Write the vector expression for the acceleration a of the mass center G of the simple

Problems 2/1 through 2/6 treat the motion of a particle which moves along the s-axis shown in the figure. Problems 2/12/6The velocity of a particle is given by 50, where v is in meters 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 2/6 treat the motion of a particle which moves along the s-axis shown in the figure. Problems 2/12/6The displacement of a particle is given by where s is in feet and t is in seconds. Plot the displacement, velocity, and acceleration as functions of time for the first 12 seconds of motion. Determine the time at which the velocity is zero.

Problems 2/1 through 2/6 treat the motion of a particle which moves along the s-axis shown in the figure. Problems 2/12/6The velocity of a particle which moves along the s-axis is given by where t is in seconds and v is in meters per second. Evaluate the displacement s, velocity v, and acceleration a when The particle is at the origin when

Problems 2/1 through 2/6 treat the motion of a particle which moves along the s-axis shown in the figure. Problems 2/12/6The velocity of a particle along the s-axis is given by where s is in millimeters and v is in millimeters per second. Determine the acceleration when s is 2 millimeters.

Problems 2/1 through 2/6 treat the motion of a particle which moves along the s-axis shown in the figure. Problems 2/12/6The position of a particle in millimeters is given by where t is in seconds. Plot the s-t and v-t relationships for the first 9 seconds. Determine the net displacement during that interval and the total distance D traveled. By inspection of the s-t relationship, what conclusion can you reach regarding the acceleration?

Problems 2/1 through 2/6 treat the motion of a particle which moves along the s-axis shown in the figure. Problems 2/12/6The velocity of a particle which moves along the s-axis is given by where t is in seconds. Calculate the displacement of the particle during the interval from to

Calculate the constant acceleration a in gs which the catapult of an aircraft carrier must provide to produce a launch velocity of 180 mi/hr in a distance of 300 ft. Assume that the carrier is at anchor.

A particle moves along a straight line with a velocity in millimeters per second given by , where t is in seconds. Calculate the net displacement and total distance D traveled during the first 6 seconds of motion.

The acceleration of a particle is given by 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 is and the initial velocity is

During a braking test, a car is brought to rest beginning from an initial speed of 60 mi/hr in a distance of 120 ft. With the same constant deceleration, what would be the stopping distance s from an initial speed of 80 mi/hr?

Ball 1 is launched with an initial vertical velocity Three seconds later, ball 2 is launched with an initial vertical velocity Determine if the balls are to collide at an altitude of 300 ft. At the instant of collision, is ball 1 ascending or descending? Problem 2/11

A projectile is fired vertically with an initial velocity of 200 m/s. Calculate the maximum altitude h reached by the projectile and the time t after firing for it to return to the ground. Neglect air resistance and take the gravitational acceleration to be constant at

A ball is thrown vertically upward with an initial speed of 80 ft/sec from the base A of a 50-ft cliff. Determine the distance h by which the ball clears the top of the cliff and the time t after release for the ball to land at B. Also, calculate the impact velocity Neglect air resistance and the small horizontal motion of the ball. Problem 2/13

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 and the speed from B to C is essentially constant, determine the time duration for the race. The effects of the small transition area at B can be neglected. Problem 2/14

Starting from rest at home plate, a baseball player runs to first base (90 ft away). He uniformly accelerates over the first 10 ft to his maximum speed, which is then maintained until he crosses first base. If the overall run is completed in 4 seconds, determine his maximum speed, the acceleration over the first 10 feet, and the time duration of the acceleration. Problem 2/15

The graph shows the displacement-time history for the rectilinear motion of a particle during an 8-second interval. Determine the average velocity during the interval and, to within reasonable limits of accuracy, find the instantaneous velocity v when Problem 2/16

The car is traveling at a constant speed km/h on the level portion of the road. When the 6-percent ( ) incline is encountered, the driver does not change the throttle setting and consequently the car decelerates at the constant rate Determine the speed of the car (a) 10 seconds after passing point A and (b) when Problem 2/16

In traveling a distance of 3 km between points A and D, a car is driven at 100 km/h from A to B for t seconds and 60 km/h from C to D also for t seconds. If the brakes are applied for 4 seconds between B and C to give the car a uniform deceleration, calculate t and the distance s between A and B. Problem 2/18

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 by which the acceleration at exceeds the average acceleration during the interval. What is the displacement during the interval? Problem 2/19 14 12 10 8 6 4 2 0 024 t, s v, m/s 6 8

A particle moves along the positive x-axis with an acceleration in meters per second squared which increases linearly with x expressed in millimeters, as shown on the graph for an interval of its motion. If the velocity of the particle at is determine the velocity at Problem 2/20

A girl rolls a ball up an incline and allows it to return to her. For the angle and ball involved, the acceleration of the ball along the incline is constant at 0.25g, directed down the incline. If the ball is released with a speed of 4 m/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 childs hand. Problem 2/21 s

A train which is traveling at 80 mi/hr applies its brakes as it reaches point A and slows down with a constant deceleration. Its decreased velocity is observed to be 60 mi/hr as it passes a point 1/2 mi beyond A. A car moving at 50 mi/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 sec and find the velocity v of the car as it reaches the crossing. Problem 2/22

Car A is traveling at a constant speed at a location where the speed limit is 100 km/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 until a speed of 160 km/h is achieved, and that speed is then maintained. Determine the distance required for the police officer to overtake car A. Neglect any nonrectilinear motion of P.

Repeat the previous problem, only now the driver of car A is traveling at as it passes P, but over the next 5 seconds, the car uniformly decelerates to the speed limit of 100 km/h, and after that the speed limit is maintained. If the motion of the police car P remains as described in the previous problem, determine the distance required for the police officer to overtake car A.

Repeat Prob. 2/23, only now the driver of car A sees and reacts very unwisely to the police car P. Car A is traveling at as it passes P, but over the next 5 seconds, the car uniformly accelerates to 150 km/h, after which that speed is maintained. If the motion of the police car P remains as described in Prob. 2/23, determine the distance required for the police officer to overtake car A.

The 14-in. spring is compressed to an 8-in. length, where it is released from rest and accelerates block A. The acceleration has an initial value of and then decreases linearly with the x-movement of the block, reaching zero when the spring regains its original 14-in. length. Calculate the time t for the block to go (a) 3 in. and (b) 6 in. Problem 2/26

A single-stage rocket is launched vertically from rest, and its thrust is programmed to give the rocket a constant upward acceleration of . If the fuel is exhausted 20 s after launch, calculate the maximum velocity and the subsequent maximum altitude h reached by the rocket.

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 , 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 sec and specify its value at time sec.

A particle starts from rest at and moves along the x-axis with the velocity history shown. Plot the corresponding acceleration and the displacement histories for the 2 seconds. Find the time t when the particle crosses the origin. Problem 2/29

A retarding force is applied to a body moving in a straight line so that, during an interval of its motion, its speed v decreases with increased position coordinate s according to the relation , where k is a constant. If the body has a forward speed of 2 in./sec and its position coordinate is 9 in. at time t  0, determine the speed v at sec.

