of \(130 \mathrm{ft} / \mathrm{sec}\). At the instant represented, a small package A is ejected from the right side of the aircraft with a horizontal velocity of \(20 \mathrm{ft} / \mathrm{sec}\) relative to the aircraft. Neglect aerodynamic effects and calculate the coordinates of the point of impact on the ground.

Cylinder A has a constant downward speed of \(1 \mathrm{~m} / \mathrm{s}\). Compute the velocity of cylinder B for (a) \(\theta=45^{\circ}\), (b) \(\theta=30^{\circ}\), and (c) \(\theta=15^{\circ}\). The spring is in tension throughout the motion range of interest, and the pulleys are connected by the cable of fixed length.

A rocket fired vertically up from the north pole achieves a velocity of \(27000 \mathrm{~km} / \mathrm{h}\) at an altitude of \(350 \mathrm{~km}\) when its fuel is exhausted. Calculate the additional vertical height h reached by the rocket before it starts its descent back to the earth. The coasting phase of its flight occurs above the atmosphere. Consult Fig. 1/1 in choosing the appropriate value of gravitational acceleration and use the mean radius of the earth from Table D/2. (Note: Launching from the earth's pole avoids considering the effect of the earth's rotation.)

The radar tracking antenna oscillates about its vertical axis according to \(\theta=\theta_0 \cos \omega t\), where \(\omega\) is the constant circular frequency and \(2 \theta_0\) is the double amplitude of oscillation. Simultaneously, the angle of elevation \(\phi\) is increasing at the constant rate \(\dot{\phi}=K\). Determine the expression for the magnitude a of the acceleration of the signal horn (a) as it passes position A and (b) as it passes the top position B, assuming that \(\theta=0\) at this instant.

A projectile is fired with the given initial conditions. Plot the r- and \(\theta\)-components of velocity and acceleration as functions of time for the time period during which the particle is in the air. State the value for each component at time \(t=9 \mathrm{~s}\).

If all frictional effects are neglected, the expression for the angular acceleration of the simple pendulum is \(\ddot{\theta}=\frac{g}{l} \cos \theta\), where g is the acceleration of gravity and l is the length of the rod OA. If the pendulum has a clockwise angular velocity \(\dot{\theta}= 2 \mathrm{rad} / \mathrm{s}\) when \(\theta=0\) at t = 0, determine the time \(t^{\prime}\) at which the pendulum passes the vertical position \(\theta=90^{\circ}\). The pendulum length is \(l=0.6 \mathrm{~m}\). Also plot the time t versus the angle \(\theta\).

A baseball is dropped from an altitude \(h=200 \mathrm{ft}\) and is found to be traveling at \(85 \mathrm{ft} / \mathrm{sec}\) when it strikes the ground. In addition to gravitational acceleration, which may be assumed constant, air resistance causes a deceleration component of magnitude \(k v^2\), where v is the speed and k is a constant. Determine the value of the coefficient k. Plot the speed of the baseball as a function of altitude y. If the baseball were dropped from a high altitude, but one at which g may still be assumed constant, what would be the terminal velocity \(v_t\) ? (The terminal velocity is that speed at which the acceleration of gravity and that due to air resistance are equal and opposite, so that the baseball drops at a constant speed.) If the baseball were dropped from \(h=200 \mathrm{ft}\) at what speed \(v^{\prime}\) would it strike the ground if air resistance were neglected?

A ship with a total displacement of 16000 metric tons (1 metric ton \(=1000 \mathrm{~kg})\) starts from rest in still water under a constant propeller thrust \(T= 250 \mathrm{kN}\). The ship develops a total resistance to motion through the water given by \(R=4.50 v^2\), where R is in kilonewtons and v is in meters per second. The acceleration of the ship is \(a=(T-R) / m\), where m equals the mass of the ship in metric tons. Plot the speed v of the ship in knots as a function of the distance s in nautical miles which the ship goes for the first 5 nautical miles from rest. Find the speed after the ship has gone 1 nautical mile. What is the maximum speed which the ship can reach?

