Solved: The time rate of change of acceleration is

a) If \(s=\left(2 t^{3}\right) \mathrm{m}\), where t is in seconds, determine v when t = 2 s. b) If v = (5s) m/s, where s is in meters, determine a at s = 1 m. c) If v = (4t + 5) m/s, where t is in seconds, determine a when t = 2 s. d) If \(a=2\mathrm{\ m}/\mathrm{s}^2\), determine v when t = 2 s if v = 0 when t = 0. e) If \(a=2\mathrm{\ m}/\mathrm{s}^2\), determine v at s = 4 m if v = 3 m/s at s = 0. f) If \(a=(s) \mathrm{m} / \mathrm{s}^{2}\), where s is in meters, determine v when s = 5 m if v = 0 at s = 4 m. g) If \(a=4\mathrm{\ m}/\mathrm{s}^2\), determine s when t = 3 s if v = 2 m/s and s = 2 m when t = 0. h) If \(a=\left(8t^2\right)\ \mathrm{m}/\mathrm{s}^2\), determine v when t = 1 s if v = 0 at t = 0. i) If \(s=\left(3t^2+2\right)\ \mathrm{m}\), determine v when t = 2 s. j) When t = 0 the particle is at A. In four seconds it travels to B, then in another six seconds it travels to C. Determine the average velocity and the average speed. The origin of the coordinate is at O.

a) Draw the s–t and a–t graphs if s = 0 when t = 0 b) Draw the a–t and v–t graphs c) Draw the v–t and s–t graphs if v = 0, s = 0 when t = 0. d) Determine s and a when t = 3 s if s = 0 when t = 0. e) Draw the v–t graph if v = 0 when t = 0. Find the equation v = f(t) for each segment. f) Determine v at s = 2 m if v = 1 m/s at s = 0. g) Determine a at s = 1 m.

Use the chain-rule and find \(\dot{y}\) and \(\ddot{y}\) in terms of x, \(\dot{x}\) and \(\ddot{x}\) if a) \(y=4 x^{2}\) b) \(y=3 e^{x}\) c) \(y=6 \sin x\)

The particle travels from A to B. Identify the three unknowns, and write the three equations needed to solve for them.

The particle travels from A to B. Identify the three unknowns, and write the three equations needed to solve for them.

The particle travels from A to B. Identify the three unknowns, and write the three equations needed to solve for them.

a) Determine the acceleration at the instant shown. b) Determine the increase in speed and the normal component of acceleration at s = 2 m. At s = 0, v = 0. c) Determine the acceleration at the instant shown. The particle has a constant speed of 2 m/s. d) Determine the normal and tangential components of acceleration at s = 0 if v = (4s + 1) m/s, where s is in meters. e) Determine the acceleration at s = 2 m if \(\dot{v}=(2s)\mathrm{\ m}/\mathrm{s}^2\), where s is in meters. At s = 0, v = 1 m/s. f. Determine the acceleration when t = 1 s if \(v=\left(4t^2+2\right)\mathrm{\ m}/\mathrm{s}\), where t is in seconds.

Initially, the car travels along a straight road with a speed of 35 m/s. If the brakes are applied and the speed of the car is reduced to 10 m/s in 15 s, determine the constant deceleration of the car.

A ball is thrown vertically upward with a speed of 15 m/s. Determine the time of flight when it returns to its original position.

A particle travels along a straight line with a velocity of \(v=\left(4t-3t^2\right)\ \mathrm{m}/\mathrm{s}\), where t is in seconds. Determine the position of the particle when t = 4 s. s = 0 when t = 0.

A particle travels along a straight line with a speed \(v=\left(0.5t^3-8t\right)\ \mathrm{m}/\mathrm{s}\), where t is in seconds. Determine the acceleration of the particle when t = 2 s.

The position of the particle is given by \(s=\left(2t^2\quad8t+6\right)\ \mathrm{m}\), where t is in seconds. Determine the time when the velocity of the particle is zero, and the total distance traveled by the particle when t = 3 s.

A particle travels along a straight line with an acceleration of \(a=(10-0.2s)\ \mathrm{m}/\mathrm{s}^2\), where s is measured in meters. Determine the velocity of the particle when s = 10 m if v = 5 m/s at s = 0.

A particle moves along a straight line such that its acceleration is \(a=\left(4 t^{2}-2\right) \mathrm{m} / \mathrm{s}^{2}\), where t is in seconds. When t = 0, the particle is located 2 m to the left of the origin, and when t = 2 s, it is 20 m to the left of the origin. Determine the position of the particle when t = 4 s.

A particle travels along a straight line with a velocity of v = (20 - 0.05s 2 ) m>s, where s is in meters. Determine the acceleration of the particle at s = 15 m.

The particle travels along a straight track such that its position is described by the s-t graph. Construct the v-t graph for the same time interval.

A van travels along a straight road with a velocity described by the graph. Construct the s-t and a-t graphs during the same period. Take s = 0 when t = 0.

A bicycle travels along a straight road where its velocity is described by the v-s graph. Construct the a-s graph for the same interval.

The sports car travels along a straight road such that its acceleration is described by the graph. Construct the v-s graph for the same interval and specify the velocity of the car when s = 10 m and s = 15 m.

The dragster starts from rest and has an acceleration described by the graph. Construct the v-t graph for the time interval \(0 \leq t \leq t^{\prime}\), where \(t^{\prime}\) is the time for the car to come to rest.

The dragster starts from rest and has a velocity described by the graph. Construct the s-t graph during the time interval \(0\le t\le15\mathrm{\ s}\). Also, determine the total distance traveled during this time interval.

If the x and y components of a particle’s velocity are \(v_x=(32t)\ \mathrm{m}/\mathrm{s}\) and \(v_y=8\mathrm{\ m}/\mathrm{s}\), determine the equation of the path y = f(x), if x = 0 and y = 0 when t = 0.

A particle is traveling along the straight path. If its position along the x axis is x = (8t) m, where t is in seconds, determine its speed when t = 2 s.

A particle is constrained to travel along the path. If \(x=\left(4t^4\right)\mathrm{\ m}\), where t is in seconds, determine the magnitude of the particle’s velocity and acceleration when t = 0.5 s.

A particle travels along a straight-line path y = 0.5x. If the x component of the particle’s velocity is \(v_x=\left(2t^2\right)\ \mathrm{m}/\mathrm{s}\), where t is in seconds, determine the magnitude of the particle’s velocity and acceleration when t = 4 s.