The deceleration of the mass center G of a car during a crash test is measured by an accelerometer with the results shown, where the distance x moved by G after impact is 0.8 m. Obtain a close approximation to the impact velocity v from the data given. Problem 2/31

A sprinter reaches his maximum speed in 2.5 seconds from rest with constant acceleration. He then maintains that speed and finishes the 100 yards in the overall time of 9.60 seconds. Determine his maximum speed Problem 2/32 100 yd t = 0 t = 2.5 sec t = 9.60 sec

If the velocity v of a particle moving along a straight line decreases linearly with its displacement s from 20 m/s to a value approaching zero at determine the acceleration a of the particle when and show that the particle never reaches the 30-m displacement. Problem 2/33

A car starts from rest with an acceleration of which decreases linearly with time to zero in 10 seconds, after which the car continues at a constant speed. Determine the time t required for the car to travel 400 m from the start.

Packages enter the 10-ft chute at A with a speed of 4 ft/sec and have a 0.3g acceleration from A to B. If the packages come to rest at C, calculate the constant acceleration a of the packages from B to C. Also find the time required for the packages to go from A to C. Problem 2/35

In an archery test, the acceleration of the arrow decreases linearly with distance s from its initial value of at A upon release to zero at B after a travel of 24 in. Calculate the maximum velocity v of the arrow. Problem 2/36

The 230,000-lb space-shuttle orbiter touches down at about 220 mi/hr. At 200 mi/hr its drag parachute deploys. At 35 mi/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 (speed v in feet per second), determine the corresponding distance traveled by the orbiter. Assume no braking from its wheel brakes. Problem 2/37

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

The body falling with speed strikes and maintains contact with the platform supported by a nest of springs. The acceleration of the body after impact is where c is a positive constant and y is measured from the original platform position. If the maximum compression of the springs is observed to be , determine the constant c. Problem 2/39

Particle 1 is subjected to an acceleration particle 2 is subjected to and particle 3 is subjected to . All three particles start at the origin with an initial velocity at time and the magnitude of k is 0.1 for all three particles (note that the units of k vary from case to case). Plot the position, velocity, and acceleration versus time for each particle over the range

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 , where K is a measure of the strength of the magnetic field. If the ball is released from rest at determine the velocity v with which it strikes the pole face. Problem 2/41

A certain lake is proposed as a landing area for large jet aircraft. The touchdown speed of 100 mi/hr upon contact with the water is to be reduced to 20 mi/hr in a distance of 1500 ft. If the deceleration is proportional to the square of the velocity of the aircraft through the water, , find the value of the design parameter K, which would be a measure of the size and shape of the landing gear vanes that plow through the water. Also find the time t elapsed during the specified interval.

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. Problem 2/43

A particle moving along the s-axis has a velocity given by where t is in seconds. When the position of the particle is given by For the first 5 seconds of motion, determine the total distance D traveled, the net displacement , and the value of s at the end of the interval.

The cone falling with a speed strikes and penetrates the block of packing material. The acceleration of the cone after impact is where c is a positive constant and y is the penetration distance. If the maximum penetration depth is observed to be determine the constant c. Problem 2/45 y v0

The acceleration of the piston in a small reciprocating engine is given in the following table in terms of the position x of the piston measured from the top of its stroke. From a plot of the data, determine to within two-significant-figure accuracy the maximum velocity reached by the piston. x, m x, m 0 4950 0.075 0.0075 4340 0.090 0.015 3740 0.105 0.030 2580 0.120 0.045 1490 0.135 0.060 476 0.150 Problem 2/46

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 where and are constants which depend on the mechanical configuration of the car. If the car has an initial velocity when the engine is disengaged, derive an expression for the distance D required for the car to coast to a stop. Problem 2/47 v

A subway train travels between two of its station stops with the acceleration schedule shown. Determine the time interval during which the train brakes to a stop with a deceleration of and find the distance s between stations. Problem 2/48

Compute the impact speed of a body released from rest at an altitude (a) Assume a constant gravitational acceleration and (b) account for the variation of g with altitude (refer to Art. 1/5). Neglect the effects of atmospheric drag. Problem 2/49

Compute the impact speed of body A which is released from rest at an altitude mi above the surface of the moon. (a) First assume a constant gravitational acceleration and (b) then account for the variation of with altitude (refer to Art. 1/5). Problem 2/50

A projectile is fired horizontally into a resisting medium with a velocity and the resulting deceleration is equal to , where c and n are constants and v is the velocity within the medium. Find the expression for the velocity v of the projectile in terms of the time t of penetration.

The horizontal motion of the plunger and shaft is arrested by the resistance of the attached disk which moves through the oil bath. If the velocity of the plunger is in the position A where and , and if the deceleration is proportional to v so that , derive expressions for the velocity v and position coordinate x in terms of the time t. Also express v in terms of x. Problem 2/52

On its takeoff roll, the airplane starts from rest and accelerates according to where is the constant acceleration resulting from the engine thrust and is the acceleration due to aerodynamic drag. If , and v is in meters per second, determine the design length of runway required for the airplane to reach the takeoff speed of 250 km/h if the drag term is (a) excluded and (b) included. Problem 2/53

A test projectile is fired horizontally into a viscous liquid with a velocity . The retarding force is proportional to the square of the velocity, so that the acceleration becomes . Derive expressions for the distance D traveled in the liquid and the corresponding time t required to reduce the velocity to Neglect any vertical motion. Problem 2/54

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 ft/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. Problem 2/55

When the effect of aerodynamic drag is included, the y-acceleration of a baseball moving vertically upward is , while the acceleration when the ball is moving downward is , where k is a positive constant and v is the speed in feet per second. If the ball is thrown upward at 100 ft/sec from essentially ground level, compute its maximum height h and its speed upon impact with the ground. Take k to be and assume that 0.002 ft g is constant.

The vertical acceleration of a certain solid-fuel rocket is given by 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 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 two miles between stations A and B. The slopes of the cubic transition curves (which are of form ) are zero at the end points. Determine the total run time t between the stations and the maximum acceleration.

At time the position vector of a particle moving in the x-y plane is By time its position vector has become Determine the magnitude of its average velocity during this interval and the angle made by the average velocity with the positive x-axis

A particle moving in the x-y plane has a velocity at time given by and at its velocity has become Calculate the magnitude aav of its average acceleration during the 0.1-s interval and the angle it makes with the x-axis.