At time t = 0, the 1.8-lb particle P is given an initial velocity \(v_0=1 \mathrm{ft} / \mathrm{sec}\) at the position \(\theta=0\) and subsequently slides along the circular path of radius \(r=1.5 \mathrm{ft}\). Because of the viscous fluid and the effect of gravitational acceleration, the tangential acceleration is \(a_t=g \cos \theta-\frac{k}{m} v\), where the constant \(k=0.2 \mathrm{lb}-\mathrm{sec} / \mathrm{ft}\) is a drag parameter. Determine and plot both \(\theta\) and \(\dot{\theta}\) as functions of the time t over the range \(0 \leq t \leq 5 \mathrm{sec}\). Determine the maximum values of \(\theta\) and \(\dot{\theta}\) and the corresponding values of t. Also determine the first time at which \(\theta=90^{\circ}\).

A projectile is launched from point A with speed \(v_0=30 \mathrm{~m} / \mathrm{s}\). Determine the value of the launch angle \(\alpha\) which maximizes the range R indicated in the figure. Determine the corresponding value of R.

A low-flying cropduster A is moving with a constant speed of \(40 \mathrm{~m} / \mathrm{s}\) in the horizontal circle of radius \(300 \mathrm{~m}\). As it passes the twelve-o'clock position shown at time t = 0, car B starts from rest from the position shown and accelerates along the straight road at the constant rate of \(3 \mathrm{~m} / \mathrm{s}^2\) until it reaches a speed of \(30 \mathrm{~m} / \mathrm{s}\), after which it maintains that constant speed. Determine the velocity and acceleration of A with respect to B and plot the magnitudes of both these quantities over the time period \(0 \leq t \leq 50 \mathrm{~s}\) as functions of both time and displacement \(s_B\) of the car. Determine the maximum and minimum values of both quantities and state the values of the time t and the displacement \(s_B\) at which they occur.

A particle P is launched from point A with the initial conditions shown. If the particle is subjected to aerodynamic drag, compute the range R of the particle and compare this with the case in which aerodynamic drag is neglected. Plot the trajectories of the particle for both cases. Use the values \(v_0= 65 \mathrm{~m} / \mathrm{s}, \theta=35^{\circ}\), and \(k=4.0 \times 10^{-3} \mathrm{~m}^{-1}\). (Note: The acceleration due to aerodynamic drag has the form \(\mathbf{a}_D=-k v^2 \mathbf{t}\), where k is a positive constant, v is the particle speed, and \(\mathbf{t}\) is the unit vector associated with the instantaneous velocity \(\mathbf{v}\) of the particle. The unit vector \(\mathbf{t}\) has the form \(\mathbf{t}=\frac{v_x \mathbf{i}+v_y \mathbf{j}}{\sqrt{v_x^2+v_y^2}}\), where \(v_x\) and \v_y\) are the instantaneous x- and y-components of particle velocity, respectively.)

By means of the control unit M, the pendulum OA is given an oscillatory motion about the vertical given by \(\theta=\theta_0 \sin \sqrt{\frac{g}{l}} t\), where \(\theta_0\) is the maximum angular displacement in radians, g is the acceleration of gravity, l is the pendulum length, and t is the time in seconds measured from an instant when OA is vertical. Determine and plot the magnitude a of the acceleration of A as a function of time and as a function of \(\theta\) over the first quarter cycle of motion. Determine the minimum and maximum values of a and the corresponding values of t and \(\theta\). Use the values \(\theta_0=\pi / 3\) radians, \(l=0.8 \mathrm{~m}\), and \(g=9.81 \mathrm{~m} / \mathrm{s}^2\). (Note: The prescribed motion is not precisely that of a freely swinging pendulum for large amplitudes.)