A particle is traveling along the parabolic path \(y=0.25 x^{2}\). If x = 8 m, \(v_x=8\mathrm{\ m}/\mathrm{s}\), and \(a_x=4\mathrm{\ m}/\mathrm{s}^2\) when t = 2 s, determine the magnitude of the particle’s velocity and acceleration at this instant.

The box slides down the slope described by the equation \(y=\left(0.05x^2\right)\ \mathrm{m}\), where x is in meters. If the box has x components of velocity and acceleration of \(v_x=-3\mathrm{\ m}/\mathrm{s}\) and \(a_x=-1.5\mathrm{\ m}/\mathrm{s}^2\) at x = 5 m, determine the y components of the velocity and the acceleration of the box at this instant.

The ball is kicked from point A with the initial velocity \(v_A=10\mathrm{\ m}/\mathrm{s}\). Determine the maximum height h it reaches.

The ball is kicked from point A with the initial velocity \(v_A=10\mathrm{\ m}/\mathrm{s}\). Determine the range R, and the speed when the ball strikes the ground.

Determine the speed at which the basketball at A must be thrown at the angle of \(30^{\circ}\) so that it makes it to the basket at B.

Water is sprayed at an angle of \(90^{\circ}\) from the slope at 20 m/s. Determine the range R.

A ball is thrown from A. If it is required to clear the wall at B, determine the minimum magnitude of its initial velocity \(\mathbf{v}_{A}\).

A projectile is fired with an initial velocity of \(v_A=150\mathrm{\ m}/\mathrm{s}\) off the roof of the building. Determine the range R where it strikes the ground at B.

The boat is traveling along the circular path with a speed of \(v=\left(0.0625t^2\right)\mathrm{\ m}/\mathrm{s}\), where t is in seconds. Determine the magnitude of its acceleration when t = 10 s.

The car is traveling along the road with a speed of v = (2 s) m/s, where s is in meters. Determine the magnitude of its acceleration when s = 10 m.

If the car decelerates uniformly along the curved road from 25 m/s at A to 15 m/s at C, determine the acceleration of the car at B.

When x = 10 ft, the crate has a speed of 20 ft/s which is increasing at \(6\mathrm{\ ft}/\mathrm{s}^2\). Determine the direction of the crate’s velocity and the magnitude of the crate’s acceleration at this instant.

If the motorcycle has a deceleration of \(a_t=-(0.001\mathrm{s})\mathrm{\ m}/\mathrm{s}^2\) and its speed at position A is 25 m/s, determine the magnitude of its acceleration when it passes point B.

The car travels up the hill with a speed of v = (0.2s) m/s, where s is in meters, measured from A. Determine the magnitude of its acceleration when it is at point s = 50 m, where \(\rho=500\mathrm{\ m}\).

The car has a speed of 55 ft/s. Determine the angular velocity \(\dot{\theta}\) of the radial line OA at this instant.

The platform is rotating about the vertical axis such that at any instant its angular position is \(\theta=\left(4t^{3/2}\right)\mathrm{\ rad}\), where t is in seconds. A ball rolls outward along the radial groove so that its position is \(r=\left(0.1t^3\right)\mathrm{\ m}\), where t is in seconds. Determine the magnitudes of the velocity and acceleration of the ball when t = 1.5 s.

Peg P is driven by the fork link OA along the curved path described by \(r=(2\theta)\ \mathrm{ft}\). At the instant \(\theta=\pi/4\mathrm{\ rad}\), the angular velocity and angular acceleration of the link are \(\dot{\theta}=3\mathrm{\ rad}/\mathrm{s}\) and \(\ddot{\theta}=1 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the magnitude of the peg’s acceleration at this instant.

Peg P is driven by the forked link OA along the path described by \(r=e^{\theta}\), where r is in meters. When \(\theta=\frac{\pi}{4}\mathrm{\ rad}\), the link has an angular velocity and angular acceleration of \(\dot{\theta}=2\mathrm{\ rad}/\mathrm{s}\) and \(\ddot{\theta}=4 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the radial and transverse components of the peg’s acceleration at this instant.

The collars are pin connected at B and are free to move along rod OA and the curved guide OC having the shape of a cardioid, \(r=[0.2(1+\cos\theta)]\mathrm{\ m}\). At \(\theta=30^{\circ}\), the angular velocity of OA is \(\dot{\theta}=3\ \mathrm{rad}/\mathrm{s}\). Determine the magnitude of the velocity of the collars at this point.

At the instant \(\theta=45^{\circ}\), the athlete is running with a constant speed of 2 m/s. Determine the angular velocity at which the camera must turn in order to follow the motion.

Determine the velocity of block D if end A of the rope is pulled down with a speed of \(v_A=3\mathrm{\ m}/\mathrm{s}\).

Determine the velocity of block A if end B of the rope is pulled down with a speed of 6 m/s.

Determine the velocity of block A if end B of the rope is pulled down with a speed of 1.5 m/s.

Determine the velocity of block A if end F of the rope is pulled down with a speed of \(v_F=3\mathrm{\ m}/\mathrm{s}\).

Determine the velocity of car A if point P on the cable has a speed of 4 m/s when the motor M winds the cable in.

Determine the velocity of cylinder B if cylinder A moves downward with a speed of \(v_A=4\mathrm{\ ft}/\mathrm{s}\).

Car A is traveling with a constant speed of 80 km/h due north, while car B is traveling with a constant speed of 100 km/h due east. Determine the velocity of car B relative to car A.

Two planes A and B are traveling with the constant velocities shown. Determine the magnitude and direction of the velocity of plane B relative to plane A.

The boats A and B travel with constant speeds of \(v_A=15\mathrm{\ m}/\mathrm{s}\) and \(v_B=10\mathrm{\ m}/\mathrm{s}\) when they leave the pier at O at the same time. Determine the distance between them when t = 4 s.

At the instant shown, cars A and B are traveling at the speeds shown. If B is accelerating at \(1200\mathrm{\ km}/\mathrm{h}^2\) while A maintains a constant speed, determine the velocity and acceleration of A with respect to B.

Starting from rest, a particle moving in a straight line has an acceleration of \(a=(2t-6)\ \mathrm{m}/\mathrm{s}^2\), where t is in seconds. What is the particle’s velocity when t = 6 s, and what is its position when t = 11 s?