The velocity of a particle moving in the x-y plane is given by at time Its average acceleration during the next 0.02 s is Determine the velocity v of the particle at and the angle between the average-acceleration vector and the velocity vector at

A particle which moves with curvilinear motion has coordinates in millimeters which vary with the time t in seconds according to and Determine the magnitudes of the velocity v and acceleration a and the angles which these vectors make with the x-axis when

The x-coordinate of a particle in curvilinear motion is given by where x is in feet and t is in seconds. The y-component of acceleration in feet per second squared is given by If the particle has y-components and when find the magnitudes of the velocity v and acceleration a when Sketch the path for the first 2 seconds of motion, and show the velocity and acceleration vectors for

The y-coordinate of a particle in curvilinear motion is given by where y is in inches and t is in seconds. Also, the particle has an acceleration in the x-direction given by If the velocity of the particle in the x-direction is when calculate the magnitudes of the velocity v and acceleration a of the particle when Construct v and a in your solution

A rocket runs out of fuel in the position shown and continues in unpowered flight above the atmosphere. If its velocity in this position was 600 mi/hr, calculate the maximum additional altitude h acquired and the corresponding time t to reach it. The gravitational acceleration during this phase of its flight is Problem 2/65

A particle moves in the x-y plane with a y-component of velocity in feet per second given by with t in seconds. The acceleration of the particle in the x-direction in feet per second squared is given by with t in seconds. When and Find the equation of the path of the particle and calculate the magnitude of the velocity v of the particle for the instant when its x-coordinate reaches 18 ft.

A roofer tosses a small tool to the ground. What minimum magnitude of horizontal velocity is required to just miss the roof corner B? Also determine the distance d. Problem 2/67

Prove the well-known result that, for a given launch speed the launch angle 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.)

Calculate the minimum possible magnitude u of the muzzle velocity which a projectile must have when fired from point A to reach a target B on the same horizontal plane 12 km away. Problem 2/69

The center of mass G of a high jumper follows the trajectory shown. Determine the component , measured in the vertical plane of the figure, of his takeoff velocity and angle 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?) Problem 2/70

The quarterback Q throws the football when the receiver R is in the position shown. The receivers velocity is constant at 10 yd/sec, and he catches passes when the ball is 6 ft above the ground. If the quarterback desires the receiver to catch the ball 2.5 sec after the launch instant shown, determine the initial speed and angle required. Problem 2/71

The water nozzle ejects water at a speed at the angle Determine where, relative to the wall base point B, the water lands. Neglect the effects of the thickness of the wall. Problem 2/72

Water is ejected from the water nozzle of Prob. 2/72 with a speed For what value of the angle will the water land closest to the wall after clearing the top? Neglect the effects of wall thickness and air resistance. Where does the water land?

A football player attempts a 30-yd field goal. If he is able to impart a velocity u of 100 ft/sec to the ball, compute the minimum angle for which the ball will clear the crossbar of the goal. (Hint: Let .) Problem 2/74 u 30 yd 10

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 with the horizontal should the pilots line of sight to the target make at the instant of release? The airplane is flying horizontally at an altitude of 100 m with a velocity of 200 km/h. Problem 2/75

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 what angle is best if the ball is to arrive at third base at essentially ground level? Problem 2/76

If the tennis player serves the ball horizontally calculate its velocity v if the center of the ball clears the 36-in. net by 6 in. 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. Problem 2/77

The basketball player likes to release his foul shots with an initial speed What value(s) of the initial angle will cause the ball to pass through the center of the rim? Neglect clearance considerations as the ball passes over the front portion of the rim. Problem 2/78

A projectile is launched with an initial speed of 200 m/s at an angle of with respect to the horizontal. Compute the range R as measured up the incline. Problem 2/79

A rock is thrown horizontally from a tower at A and hits the ground 3.5 s later at B. The line of sight from A to B makes an angle of with the horizontal. Compute the magnitude of the initial velocity u of the rock. Problem 2/80

The muzzle velocity of a long-range rifle at A is Determine the two angles of elevation which will permit the projectile to hit the mountain target B. Problem 2/81

A projectile is launched with a speed 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 . Problem 2/82

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. Problem 2/83 v0 = 120 m/s = 40 20 B s A 800 m

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 angle with the horizontal, determine the range of launch speeds for which the ball will land inside the box. Problem 2/84

Ball bearings leave the horizontal trough with a velocity of magnitude u and fall through the 70-mmdiameter hole as shown. Calculate the permissible range of u which will enable the balls to enter the hole. Take the dashed positions to represent the limiting conditions. Problem 2/85

A horseshoe player releases the horseshoe at A with an initial speed Determine the range for the launch angle for which the shoe will strike the 14-in. vertical stake. Problem 2/86

A fireworks shell is launched vertically from point A with speed sufficient to reach a maximum altitude of 500 ft. A steady horizontal wind causes a constant horizontal acceleration of but does not affect the vertical motion. Determine the deviation at the top of the trajectory caused by the wind. Problem 2/87

Consider the fireworks shell of the previous problem. What angle compensates for the wind in that the shell peaks directly over the launch point A? All other information remains as stated in the previous problem, including the fact that the initial launch velocity if vertical would result in a maximum altitude of 500 ft. What is the maximum height h possible in this problem? Problem 2/88

Determine the location h of the spot toward which the pitcher must throw if the ball is to hit the catchers mitt. The ball is released with a speed of 40 m/s. Problem 2/89

The pilot of an airplane pulls into a steep 45 climb at 300 km/h and releases a package at position A. Calculate the horizontal distance s and the time t from the point of release to the point at which the package strikes the ground. Problem 2/90

Compare the slant range and flight time for the depicted projectile with the range R and flight time t for a projectile (launched with speed and inclination angle ) which flies over a horizontal surface. Evaluate your four results for Problem 2/91

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 of the total flight time that the projectile is above the level where is a fraction which can vary from zero to 1. State the value of for Problem 2/92

A projectile is ejected into an experimental fluid at time The initial speed is and the angle to the horizontal is . The drag on the projectile results in an acceleration term 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. Problem 2/93

An experimental fireworks shell is launched vertically from point A with an initial velocity of magnitude In addition to the acceleration due to gravity, an internal thrusting mechanism causes a constant acceleration component of 2g in the direction shown for the first 2 seconds of flight, after which the thruster ceases to function. Determine the maximum height h achieved, the total flight time, the net horizontal displacement from point A, and plot the entire trajectory. Neglect any acceleration due to aerodynamics. Problem 2/94

A projectile is launched with speed from point A. Determine the launch angle which results in the maximum range R up the incline of angle (where ). Evaluate your results for and Problem 2/95

A projectile is launched from point A with the initial conditions shown in the figure. Determine the x- and y-coordinates of the point of impact. Problem 2/96

Determine the maximum speed for each car if the normal acceleration is limited to 0.88g. The roadway is unbanked and level. Problem 2/97

A car is traveling around a circular track of 800-ft radius. If the magnitude of its total acceleration is at the instant when its speed is 45 mi/hr, determine the rate at which the car is changing its speed.