If a particle has an initial velocity of \(v_0=12\mathrm{\ ft}/\mathrm{s}\) to the right, at \(s_{0}=0\), determine its position when t = 10 s, if \(a=2\ \mathrm{ft}/\mathrm{s}^2\) to the left.

A particle travels along a straight line with a velocity \(v=\left(12-3t^2\right)\ \mathrm{m}/\mathrm{s}\), where t is in seconds. When t = 1 s, the particle is located 10 m to the left of the origin. Determine the acceleration when t = 4 s, the displacement from t = 0 to t = 10 s, and the distance the particle travels during this time period.

A particle travels along a straight line with a constant acceleration. When s = 4 ft, v = 3 ft/s and when s = 10 ft, v = 8 ft/s. Determine the velocity as a function of position.

The velocity of a particle traveling in a straight line is given by \(v=\left(6t-3t^2\right)\mathrm{\ m}/\mathrm{s}\), where t is in seconds. If s = 0 when t = 0, determine the particle’s deceleration and position when t = 3 s. How far has the particle traveled during the 3-s time interval, and what is its average speed?

The position of a particle along a straight line is given by \(s=\left(1.5t^3-13.5t^2+22.5t\right)\ \mathrm{ft}\), where t is in seconds. Determine the position of the particle when t = 6 s and the total distance it travels during the 6-s time interval. Hint: Plot the path to determine the total distance traveled.

A particle moves along a straight line such that its position is defined by \(s=\left(t^2-6t+5\right)\ \mathrm{m}\). Determine the average velocity, the average speed, and the acceleration of the particle when t = 6 s.

A particle is moving along a straight line such that its position is defined by \(s=\left(10t^2+20\right)\mathrm{\ mm}\), where t is in seconds. Determine (a) the displacement of the particle during the time interval from t = 1 s to t = 5 s, (b) the average velocity of the particle during this time interval, and (c) the acceleration when t = 1 s.

The acceleration of a particle as it moves along a straight line is given by \(a=(2t-1)\mathrm{\ m}/\mathrm{s}^2\), where t is in seconds. If s = 1 m and v = 2 m/s when t = 0, determine the particle’s velocity and position when t = 6 s. Also, determine the total distance the particle travels during this time period.

A particle moves along a straight line with an acceleration of \(a=5/\left(3s^{1/3}+s^{5/2}\right)\ \mathrm{m}/\mathrm{s}^2\), where s is in meters. Determine the particle’s velocity when s = 2 m, if it starts from rest when s = 1 m. Use a numerical method to evaluate the integral.

A particle travels along a straight-line path such that in 4 s it moves from an initial position \(s_A=-8\mathrm{\ m}\) to a position \(s_B=+3\mathrm{\ m}\). Then in another 5 s it moves from \(s_B\) to \(s_C=-6\mathrm{\ m}\). Determine the particle’s average velocity and average speed during the 9-s time interval.

Traveling with an initial speed of 70 km>h, a car accelerates at \(6000\mathrm{\ km}/\mathrm{h}^2\) along a straight road. How long will it take to reach a speed of 120 km/h? Also, through what distance does the car travel during this time?

Tests reveal that a normal driver takes about 0.75 s before he or she can react to a situation to avoid a collision. It takes about 3 s for a driver having 0.1% alcohol in his system to do the same. If such drivers are traveling on a straight road at 30 mph (44 ft/s) and their cars can decelerate at \(2\mathrm{\ ft}/\mathrm{s}^2\), determine the shortest stopping distance d for each from the moment they see the pedestrians. Moral: If you must drink, please don’t drive!

The position of a particle along a straight-line path is defined by \(s=\left(t^3-6t^2-15t+7\right)\mathrm{\ ft}\), where t is in seconds. Determine the total distance traveled when t = 10 s. What are the particle’s average velocity, average speed, and the instantaneous velocity and acceleration at this time?

A particle is moving with a velocity of \(v_{0}\) when s = 0 and t = 0. If it is subjected to a deceleration of \(a=-k v^{3}\), where k is a constant, determine its velocity and position as functions of time.

A particle is moving along a straight line with an initial velocity of 6 m/s when it is subjected to a deceleration of \(a=\left(-1.5v^{1/2}\right)\ \mathrm{m}/\mathrm{s}^2\), where v is in m/s. Determine how far it travels before it stops. How much time does this take?

Car B is traveling a distance d ahead of car A. Both cars are traveling at 60 ft/s when the driver of B suddenly applies the brakes, causing his car to decelerate at \(12\mathrm{\ ft}/\mathrm{s}^2\). It takes the driver of car A 0.75 s to react (this is the normal reaction time for drivers). When he applies his brakes, he decelerates at \(15\mathrm{\ ft}/\mathrm{s}^2\). Determine the minimum distance d between the cars so as to avoid a collision.

The acceleration of a rocket traveling upward is given by \(a=(6+0.02\mathrm{s})\mathrm{\ m}/\mathrm{s}^2\), where s is in meters. Determine the time needed for the rocket to reach an altitude of s = 100 m. Initially, v = 0 and s = 0 when t = 0.

A train starts from rest at station A and accelerates at \(0.5\mathrm{\ m}/\mathrm{s}^2\) for 60 s. Afterwards it travels with a constant velocity for 15 min. It then decelerates at \(1\mathrm{\ m}/\mathrm{s}^2\) until it is brought to rest at station B. Determine the distance between the stations.

The velocity of a particle traveling along a straight line is \(v=\left(3t^2-6t\right)\ \mathrm{ft}/\mathrm{s}\), where t is in seconds. If s = 4 ft when t = 0, determine the position of the particle when t = 4 s. What is the total distance traveled during the time interval t = 0 to t = 4 s? Also, what is the acceleration when t = 2 s?

A freight train travels at \(v=60\left(1-e^{-t}\right)\ \mathrm{ft}/\mathrm{s}\), where t is the elapsed time in seconds. Determine the distance traveled in three seconds, and the acceleration at this time.

A sandbag is dropped from a balloon which is ascending vertically at a constant speed of 6 m/s. If the bag is released with the same upward velocity of 6 m/s when t = 0 and hits the ground when t = 8 s, determine the speed of the bag as it hits the ground and the altitude of the balloon at this instant.

A particle is moving along a straight line such that its acceleration is defined as \(a=(-2v)\mathrm{\ m}/\mathrm{s}^2\), where v is in meters per second. If v = 20 m/s when s = 0 and t = 0, determine the particle’s position, velocity, and acceleration as functions of time.

The acceleration of a particle traveling along a straight line is \(a=\frac{1}{4}s^{1/2}\mathrm{\ m}/\mathrm{s}^2\), where s is in meters. If v = 0, s = 1 m when t = 0, determine the particle’s velocity at s = 2 m.

If the effects of atmospheric resistance are accounted for, a freely falling body has an acceleration defined by the equation \(a=9.81\left[1-v^2\left(10^{-4}\right)\right]\ \mathrm{m}/\mathrm{s}^2\), where v is in m/s and the positive direction is downward. If the body is released from rest at a very high altitude, determine (a) the velocity when t = 5 s, and (b) the body’s terminal or maximum attainable velocity (as \(t \rightarrow \infty\)).

The acceleration of a particle along a straight line is defined by \(a=(2t-9)\mathrm{\ m}/\mathrm{s}^2\), where t is in seconds. At t = 0, s = 1 m and v = 10 m/s. When t = 9 s, determine (a) the particle’s position, (b) the total distance traveled, and (c) the velocity.

When a particle falls through the air, its initial acceleration a = g diminishes until it is zero, and thereafter it falls at a constant or terminal velocity \(v_{f}\). If this variation of the acceleration can be expressed as \(a=\left(g / v_{f}^{2}\right)\left(v_{f}^{2}-v^{2}\right)\), determine the time needed for the velocity to become \(v=v_{f} / 2\). Initially the particle falls from rest.

Two particles A and B start from rest at the origin s = 0 and move along a straight line such that \(a_A=(6t-3)\ \mathrm{ft}/\mathrm{s}^2\) and \(a_B=\left(12t^2-8\right)\mathrm{\ ft}/\mathrm{s}^2\), where t is in seconds. Determine the distance between them when t = 4 s and the total distance each has traveled in t = 4 s.

A ball A is thrown vertically upward from the top of a 30-m-high building with an initial velocity of 5 m/s. At the same instant another ball B is thrown upward from the ground with an initial velocity of 20 m/s. Determine the height from the ground and the time at which they pass.

A sphere is fired downwards into a medium with an initial speed of 27 m/s. If it experiences a deceleration of \(a=(-6t)\mathrm{\ m}/\mathrm{s}^2\), where t is in seconds, determine the distance traveled before it stops.

The velocity of a particle traveling along a straight line is \(v=v_{0}-k s\), where k is constant. If s = 0 when t = 0, determine the position and acceleration of the particle as a function of time.

Ball A is thrown vertically upwards with a velocity of \(v_{0}\). Ball B is thrown upwards from the same point with the same velocity t seconds later. Determine the elapsed time \(t<2 v_{0} / g\) from the instant ball A is thrown to when the balls pass each other, and find the velocity of each ball at this instant.

As a body is projected to a high altitude above the earth’s surface, the variation of the acceleration of gravity with respect to altitude y must be taken into account. Neglecting air resistance, this acceleration is determined from the formula \(a=-g_{0}\left[R^{2} /(R+y)^{2}\right]\), where \(g_{0}\) is the constant gravitational acceleration at sea level, R is the radius of the earth, and the positive direction is measured upward. If \(g_0=9.81\mathrm{\ m}/\mathrm{s}^2\) and R = 6356 km, determine the minimum initial velocity (escape velocity) at which a projectile should be shot vertically from the earth’s surface so that it does not fall back to the earth. Hint: This requires that v = 0 as \(y \rightarrow \infty\).

Accounting for the variation of gravitational acceleration a with respect to altitude y (see Prob. 12–36), derive an equation that relates the velocity of a freely falling particle to its altitude. Assume that the particle is released from rest at an altitude \(y_{0}\) from the earth’s surface. With what velocity does the particle strike the earth if it is released from rest at an altitude \(y_0=500\mathrm{\ km}\)? Use the numerical data in Prob. 12–33.

A freight train starts from rest and travels with a constant acceleration of \(0.5\ \mathrm{ft}/\mathrm{s}^2\). After a time \(t^{\prime}\) it maintains a constant speed so that when t = 160 s it has traveled 2000 ft. Determine the time \(t^{\prime}\) and draw the v–t graph for the motion.

The s–t graph for a train has been experimentally determined. From the data, construct the v–t and a–t graphs for the motion; \(0\le t\le40\ s\). For \(0\le t\le30\ s\), the curve is \(s=\left(0.4t^2\right)\ \mathrm{m}\), and then it becomes straight for \(t\ge30\mathrm{\ s}\).

Two rockets start from rest at the same elevation. Rocket A accelerates vertically at \(20\mathrm{\ m}/\mathrm{s}^2\) for 12 s and then maintains a constant speed. Rocket B accelerates at \(15\mathrm{\ m}/\mathrm{s}^2\) until reaching a constant speed of 150 m/s. Construct the a–t, v–t, and s–t graphs for each rocket until t = 20 s. What is the distance between the rockets when t = 20 s?

A particle starts from s = 0 and travels along a straight line with a velocity \(v=\left(t^2-4t+3\right)\ \mathrm{m}/\mathrm{s}\), where t is in seconds. Construct the v–t and a–t graphs for the time interval \(0\le t\le4\mathrm{\ s}\).

If the position of a particle is defined by \(s=[2\sin(\pi/5)t+4]\mathrm{\ m}\), where t is in seconds, construct the s-t, v-t, and a-t graphs for \(0\le t\le10\mathrm{\ s}\).

An airplane starts from rest, travels 5000 ft down a runway, and after uniform acceleration, takes off with a speed of 162 mi/h. It then climbs in a straight line with a uniform acceleration of \(3\ \mathrm{ft}/\mathrm{s}^2\) until it reaches a constant speed of 220 mi/h. Draw the s–t, v–t, and a–t graphs that describe the motion.

The elevator starts from rest at the first floor of the building. It can accelerate at \(5\ \mathrm{ft}/\mathrm{s}^2\) and then decelerate at \(2\ \mathrm{ft}/\mathrm{s}^2\). Determine the shortest time it takes to reach a floor 40 ft above the ground. The elevator starts from rest and then stops. Draw the a–t, v–t, and s–t graphs for the motion.

The velocity of a car is plotted as shown. Determine the total distance the car moves until it stops (t = 80 s). Construct the a–t graph.

The motion of a jet plane just after landing on a runway is described by the a–t graph. Determine the time \(t^{\prime}\) when the jet plane stops. Construct the v–t and s–t graphs for the motion. Here s = 0, and v = 300 ft/s when t = 0.