Six acceleration vectors are shown for the car whose velocity vector is directed forward. For each acceleration vector describe in words the instantaneous motion of the car. Problem 2/99

The driver of the truck has an acceleration of 0.4g 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 m, and the center of mass G of the driver (considered a particle) is 2 m above the road. Calculate the speed v of the truck. Problem 2/100

A bicycle is placed on a service rack with its wheels hanging free. As part of a bearing test, the front wheel is spun at the rate Assume that this rate is constant and determine the speed v and magnitude a of the acceleration of point A. Problem 2/101

A ship which moves at a steady 20-knot speed (1 executes a turn to port by changing its compass heading at a constant counterclockwise rate. If it requires 60 s to alter course calculate the magnitude of the acceleration a of the ship during the turn.

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

The two cars A and B enter an unbanked and level turn. They cross line C-C simultaneously, and each car has the speed corresponding to a maximum normal acceleration of 0.9g in the turn. Determine the elapsed time for each car between its two crossings of line C-C. What is the relative position of the two cars as the second car exits the turn? Assume no speed changes throughout. Problem 2/104

Revisit the two cars of the previous problem, only now the track has variable bankinga concept shown in the figure. Car A is on the unbanked portion of the track and its normal acceleration remains at 0.9g. Car B is on the banked portion of the track and its normal acceleration is limited to 1.12g. If the cars approach line C-C with speeds equal to the respective maxima in the turn, determine the time for each car to negotiate the turn as delimited by line C-C. What is the relative position of the two cars as the second car exits the turn? Assume no speed changes throughout. Problem 2/105 A B

A particle moves along the curved path shown. If the particle has a speed of 40 ft/sec at \(A\) at time \(t_{A}\) and a speed of 44 ft/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.

The speed of a car increases uniformly with time from 50 km/h at A to 100 km/h at B during 10 seconds. The radius of curvature of the hump at A is 40 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 of the dip in the road at B. The mass center of the car is 0.6 m from the road. Problem 2/107

The figure shows two possible paths for negotiating an unbanked turn on a horizontal portion of a race course. Path A-A follows the centerline of the road and has a radius of curvature , while path B-B uses the width of the road to good advantage in increasing the radius of curvature to . If the drivers limit their speeds in their curves so that the lateral acceleration does not exceed 0.8g, determine the maximum speed for each path. Problem 2/108

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 earths surface at latitude 40 north. The mean diameter of the earth is 12 742 km and its angular velocity is . Problem 2/109 N S P 40 0.7292(10

A satellite travels with constant speed v in a circular orbit 320 km above the earths surface. Calculate v knowing that the acceleration of the satellite is the gravitational acceleration at its altitude. (Note: Review Art. 1/5 as necessary and use the mean value of g and the mean value of the earths radius. Also recognize that v is the magnitude of the velocity of the satellite with respect to the center of the earth.)

The car is traveling at a speed of 60 mi/hr as it approaches point A. Beginning at A, the car decelerates at a constant until it gets to point B, after which its constant rate of decrease of speed is as it rounds the interchange ramp. Determine the magnitude of the total car acceleration (a) just before it gets to B, (b) just after it passes B, and (c) at point C. Problem 2/111

Write the vector expression for the acceleration a of the mass center G of the simple pendulum in both n-t and x-y coordinates for the instant when if  2 rad/sec and  4.025 rad/sec2 . Problem 2/112 x y n t 4

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 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 . Recall that you would be unaware of a gravitational field if you were in a nonrotating spacecraft in a circular orbit around the earth. Problem 2/113

Magnetic tape is being transferred from reel A to reel B and passes around idler pulleys C and D. At a certain instant, point on the tape is in contact with pulley C and point is in contact with pulley D. If the normal component of acceleration of is and the tangential component of acceleration of is at this instant, compute the corresponding speed v of the tape, the magnitude of the total acceleration of , and the magnitude of the total acceleration of . Problem 2/114

The car C increases its speed at the constant rate of as it rounds the curve shown. If the magnitude of the total acceleration of the car is at the point A where the radius of curvature is 200 m, compute the speed v of the car at this point. Problem 2/115

A football player releases a ball with the initial conditions shown in the figure. Determine the radius of curvature of the trajectory (a) just after release and (b) at the apex. For each case, compute the time rate of change of the speed. Problem 2/116

For the football of the previous problem, determine the radius of curvature of the path and the time rate of change of the speed at times sec and sec, where is the time of release from the quarterbacks hand

A particle moving in the x-y plane has a position vector given by , where r is in inches and t is in seconds. Calculate the radius of curvature of the path for the position of the particle when sec. Sketch the velocity v and the curvature of the path for this particular instant.

The design of a camshaft-drive system of a fourcylinder automobile engine is shown. As the engine is revved up, the belt speed v changes uniformly from 3 m/s to 6 m/s over a two-second interval. Calculate the magnitudes of the accelerations of points and halfway through this time interval. Problem 2/119

A small particle P starts from point O with a negligible speed and increases its speed to a value , where y is the vertical drop from O. When , determine the n-component of acceleration of the particle. (See Art. C/10 for the radius of curvature.) Problem 2/120 v

At a certain point in the reentry of the space shuttle into the earths atmosphere, the total acceleration of the shuttle may be represented by two components. One component is the gravitational acceleration at this altitude. The second component equals due to atmospheric resistance and is directed opposite to the velocity. The shuttle is at an altitude of 48.2 km and has reduced its orbital velocity of 28 300 km/h to 15 450 km/h in the direction . For this instant, calculate the radius of curvature of the path and the rate at which the speed is changing. Problem 2/121

The particle P starts from rest at point A at time 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 point B, (b) just after point B, and (c) as it passes point C. State your directions relative to the x-axis shown (CCW positive). Problem 2/122 x A B P C 3 m 3

For the conditions of the previous problem, determine the magnitude and direction of the total acceleration of the particle P at times and

Race car A follows path a-a while race car B follows path b-b on the unbanked track. If each car has a constant speed limited to that corresponding to a lateral (normal) acceleration of 0.8g, determine the times and for both cars to negotiate the turn as delimited by the line C-C. Problem 2/124

The mine skip is being hauled to the surface over the curved track by the cable wound around the 30-in. drum, which turns at the constant clockwise speed of 120 rev/min. The shape of the track is designed so that , where x and y are in feet. Calculate the magnitude of the total acceleration of the skip as it reaches a level of 2 ft below the top. Neglect the dimensions of the skip compared with those of the path. Recall that the radius of curvature is given by Problem 2/125

An earth satellite which moves in the elliptical equatorial orbit shown has a velocity v in space of 17 970 km/h when it passes the end of the semiminor axis at A. The earth has an absolute surface value of g of 9.821 m/s2 and has a radius of 6371 km. Determine the radius of curvature of the orbit at A. Problem 2/126

A particle which moves in two-dimensional curvilinear motion has coordinates in millimeters which vary with time t in seconds according to and . For time , determine the radius of curvature of the particle path and the magnitudes of the normal and tangential accelerations.

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 km/h, what cone spacing L should be used? Problem 2/128

The pin P is constrained to move in the slotted guides which move at right angles to one another. At the instant represented, A has a velocity to the right of 0.2 m/s which is decreasing at the rate of 0.75 m/s each second. At the same time, B is moving down with a velocity of 0.15 m/s which is decreasing at the rate of 0.5 m/s each second. For this instant determine the radius of curvature of the path followed by P. Is it possible to also determine the time rate of change of ? Problem 2/129

A particle which moves with curvilinear motion has coordinates in meters which vary with time t in seconds according to and . Determine the coordinates of the center of curvature C at time .

The position of the slider P in the rotating slotted arm OA is controlled by a power screw as shown. At the instant represented,  8 rad/s and  20 rad/s2 . Also at this same instant, , , and . For this instant determine the r- and -components of the acceleration of P. Problem 2/131

A model airplane flies over an observer O with constant speed in a straight line as shown. Determine the signs (plus, minus, or zero) for r, , , , , and for each of the positions A, B, and C. Problem 2/132 x y B O

A car P travels along a straight road with a constant speed . At the instant when the angle , determine the values of in ft/sec and in deg/sec. Problem 2/133

The sphere P travels in a straight line with speed . For the instant depicted, determine the corresponding values of and as measured relative to the fixed Oxy coordinate system. Problem 2/134

If the 10-m/s speed of the previous problem is constant, determine the values of and at the instant shown

As the hydraulic cylinder rotates around O, the exposed length l of the piston rod P is controlled by the action of oil pressure in the cylinder. If the cylinder rotates at the constant rate and l is decreasing at the constant rate of 150 mm/s, calculate the magnitudes of the velocity v and acceleration a of end B when . Problem 2/136

The drag racer P starts from rest at the start line S and then accelerates along the track. When it has traveled 100 m, its speed is 45 m/s. For that instant, determine the values of and relative to axes fixed to an observer O in the grandstand G as shown. Problem 2/137

In addition to the information supplied in the previous problem, it is known that the drag racer is accelerating forward at when it has traveled 100 m from the start line S. Determine the corresponding values of and .

An internal mechanism is used to maintain a constant angular rate about the z-axis of the spacecraft as the telescopic booms are extended at a constant rate. The length l is varied from essentially zero to 3 m. The maximum acceleration to which the sensitive experiment modules P may be subjected is . Determine the maximum allowable boom extension rate . Problem 2/139

The radial position of a fluid particle P in a certain centrifugal pump with radial vanes is approximated by cosh Kt, where t is time and 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, and K. Problem 2/140

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 , where is in radians and t is in seconds. The slider is at when and thereafter is drawn inward at the constant rate of 0.2 m/s. Determine the magnitude and direction (expressed by the angle relative to the x-axis) of the velocity and acceleration of the slider when . Problem 2/141

The piston of the hydraulic cylinder gives pin A a constant velocity in the direction shown for an interval of its motion. For the instant when , determine , , , and where . Problem 2/142

The rocket is fired vertically and tracked by the radar station shown. When reaches 60, other corresponding measurements give the values , , and . Calculate the magnitudes of the velocity and acceleration of the rocket at this position. Problem 2/143

A hiker pauses to watch a squirrel P run up a partially downed tree trunk. If the squirrels speed is when the position , determine the corresponding values of and . Problem 2/144

A jet plane flying at a constant speed v at an altitude is being tracked by radar located at O directly below the line of flight. If the angle is decreasing at the rate of 0.020 rad/s when , determine the value of at this instant and the magnitude of the velocity v of the plane. Problem 2/145

A projectile is launched from point A with the initial conditions shown. With the conventional definitions of r- and -coordinates relative to the Oxy coordinate system, determine r, , , , , and at the instant just alter launch. Neglect aerodynamic drag. Problem 2/146 x y v0 O A d

Instruments located at O are part of the groundtraffic control system for a major airport. At a certain instant during the takeoff roll of the aircraft P, the sensors indicate the angle and the range rate . Determine the corresponding speed v of the aircraft and the value of . Problem 2/147

In addition to the information supplied in the previous problem, the sensors at O indicate that . Determine the corresponding acceleration a of the aircraft and the value of .

The cam is designed so that the center of the roller A which follows the contour moves on a limaon defined by , where . If the cam does not rotate, determine the magnitude of the total acceleration of A in terms of if the slotted arm revolves with a constant counterclockwise angular rate . Problem 2/149 r

The slotted arm OA forces the small pin to move in the fixed spiral guide defined by . Arm OA starts from rest at and has a constant counterclockwise angular acceleration . Determine the magnitude of the acceleration of the pin P when . Problem 2/150

A rocket is tracked by radar from its launching point A. When it is 10 seconds into its flight, the following radar measurements are recorded: , , , , , and . For this instant determine the angle between the horizontal and the direction of the trajectory of the rocket and find the magnitudes of its velocity v and acceleration a. Problem 2/151

For an interval of motion the drum of radius b turns clockwise at a constant rate in radians per second and causes the carriage P to move to the right as the unwound length of the connecting cable is shortened. Use polar coordinates r and and derive expressions for the velocity v and acceleration a of P in the horizontal guide in terms of the angle . Check your solution by a direct differentiation with time of the relation . Problem 2/152

Car A is moving with constant speed v on the straight and level highway. The police officer in the stationary car P attempts to measure the speed v with radar. If the radar measures line-of sight velocity, what velocity will the officer observe? Evaluate your general expression for the values , , and , and draw any appropriate conclusions. Problem 2/153

The hydraulic cylinder gives pin A a constant velocity along its axis for an interval of motion and, in turn, causes the slotted arm to rotate about O. Determine the values of , , and for the instant when . (Hint: Recognize that all acceleration components are zero when the velocity is constant.) Problem 2/154

The particle P moves along the parabolic surface shown. When , the particle speed is . For this instant determine the corresponding values of r, , , and . Both x and y are in meters. Problem 2/155 x

The member OA of the industrial robot telescopes and pivots about the fixed axis at point O. At the instant shown, , , , , , a n d . 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. Problem 2/156

The robot arm is elevating and extending simultaneously. At a given instant, , constant, , , and . Compute the magnitudes of the velocity v and acceleration a of the gripped part P. In addition, express v and a in terms of the unit vectors i and j. Problem 2/157 l 0.75 m

During a portion of a vertical loop, an airplane flies in an arc of radius with a constant speed . When the airplane is at A, the angle made by v with the horizontal is , and radar tracking gives and . Calculate , , , and for this instant. Problem 2/158

The particle P starts from rest at point O at time , and then undergoes a constant tangential acceleration as it negotiates the circular slot in the counterclockwise direction. Determine r, , , and as functions of time over the first revolution. Problem 2/159

The low-flying aircraft P is traveling at a constant speed of 360 km/h in the holding circle of radius 3 km. For the instant shown, determine the quantities r, , , , , and relative to the fixed x-y coordinate system, which has its origin on a mountaintop at O. Treat the system as two-dimensional. Problem 2/160