The v–t graph for a particle moving through an electric field from one plate to another has the shape shown in the figure. The acceleration and deceleration that occur are constant and both have a magnitude of \(4\mathrm{\ m}/\mathrm{s}^2\). If the plates are spaced 200 mm apart, determine the maximum velocity \(v_{\max }\) and the time \(t^{\prime}\) for the particle to travel from one plate to the other. Also draw the s–t graph. When \(t=t^{\prime} / 2\) the particle is at s = 100 mm.

The v–t graph for a particle moving through an electric field from one plate to another has the shape shown in the figure, where \(t{\prime}=0.2\mathrm{\ s}\) and \(v_{\max}=10\mathrm{\ m}/\mathrm{s}\). Draw the s–t and a–t graphs for the particle. When \(t=t{\prime} / 2\) the particle is at s = 0.5 m.

The a–s graph for a rocket moving along a straight track has been experimentally determined. If the rocket starts at s = 0 when v = 0, determine its speed when it is at s = 75 ft, and 125 ft, respectively. Use Simpson’s rule with n = 100 to evaluate v at s = 125 ft.

A two-stage rocket is fired vertically from rest at s = 0 with the acceleration as shown. After 30 s the first stage, A, burns out and the second stage, B, ignites. Plot the v–t and s–t graphs which describe the motion of the second stage for \(0\le t\le60\mathrm{\ s}\)s.

The race car starts from rest and travels along a straight road until it reaches a speed of 26 m/s in 8 s as shown on the v–t graph. The flat part of the graph is caused by shifting gears. Draw the a–t graph and determine the maximum acceleration of the car.

The jet car is originally traveling at a velocity of 10 m/s when it is subjected to the acceleration shown. Determine the car’s maximum velocity and the time \(t{\prime}\) when it stops. When t = 0, s = 0.

The car starts from rest at s = 0 and is subjected to an acceleration shown by the a–s graph. Draw the v–s graph and determine the time needed to travel 200 ft.

The v–t graph for a train has been experimentally determined. From the data, construct the s–t and a–t graphs for the motion for \(0\le t\le180\mathrm{\ s}\). When t = 0, s = 0.

A motorcycle starts from rest at s = 0 and travels along a straight road with the speed shown by the v–t graph. Determine the total distance the motorcycle travels until it stops when t = 15 s. Also plot the a–t and s–t graphs.

A motorcycle starts from rest at s = 0 and travels along a straight road with the speed shown by the v–t graph. Determine the motorcycle’s acceleration and position when t = 8 s and t = 12 s.

The v–t graph for the motion of a car as it moves along a straight road is shown. Draw the s–t and a–t graphs. Also determine the average speed and the distance traveled for the 15-s time interval. When t = 0, s = 0.

An airplane lands on the straight runway, originally traveling at 110 ft/s when s = 0. If it is subjected to the decelerations shown, determine the time \(t{\prime}\) needed to stop the plane and construct the s–t graph for the motion.

Starting from rest at s = 0, a boat travels in a straight line with the acceleration shown by the a–s graph. Determine the boat’s speed when s = 50 ft, 100 ft, and 150 ft.

Starting from rest at s = 0, a boat travels in a straight line with the acceleration shown by the a–s graph. Construct the v–s graph.

A two-stage rocket is fired vertically from rest with the acceleration shown. After 15 s the first stage A burns out and the second stage B ignites. Plot the v–t and s–t graphs which describe the motion of the second stage for \(0\le t\le40\mathrm{\ s}\).

The speed of a train during the first minute has been recorded as follows: \(\begin{array}{ccccc} t \ (\mathrm{s}) & 0 & 20 & 40 & 60 \\ \hline v \ (\mathrm{m} / \mathrm{s}) & 0 & 16 & 21 & 24 \end{array}\) Plot the v–t graph, approximating the curve as straight-line segments between the given points. Determine the total distance traveled.

A man riding upward in a freight elevator accidentally drops a package off the elevator when it is 100 ft from the ground. If the elevator maintains a constant upward speed of 4 ft/s, determine how high the elevator is from the ground the instant the package hits the ground. Draw the v–t curve for the package during the time it is in motion. Assume that the package was released with the same upward speed as the elevator.

Two cars start from rest side by side and travel along a straight road. Car A accelerates at \(4\mathrm{\ m}/\mathrm{s}^2\) for 10 s and then maintains a constant speed. Car B accelerates at \(5\mathrm{\ m}/\mathrm{s}^2\) until reaching a constant speed of 25 m/s and then maintains this speed. Construct the a–t, v–t, and s–t graphs for each car until t = 15 s. What is the distance between the two cars when t = 15 s?

If the position of a particle is defined as \(s=\left(5t-3t^2\right)\ \mathrm{ft}\), where t is in seconds, construct the s–t, v–t, and a–t graphs for \(0\le t\le10\mathrm{\ s}\).

From experimental data, the motion of a jet plane while traveling along a runway is defined by the v–t graph. Construct the s–t and a–t graphs for the motion. When t = 0, s = 0.

The motion of a train is described by the a–s graph shown. Draw the v–s graph if v = 0 at s = 0.

The jet plane starts from rest at s = 0 and is subjected to the acceleration shown. Determine the speed of the plane when it has traveled 1000 ft. Also, how much time is required for it to travel 1000 ft?

The boat travels along a straight line with the speed described by the graph. Construct the s–t and a–s graphs. Also, determine the time required for the boat to travel a distance s = 400 m if s = 0 when t = 0.

The v–s graph of a cyclist traveling along a straight road is shown. Construct the a–s graph.

The v–s graph for a test vehicle is shown. Determine its acceleration when s = 100 m and when s = 175 m.

If the velocity of a particle is defined as \(\mathbf{v}(t)=\left\{0.8t^2\mathbf{i}+12t^{1/2}\mathbf{j}+5\mathbf{k}\right\}\mathrm{\ m}/\mathrm{s}\), determine the magnitude and coordinate direction angles \(\alpha\), \(\beta\), \(\gamma\) of the particle’s acceleration when t = 2 s.

The velocity of a particle is v = {3i + (6 - 2t)j} m/s, where t is in seconds. If r = 0 when t = 0, determine the displacement of the particle during the time interval t = 1 s to t = 3 s.

A particle, originally at rest and located at point (3 ft, 2 ft, 5 ft), is subjected to an acceleration of \(\mathbf{a}=\left\{6t\mathbf{i}+12t^2\mathbf{k}\right\}\ \mathrm{ft}/\mathrm{s}^2\). Determine the particle’s position (x, y, z) at t = 1 s.

The velocity of a particle is given by \(v=\left\{16 t^{2} \mathbf{i}+\right.\left.4 t^{3} \mathrm{j}+(5 t+2) \mathbf{k}\right\} \mathrm{\ m} / \mathrm{s}\), where t is in seconds. If the particle is at the origin when t = 0, determine the magnitude of the particle’s acceleration when t = 2 s. Also, what is the x, y, z coordinate position of the particle at this instant?

The water sprinkler, positioned at the base of a hill, releases a stream of water with a velocity of 15 ft/s as shown. Determine the point B(x, y) where the water strikes the ground on the hill. Assume that the hill is defined by the equation \(y=\left(0.05x^2\right)\ \mathrm{ft}\) and neglect the size of the sprinkler.

A particle, originally at rest and located at point (3 ft, 2 ft, 5 ft), is subjected to an acceleration \(\mathbf{a}=\left\{6t\mathbf{i}+12t^2\mathbf{k}\right\}\ \mathrm{ft}/\mathrm{s}^2\). Determine the particle’s position (x, y, z) when t = 2 s.

A particle travels along the curve from A to B in 2 s. It takes 4 s for it to go from B to C and then 3 s to go from C to D. Determine its average speed when it goes from A to D.

A particle travels along the curve from A to B in 5 s. It takes 8 s for it to go from B to C and then 10 s to go from C to A. Determine its average speed when it goes around the closed path.

The position of a crate sliding down a ramp is given by \(x=\left(0.25t^3\right)\mathrm{\ m}\), \(y=\left(1.5t^2\right)\mathrm{\ m}\), \(z=\left(6-0.75t^{5/2}\right)\mathrm{\ m}\), where t is in seconds. Determine the magnitude of the crate’s velocity and acceleration when t = 2 s.

A rocket is fired from rest at x = 0 and travels along a parabolic trajectory described by \(y^{2}=\left|120\left(10^{3}\right) x\right| \mathrm{m}\). If the x component of acceleration is \(a_x=\left(\frac{1}{4}t^2\right)\ \mathrm{m}/\mathrm{s}^2\), where t is in seconds, determine the magnitude of the rocket’s velocity and acceleration when t = 10 s.

The particle travels along the path defined by the parabola \(y=0.5 x^{2}\). If the component of velocity along the x axis is \(v_x=(5t)\mathrm{\ ft}/\mathrm{s}\), where t is in seconds, determine the particle’s distance from the origin O and the magnitude of its acceleration when t = 1 s. When t = 0, x = 0, y = 0.

The motorcycle travels with constant speed \(v_{0}\) along the path that, for a short distance, takes the form of a sine curve. Determine the x and y components of its velocity at any instant on the curve.

A particle travels along the curve from A to B in 1 s. If it takes 3 s for it to go from A to C, determine its average velocity when it goes from B to C.

The roller coaster car travels down the helical path at constant speed such that the parametric equations that define its position are x = c sin kt, y = c cos kt, z = h ? bt, where c, h, and b are constants. Determine the magnitudes of its velocity and acceleration.

Pegs A and B are restricted to move in the elliptical slots due to the motion of the slotted link. If the link moves with a constant speed of 10 m/s, determine the magnitude of the velocity and acceleration of peg A when x = 1m.

The van travels over the hill described by \(y=\left(-1.5\left(10^{-3}\right)\ x^2+15\right)\ \mathrm{ft}\). If it has a constant speed of 75 ft/s, determine the x and y components of the van’s velocity and acceleration when x = 50 ft.

The flight path of the helicopter as it takes off from A is defined by the parametric equations \(x=\left(2t^2\right)\mathrm{\ m}\) and \(y=\left(0.04t^3\right)\ \mathrm{m}\), where t is the time in seconds. Determine the distance the helicopter is from point A and the magnitudes of its velocity and acceleration when t = 10 s.

Determine the minimum initial velocity \(v_{0}\) and the corresponding angle \(\theta_{0}\) at which the ball must be kicked in order for it to just cross over the 3-m high fence.

The catapult is used to launch a ball such that it strikes the wall of the building at the maximum height of its trajectory. If it takes 1.5 s to travel from A to B, determine the velocity \(\(\mathbf{v}_{A}\)\) at which it was launched, the angle of release \(\theta\), and the height h.

Neglecting the size of the ball, determine the magnitude \(v_A\) of the basketball’s initial velocity and its velocity when it passes through the basket.

The girl at A can throw a ball at \(v_A=10\mathrm{\ m}/\mathrm{s}\). Calculate the maximum possible range \(R=R_{\max }\) and the associated angle \(\theta\) at which it should be thrown. Assume the ball is caught at B at the same elevation from which it is thrown.

Show that the girl at A can throw the ball to the boy at B by launching it at equal angles measured up or down from a \(45^{\circ}\) inclination. If \(v_A=10\mathrm{\ m}/\mathrm{s}\), determine the range R if this value is \(15^{\circ}\), i.e., \(\theta_{1}=45^{\circ}-15^{\circ}=30^{\circ}\) and \theta_{2}=45^{\circ}+15^{\circ}=60^{\circ}. Assume the ball is caught at the same elevation from which it is thrown.

The ball at A is kicked with a speed \(v_A=80\mathrm{\ ft}/\mathrm{s}\) and at an angle \(\theta_{A}=30^{\circ}\). Determine the point (x, –y) where it strikes the ground. Assume the ground has the shape of a parabola as shown.

The ball at A is kicked such that \(\theta_{A}=30^{\circ}\). If it strikes the ground at B having coordinates x = 15 ft, y = -9 ft, determine the speed at which it is kicked and the speed at which it strikes the ground.

A golf ball is struck with a velocity of 80 ft/s as shown. Determine the distance d to where it will land.

A golf ball is struck with a velocity of 80 ft/s as shown. Determine the speed at which it strikes the ground at B and the time of flight from A to B.

The basketball passed through the hoop even though it barely cleared the hands of the player B who attempted to block it. Neglecting the size of the ball, determine the magnitude \(v_A\) of its initial velocity and the height h of the ball when it passes over player B.

It is observed that the skier leaves the ramp A at an angle \(\theta_{A}=25^{\circ}\) with the horizontal. If he strikes the ground at B, determine his initial speed \(v_A\) and the time of flight \(t_{A B}\).