Pin A moves in a circle of 90-mm radius as crank AC revolves at the constant rate . The slotted link rotates about point O as the rod attached to A moves in and out of the slot. For the position , determine , , , and . Problem 2/161

A fireworks shell P fired in a vertical trajectory has a y-acceleration given by , where the latter term is due to aerodynamic drag. If the speed of the shell is 15 m/s at the instant shown, determine the corresponding values of r, , , , , and . The drag parameter k has a constant value of . Problem 2/162

An earth satellite traveling in the elliptical orbit shown has a velocity 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 directed from A to O. For position A calculate the values of , , , and . Problem 2/163

At time , the baseball player releases a ball with the initial conditions shown in the figure. Determine the quantities r, , , , , and , all relative to the coordinate system shown, at time . Problem 2/164

The velocity and acceleration of a particle are given for a certain instant by and . Determine the angle between v and a, , and the radius of curvature in the osculating plane

A projectile is launched from point O with an initial speed 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. Problem 2/166

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 m/s, and the component of their acceleration measured along the tangent to the path is g cos at this point. The effective radius of the cylindrical helix is 5 m, and the helix angle is . Compute the magnitude of the acceleration of the passengers as they pass position A. Problem 2/167

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. Problem 2/168

The rotating element in a mixing chamber is given a periodic axial movement while it is rotating at the constant angular velocity . 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. Problem 2/169

The vertical shaft of the industrial robot rotates at the constant rate . The length h of the vertical shaft has a known time history, and this is true of its time derivatives and as well. Likewise, the values of l, , and are known. Determine the magnitudes of the velocity and acceleration of point P. The lengths and are fixed. Problem 2/170

The car A is ascending a parking-garage ramp in the form of a cylindrical helix of 24-ft radius rising 10 ft for each half turn. At the position shown the car has a speed of 15 mi/hr, which is decreasing at the rate of 2 mi/hr per second. Determine the r-, -, and z-components of the acceleration of the car. Problem 2/171

An aircraft takes off at A and climbs at a steady angle with a slope of 1 to 2 in the vertical y-z plane at a constant speed . The aircraft is tracked by radar at O. For the position B, determine the values of , , and . Problem 2/172

For the conditions of Prob. 2/172, find the values of , , and for the radar tracking coordinates as the aircraft passes point B. Use the results cited for Prob. 2/172.

The rotating nozzle sprays a large circular area and turns with the constant angular rate . Particles of water move along the tube at the constant rate relative to the tube. Write expressions for the magnitudes of the velocity and acceleration of a water particle P for a given position l in the rotating tube. Problem 2/174 z l P

The small block P travels with constant speed v in the circular path of radius r on the inclined surface. If at time , determine the x-, y-, and z-components of velocity and acceleration as functions of time. Problem 2/175

An aircraft is flying in a horizontal circle of radius b with a constant speed u at an altitude h. A radar tracking unit is located at C. Write expressions for the components of the velocity of the aircraft in the spherical coordinates of the radar station for a given position . Problem 2/176

The base structure of the firetruck ladder rotates about a vertical axis through O with a constant angular velocity . At the same time, the ladder unit OB elevates at a constant rate 7 , and section AB of the ladder extends from within section OA at the constant rate of 0.5 m/s. At the instant under consideration, , , and . Determine the magnitudes of the velocity and acceleration of the end B of the ladder. Problem 2/177

The member OA of the industrial robot telescopes. At the instant represented, , , , , , and . The base of the robot is revolving at the constant rate . Calculate the magnitudes of the velocity and acceleration of joint A. Problem 2/178

Consider the industrial robot of the previous problem. The telescoping member OA is now fixed in length at 0.9 m. The other conditions remain at , , , , , and angle OAP is locked at 105. Determine the magnitudes of the velocity and acceleration of the end point P

In a design test of the actuating mechanism for a telescoping antenna on a spacecraft, the supporting shaft rotates about the fixed z-axis with an angular rate . Determine the R-, -, and -components of the acceleration a of the end of the antenna at the instant when and if the rates rad/s, , and are constant during the motion. Problem 2/180 L R y x z

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 . The cars rise and fall with the track according to the relation . Find the R-, -, and -components of the velocity v of each car as it passes the position . Problem 2/181

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 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 is constant. Determine the expression for the radial acceleration of the particle for any value of . Problem 2/182

Car A rounds a curve of 150-m radius at a constant speed of 54 km/h. At the instant represented, car B is moving at 81 km/h but is slowing down at the rate of . Determine the velocity and acceleration of car A as observed from car B. Problem 2/183

For the instant represented, car A is rounding the circular curve at a constant speed of 30 mi/hr, while car B is slowing down at the rate of 5 mi/hr per second. Determine the magnitude of the acceleration that car A appears to have to an observer in car B. Problem 2/184

The passenger aircraft B is flying east with a velocity . A military jet traveling south with a velocity passes under B at a slightly lower altitude. What velocity does A appear to have to a passenger in B, and what is the direction of that apparent velocity? Problem 2/185

A marathon participant R is running north at a speed . A wind is blowing in the direction shown at a speed . (a) Determine the velocity of the wind relative to the runner. (b) Repeat for the case when the runner is moving directly to the south at the same speed. Express all answers both in terms of the unit vectors i and j and as magnitudes and compass directions. Problem 2/186 N 35 x y R

A small aircraft A is about to land with an airspeed of 80 mi/hr. If the aircraft is encountering a steady side wind of speed as shown, at what angle should the pilot direct the aircraft so that the absolute velocity is parallel to the runway? What is the speed at touchdown? Problem 2/187

The car A has a forward speed of 18 km/h and is accelerating at . 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 of the Ferris wheel is constant. Problem 2/188

A small ship capable of making a speed of 6 knots through still water maintains a heading due east while being set to the south by an ocean current. The actual course of the boat is from A to B, a distance of 10 nautical miles that requires exactly 2 hours. Determine the speed of the current and its direction measured clockwise from the north. Problem 2/189

Hockey player A carries the puck on his stick and moves in the direction shown with a speed . In passing the puck to his stationary teammate B, by what angle should the direction of his shot trail the line of sight if he launches the puck with a speed of 7 m/s relative to himself? Problem 2/190 45 vA

A ferry is moving due east and encounters a southwest wind of speed as shown. The experienced ferry captain wishes to minimize the effects of the wind on the passengers who are on the outdoor decks. At what speed should he proceed? Problem 2/191

A drop of water falls with no initial speed from point A of a highway overpass. After dropping 6 m, it strikes the windshield at point B of a car which is traveling at a speed of 100 km/h on the horizontal road. If the windshield is inclined 50 from the vertical as shown, determine the angle relative to the normal n to the windshield at which the water drop strikes. Problem 2/192

While scrambling directly toward the sideline at a speed 20 ft/sec, the football quarterback Q throws a pass toward the stationary receiver R. At what angle should the quarterback release the ball? The speed of the ball relative to the quarterback is 60 ft/sec. Treat the problem as two-dimensional. Problem 2/193

The speedboat B is cruising to the north at 75 mi/hr when it encounters an eastward current of speed but does not change its heading (relative to the water). Determine the subsequent velocity of the boat relative to the wind and express your result as a magnitude and compass direction. The current affects the motion of the boat; the southwesterly wind of speed does not. Problem 2/194 vC vW N B

Starting from the relative position shown, aircraft B is to rendezvous with the refueling tanker A. If B is to arrive in close proximity to A in a two-minute time interval, what absolute velocity vector should B acquire and maintain? The velocity of the tanker A is 300 mi/hr along the constant-altitude path shown. Problem 2/195

Airplane A is flying horizontally with a constant speed of 200 km/h and is towing the glider B, which is gaining altitude. If the tow cable has a length and is increasing at the constant rate of 5 degrees per second, determine the magnitudes of the velocity v and acceleration a of the glider for the instant when . Problem 2/196

If the airplane in Prob. 2/196 is increasing its speed in level flight at the rate of 5 km/h each second and is unreeling the glider tow cable at the constant rate while remains constant, determine the magnitude of the acceleration of the glider B.

The spacecraft S approaches the planet Mars along a trajectory b-b in the orbital plane of Mars with an absolute velocity of 19 km/s. Mars has a velocity of 24.1 km/s along its trajectory a-a. Determine the angle between the line of sight S-M and the trajectory b-b when Mars appears from the spacecraft to be approaching it head on. Problem 2/198

Two ships A and B are moving with constant speeds and , respectively, along straight intersecting courses. The navigator of ship B notes the time rates of change of the separation distance r between the ships and the bearing angle . Show that and . Problem 2/199

Airplane A is flying north with a constant horizontal velocity of 500 km/h. Airplane B is flying southwest at the same altitude with a velocity of 500 km/h. From the frame of reference of A, determine the magnitude of the apparent or relative velocity of B. Also find the magnitude of the apparent velocity with which B appears to be moving sideways or normal to its centerline. Would the results be different if the two airplanes were flying at different but constant altitudes? Problem 2/200

In Prob. 2/200 if aircraft A is accelerating in its northward direction at the rate of 3 km/h each second while aircraft B is slowing down at the rate of 4 km/h each second in its southwesterly direction, determine the acceleration in which B appears to have to an observer in A and specify its direction ( ) measured clockwise from the north.

The shuttle orbiter A is in a circular orbit of altitude 200 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 the shuttle A. Use for the surface-level gravitational acceleration and mi for the radius of the earth. Problem 2/202

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 in the direction shown. The quarterback releases the ball with a horizontal velocity of 100 ft/sec at the instant the receiver passes point P. Determine the angle 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. Problem 2/203

The aircraft A with radar detection equipment is flying horizontally at an altitude of 12 km and is increasing its speed at the rate of 1.2 m/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 km. If A has a speed of 1000 km/h at the instant when , determine the values of and at this same instant if B has a constant speed of 1500 km/h. Problem 2/204

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 . The airplane has the same constant speed , and after a period of level flight is just beginning to follow the circular path shown of radius . (a) Determine the velocity and acceleration of the airplane relative to the skydiver. (b) Determine the time rate of change of the speed of the airplane and the radius of curvature of its path, both as observed by the nonrotating skydiver. Problem 2/205

A batter hits the baseball A with an initial velocity of directly toward fielder B at an angle of 30 to the horizontal; the initial position of the ball is 3 ft above ground level. Fielder B requires 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 ft. Determine the velocity of the ball relative to the fielder at the instant the catch is made. Problem 2/206

If block B has a leftward velocity of 1.2 m/s, determine the velocity of cylinder A. Problem 2/207

At a certain instant, the velocity of cylinder B is 1.2 m/s down and its acceleration is up. Determine the corresponding velocity and acceleration of block A. Problem 2/208

Cylinder B has a downward velocity in feet per second given by , where t is in seconds. Calculate the acceleration of A when sec. Problem 2/209

Determine the constraint equation which relates the accelerations of bodies A and B. Assume that the upper surface of A remains horizontal. Problem 2/210

Determine the vertical rise h of the load W during 5 seconds if the hoisting drum wraps cable around it at the constant rate of 320 mm/s. Problem 2/211

A truck equipped with a power winch on its front end pulls itself up a steep incline with the cable and pulley arrangement shown. If the cable is wound up on the drum at the constant rate of 40 mm/s, how long does it take for the truck to move 4 m up the incline? Problem 2/212

For the pulley system shown, each of the cables at A and B is given a velocity of 2 m/s in the direction of the arrow. Determine the upward velocity v of the load m. Problem 2/213

Determine the relationship which governs the velocities of the two cylinders A and B. Express all velocities as positive down. How many degrees of freedom are present? Problem 2/214

The pulley system of the previous problem is modi- vA fied as shown with the addition of a fourth pulley and a third cylinder C. Determine the relationship which governs the velocities of the three cylinders, and state the number of degrees of freedom. Express all velocities as positive down. Problem 2/215

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. Problem 2/216

Determine an expression for the velocity of the cart A down the incline in terms of the upward velocity of cylinder B. Problem 2/217

Under the action of force P, the constant acceleration of block B is up the incline. For the instant when the velocity of B is 3 ft/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. Problem 2/218 A 20 B C

The small sliders A and B are connected by the rigid slender rod. If the velocity of slider B is 2 m/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. Problem 2/219

The power winches on the industrial scaffold enable it to be raised or lowered. For rotation in the senses indicated, the scaffold is being raised. If each drum has a diameter of 200 mm and turns at the rate of 40 rev/min. determine the upward velocity v of the scaffold. Problem 2/220

Collars A and B slide along the fixed right-angle rods and are connected by a cord of length L. Determine the acceleration of collar B as a function of y if collar A is given a constant upward velocity . Problem 2/221

Collars A and B slide along the fixed rods and are connected by a cord of length L. If collar A has a velocity to the right, express the velocity of B in terms of , , and s. Problem 2/222 s B A L

The particle A is mounted on a light rod pivoted at 2/224 O and therefore is constrained to move in a circular arc of radius r. Determine the velocity of A in terms of the downward velocity of the counterweight for any angle . Problem 2/223

The rod of the fixed hydraulic cylinder is moving to the left with a constant speed . Determine the corresponding velocity of slider B when . The length of the cord is 1050 mm, and the effects of the radius of the small pulley A may be neglected. Problem 2/224

With all conditions of Prob. remaining the same, determine the acceleration of slider B at the instant when .

Neglect the diameter of the small pulley attached to body A and determine the magnitude of the total velocity of B in terms of the velocity which body A has to the right. Assume that the cable between B and the pulley remains vertical and solve for a given value of x. Problem 2/226 h x

The position s of a particle along a straight line is given by , where s is in meters and t is the time in seconds. Determine the velocity v when the acceleration is .