It is observed that the skier leaves the ramp A at an angle \(\theta_{A}=25^{\circ}\) with the horizontal. If he strikes the ground at B, determine his initial speed \(v_A\) and the speed at which he strikes the ground.

Determine the horizontal velocity \(v_A\) of a tennis ball at A so that it just clears the net at B. Also, find the distance s where the ball strikes the ground.

The missile at A takes off from rest and rises vertically to B, where its fuel runs out in 8 s. If the acceleration varies with time as shown, determine the missile’s height \(h_{B}\) and speed \(v_{B}\). If by internal controls the missile is then suddenly pointed \(45^{\circ}\) as shown, and allowed to travel in free flight, determine the maximum height attained, \(h_{C}\), and the range R to where it crashes at D.

The projectile is launched with a velocity \(\mathbf{v}_{0}\). Determine the range R, the maximum height h attained, and the time of flight. Express the results in terms of the angle \(\theta\) and \(v_{0}\). The acceleration due to gravity is g.

The drinking fountain is designed such that the nozzle is located from the edge of the basin as shown. Determine the maximum and minimum speed at which water can be ejected from the nozzle so that it does not splash over the sides of the basin at B and C.

If the dart is thrown with a speed of 10 m/s, determine the shortest possible time before it strikes the target. Also, what is the corresponding angle \(\theta_{A}\) at which it should be thrown, and what is the velocity of the dart when it strikes the target?

If the dart is thrown with a speed of 10 m/s, determine the longest possible time when it strikes the target. Also, what is the corresponding angle \(\theta_{A}\) at which it should be thrown, and what is the velocity of the dart when it strikes the target?

The man at A wishes to throw two darts at the target at B so that they arrive at the same time. If each dart is thrown with a speed of 10 m/s, determine the angles \(\theta_{C}\) and \(\theta_{D}\) at which they should be thrown and the time between each throw. Note that the first dart must be thrown at \(\theta_{C}\left(>\theta_{D}\right)\), then the second dart is thrown at \(\theta_{D}\).

The velocity of the water jet discharging from the orifice can be obtained from \(v=\sqrt{2\ gh}\), where h = 2 m is the depth of the orifice from the free water surface. Determine the time for a particle of water leaving the orifice to reach point B and the horizontal distance x where it hits the surface.

The snowmobile is traveling at 10 m/s when it leaves the embankment at A. Determine the time of flight from A to B and the range R of the trajectory.

The fireman wishes to direct the flow of water from his hose to the fire at B. Determine two possible angles \(\theta_{1}\) and \(\theta_{2}\) at which this can be done. Water flows from the hose at \(v_A=80\mathrm{\ ft}/\mathrm{s}\).

The baseball player A hits the baseball at \(v_A=40\mathrm{\ ft}/\mathrm{s}\) and \(\theta_{A}=60^{\circ}\) from the horizontal. When the ball is directly overhead of player B he begins to run under it. Determine the constant speed at which B must run and the distance d in order to make the catch at the same elevation at which the ball was hit.

The catapult is used to launch a ball such that it strikes the wall of the building at the maximum height of its trajectory. If it takes 1.5 s to travel from A to B, determine the velocity \(\mathbf{v}_{A}\) at which it was launched, the angle of release \(\theta\), and the height h.

An automobile is traveling on a curve having a radius of 800 ft. If the acceleration of the automobile is \(5\mathrm{\ ft}/\mathrm{s}^2\), determine the constant speed at which the automobile is traveling.

Determine the maximum constant speed a race car can have if the acceleration of the car cannot exceed \(7.5\mathrm{\ m}/\mathrm{s}^2\) while rounding a track having a radius of curvature of 200 m.

A boat has an initial speed of 16 ft/s. If it then increases its speed along a circular path of radius \(\rho=80\ \mathrm{ft}\) at the rate of \(\dot{\mathrm{v}}=(1.5\mathrm{s})\mathrm{\ ft}/\mathrm{s}\), where s is in feet, determine the time needed for the boat to travel s = 50 ft.

The position of a particle is defined by \(\mathbf{r}=\left\{4(t-\sin t)\mathbf{i}+\left(2t^2-3\right)\mathbf{j}\right\}\ \mathrm{m}\), where t is in seconds and the argument for the sine is in radians. Determine the speed of the particle and its normal and tangential components of acceleration when t = 1 s.

The automobile has a speed of 80 ft/s at point A and an acceleration having a magnitude of \(10\ \mathrm{ft}/\mathrm{s}^2\), acting in the direction shown. Determine the radius of curvature of the path at point A and the tangential component of acceleration.

The automobile is originally at rest at s = 0. If its speed is increased by \(\dot{v}=\left(0.05t^2\right)\ \mathrm{ft}/\mathrm{s}^2\), where t is in seconds, determine the magnitudes of its velocity and acceleration when t = 18 s.

The automobile is originally at rest s = 0. If it then starts to increase its speed at \(\dot{v}=\left(0.05t^2\right)\ \mathrm{ft}/\mathrm{s}^2\), where t is in seconds, determine the magnitudes of its velocity and acceleration at s = 550 ft.

The two cars A and B travel along the circular path at constant speeds \(v_A=80\ \mathrm{ft}/\mathrm{s}\) and \(v_B=100\ \mathrm{ft}/\mathrm{s}\), respectively. If they are at the positions shown when t = 0, determine the time when the cars are side by side, and the time when they are \(90^{\circ}\) apart.

Cars A and B are traveling around the circular race track. At the instant shown, A has a speed of 60 ft/s and is increasing its speed at the rate of \(15\mathrm{\ ft}/\mathrm{s}^2\) until it travels for a distance of \(100\pi\mathrm{\ ft}\), after which it maintains a constant speed. Car B has a speed of 120 ft/s and is decreasing its speed at \(15\mathrm{\ ft}/\mathrm{s}^2\) until it travels a distance of \(65\pi\mathrm{\ ft}\), after which it maintains a constant speed. Determine the time when they come side by side.

The satellite S travels around the earth in a circular path with a constant speed of 20 Mm/h. If the acceleration is \(2.5\mathrm{\ m}/\mathrm{s}^2\), determine the altitude h. Assume the earth’s diameter to be 12 713 km.

The car travels along the circular path such that its speed is increased by \(a_{\mathrm{t}}=\left(0.5e^t\right)\ \mathrm{m}/\mathrm{s}^2\), where t is in seconds. Determine the magnitudes of its velocity and acceleration after the car has traveled s = 18 m starting from rest. Neglect the size of the car.