While scrambling directly toward the sideline, the football quarterback Q throws a pass toward the stationary receiver R. At what speed 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 ft/sec. What is the absolute speed of the ball? Treat the problem as two-dimensional. Problem 2/228

A golfer is out of bounds and in a gulley. For the initial conditions shown, determine the coordinates of the point of first impact of the golf ball. The camera platform B is in the plane of the trajectory. Problem 2/229

At time a small ball is projected from point A with a velocity of 200 ft/sec at the angle. Neglect atmospheric resistance and determine the two times and when the velocity of the ball makes an angle of with the horizontal x-axis. Problem 2/230

The third stage of a rocket is injected by its booster with a velocity u of 15 000 km/h at A into an unpowered coasting flight to B. At B its rocket motor is ignited when the trajectory makes an angle of with the horizontal. Operation is effectively above the atmosphere, and the gravitational acceleration during this interval may be taken as , constant in magnitude and direction. Determine the time t to go from A to B. (This quantity is needed in the design of the ignition control system.) Also determine the corresponding increase h in altitude. Problem 2/231

The small cylinder is made to move along the rotating rod with a motion between and given by , where t is the time counted from the instant the cylinder passes the position and is the period of the oscillation (time for one complete oscillation). Simultaneously, the rod rotates about the vertical at the constant angular rate . Determine the value of r for which the radial (r-direction) acceleration is zero. Problem 2/232

Rotation of the arm PO is controlled by the horizontal motion of the vertical slotted link. If and when in., determine and for this instant. Problem 2/233

In case (a), the baseball player stands relatively stationary and throws the ball with the initial conditions shown. In case (b), he runs with speed as he launches the ball with the same conditions relative to himself. What is the additional range of the ball in case (b)? Compare the two flight times. Problem 2/234

A small projectile is fired from point O with an initial velocity at the angle of from the horizontal as shown. Neglect atmospheric resistance and any change in g and compute the radius of curvature of the path of the projectile 30 seconds after the firing. Problem 2/235

The motion of pin P is controlled by the two moving slots A and B in which the pin slides. If B has a velocity to the right while A has an upward velocity , determine the magnitude of the velocity of the pin. Problem 2/236

The angular displacement of the centrifuge is given by rad, where t is in seconds and is the startup time. If the person loses consciousness at an acceleration level of 10g, determine the time t at which this would occur. Verify that the tangential acceleration is negligible as the normal acceleration approaches 10g. Problem 2/237

For the instant represented the particle P has a velocity in the direction shown and has acceleration components and . Determine , , , , and the radius of curvature 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.) Problem 2/238

As part of a training exercise, the pilot of aircraft A adjusts her airspeed (speed relative to the wind) to 220 km/h while in the level portion of the approach path and thereafter holds her absolute speed constant as she negotiates the glide path. The absolute speed of the aircraft carrier is 30 km/h and that of the wind is 48 km/h. What will be the angle of the glide path with respect to the horizontal as seen by an observer on the ship? Problem 2/239

A small aircraft is moving in a horizontal circle with a constant speed of 130 ft/sec. At the instant represented, a small package A is ejected from the right side of the aircraft with a horizontal velocity of 20 ft/sec relative to the aircraft. Neglect aerodynamic effects and calculate the coordinates of the point of impact on the ground. Problem 2/240

Car A negotiates a curve of 60-m radius at a constant speed of 50 km/h. When A passes the position shown, car B is 30 m from the intersection and is accelerating south toward the intersection at the rate of . Determine the acceleration which A appears to have when observed by an occupant of B at this instant. Problem 2/241

Particle P moves along the curved path shown. At the instant represented, , , and the velocity v makes an angle with the horizontal x-axis and has a magnitude of 3.2 m/s. If the y- and r-components of the acceleration of P are and , respectively, at this position, determine the corresponding radius of curvature of the path and the x-component of the acceleration of the particle. Solve graphically or analytically. Problem 2/242

At the instant depicted, assume that the particle P, which moves on a curved path, is 80 m from the pole O and has the velocity v and acceleration a as indicated. Determine the instantaneous values of , , , , the n- and t-components of acceleration, and the radius of curvature . Problem 2/243

The radar tracking antenna oscillates about its vertical axis according to , where is the constant circular frequency and is the double amplitude of oscillation. Simultaneously, the angle of elevation is increasing at the constant rate . 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 at this instant. Problem 2/244

The rod of the fixed hydraulic cylinder is moving to the left with a constant speed . Determine the corresponding velocity of slider B when . The length of the cord is 1600 mm, and the effects of the radius of the small pulley at A may be neglected. Problem 2/245

With all conditions of Prob. 2/245 remaining the same, determine the acceleration of slider B at the instant when .

Two particles A and B start from rest at and move along parallel paths according to and , where and are in meters and t is in seconds counted from the start. Determine the time t (where ) when both particles have the same displacement and calculate this displacement x.

A baseball is dropped from an altitude and is found to be traveling at 85 ft/sec when it strikes the ground. In addition to gravitational acceleration, which may be assumed constant, air resistance causes a deceleration component of magnitude , 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 ? (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 at what speed would it strike the ground if air resistance were neglected?

The slotted arm is fixed and the four-lobe cam rotates counterclockwise at the constant speed of 2 revolutions per second. The distance 12 , where r is millimeters and is in radians. Plot the radial velocity and the radial acceleration of pin P versus from to . State the acceleration of pin P for (a) , (b) , and (c) . Problem 2/249

At time , the 1.8-lb particle P is given an initial velocity at the position and subsequently slides along the circular path of radius . Because of the viscous fluid and the effect of gravitational acceleration, the tangential acceleration is , where the constant is a drag parameter. Determine and plot both and as functions of the time t over the range . Determine the maximum values of and and the corresponding values of t. Also determine the first time at which . Problem 2/250

A low-flying cropduster A is moving with a constant speed of 40 m/s in the horizontal circle of radius 300 m. As it passes the twelve-oclock position shown at time , car B starts from rest from the position shown and accelerates along the straight road at the constant rate of until it reaches a speed of 30 m/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 s as functions of both time and displacement of the car. Determine the maximum and minimum values of both quantities and state the values of the time t and the displacement at which they occur. Problem 2/251

A projectile is launched from point A with speed . Determine the value of the launch angle which maximizes the range R indicated in the figure. Determine the corresponding value R. Problem 2/252

By means of the control unit M, the pendulum OA is given an oscillatory motion about the vertical given by , where 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 over the first quarter cycle of motion. Determine the minimum and maximum values of a and the corresponding values of t and . Use the values radians, , and . (Note: The prescribed motion is not precisely that of a freely swinging pendulum for large amplitudes.) Problem 2/253

The guide with the vertical slot is given a horizontal oscillatory motion according to , where x is in inches and t is in seconds. The oscillation causes the pin P to move in the fixed parabolic slot whose shape is given by , with y also in inches. Plot the magnitude v of the velocity of the pin as a function of time during the interval required for pin P to go from the center to the extremity in. Find and locate the maximum value of v and verify your results analytically. Problem 2/254