The car passes point A with a speed of 25 m/s after which its speed is defined by v = (25 - 0.15s) m/s. Determine the magnitude of the car’s acceleration when it reaches point B, where s = 51.5 m and x = 50 m.

If the car passes point A with a speed of 20 m/s and begins to increase its speed at a constant rate of \(a_t=0.5\mathrm{\ m}/\mathrm{s}^2\), determine the magnitude of the car’s acceleration when s = 101.68 m and x = 0.

The motorcycle is traveling at 1 m/s when it is at A. If the speed is then increased at \(\dot{v}=0.1\mathrm{\ m}/\mathrm{s}^2\), determine its speed and acceleration at the instant t = 5 s.

The box of negligible size is sliding down along a curved path defined by the parabola \(y=0.4 x^{2}\). When it is at \(A\left(x_A=2\mathrm{\ m},\ y_A=1.6\mathrm{\ m}\right)\), the speed is v = 8 m/s and the increase in speed is \(dv/dt=4\mathrm{\ m}/\mathrm{s}^2\). Determine the magnitude of the acceleration of the box at this instant.

The car travels around the circular track having a radius of r = 300 m such that when it is at point A it has a velocity of 5 m/s, which is increasing at the rate of \(\dot{v}=(0.06t)\mathrm{\ m}/\mathrm{s}^2\), where t is in seconds. Determine the magnitudes of its velocity and acceleration when it has traveled one-third the way around the track.

The car travels around the portion of a circular track having a radius of r = 500 ft such that when it is at point A it has a velocity of 2 ft/s, which is increasing at the rate of \(\dot{v}=(0.002t)\mathrm{\ ft}/\mathrm{s}^2\), where t is in seconds. Determine the magnitudes of its velocity and acceleration when it has traveled three-fourths the way around the track.

At a given instant the train engine at E has a speed of 20 m/s and an acceleration of \(14\mathrm{\ m}/\mathrm{s}^2\) acting in the direction shown. Determine the rate of increase in the train’s speed and the radius of curvature \(\rho\) of the path.

The car has an initial speed \(v_0=20\mathrm{\ m}/\mathrm{s}\). If it increases its speed along the circular track at s = 0, \(a_t=(0.8\mathrm{s})\ \mathrm{m}/\mathrm{s}^2\), where s is in meters, determine the time needed for the car to travel s = 25 m.

The car starts from rest at s = 0 and increases its speed at \(a_t=4\mathrm{\ m}/\mathrm{s}^2\). Determine the time when the magnitude of acceleration becomes \(20\mathrm{\ m}/\mathrm{s}^2\). At what position s does this occur?

A boat is traveling along a circular curve having a radius of 100 ft. If its speed at t = 0 is 15 ft/s and is increasing at \(\dot{v}=(0.8t)\mathrm{\ ft}/\mathrm{s}^2\), determine the magnitude of its acceleration at the instant t = 5 s.

A boat is traveling along a circular path having a radius of 20 m. Determine the magnitude of the boat’s acceleration when the speed is v = 5 m/s and the rate of increase in the speed is \(\dot{v}=2\mathrm{\ m}/\mathrm{s}^2\).

Starting from rest, a bicyclist travels around a horizontal circular path, \(\rho=10\mathrm{\ m}\), at a speed of \(v=\left(0.09t^2+0.1t\right)\mathrm{\ m}/\mathrm{s}\), where t is in seconds. Determine the magnitudes of his velocity and acceleration when he has traveled s = 3 m.

A particle travels around a circular path having a radius of 50 m. If it is initially traveling with a speed of 10 m/s and its speed then increases at a rate of \(\dot{v}=(0.05v)\mathrm{\ m}/\mathrm{s}^2\), determine the magnitude of the particle’s acceleration four seconds later.

The motorcycle is traveling at a constant speed of 60 km/h. Determine the magnitude of its acceleration when it is at point A.

When t = 0, the train has a speed of 8 m/s, which is increasing at \(0.5\mathrm{\ m}/\mathrm{s}^2\). Determine the magnitude of the acceleration of the engine when it reaches point A, at t = 20 s. Here the radius of curvature of the tracks is \(\rho_A=400\mathrm{\ m}\).

At a given instant the jet plane has a speed of 550 m/s and an acceleration of \(50\mathrm{\ m}/\mathrm{s}^2\) acting in the direction shown. Determine the rate of increase in the plane’s speed, and also the radius of curvature \(\rho\) of the path.

The ball is ejected horizontally from the tube with a speed of 8 m/s. Find the equation of the path, y = f(x), and then find the ball’s velocity and the normal and tangential components of acceleration when t = 0.25 s.

The motorcycle is traveling at 40 m/s when it is at A. If the speed is then decreased at \(\dot{v}=-(0.05\mathrm{\ s})\ \mathrm{m}/\mathrm{s}^2\), where s is in meters measured from A, determine its speed and acceleration when it reaches B.

Cars move around the “traffic circle” which is in the shape of an ellipse. If the speed limit is posted at 60 km/h, determine the minimum acceleration experienced by the passengers.

Cars move around the “traffic circle” which is in the shape of an ellipse. If the speed limit is posted at 60 km/h, determine the maximum acceleration experienced by the passengers.

A package is dropped from the plane which is flying with a constant horizontal velocity of \(v_A=150\ \mathrm{ft}/\mathrm{s}\). Determine the normal and tangential components of acceleration and the radius of curvature of the path of motion (a) at the moment the package is released at A, where it has a horizontal velocity of \(v_A=150\mathrm{\ ft}/\mathrm{s}\), and (b) just before it strikes the ground at B.

The race car has an initial speed \(v_A=15\mathrm{\ m}/\mathrm{s}\) at A. If it increases its speed along the circular track at the rate \(a_t=(0.4\mathrm{s})\mathrm{\ m}/\mathrm{s}^2\), where s is in meters, determine the time needed for the car to travel 20 m. Take \(\rho=150\mathrm{\ m}\).

The motorcycle travels along the elliptical track at a constant speed v. Determine its greatest acceleration if a > b.

The motorcycle travels along the elliptical track at a constant speed v. Determine its smallest acceleration if a > b.

Particles A and B are traveling counter-clockwise around a circular track at a constant speed of 8 m/s. If at the instant shown the speed of A begins to increase by \(\left(a_t\right)_A=\left(0.4s_A\right)\mathrm{\ m}/\mathrm{s}^2\), where \(s_{A}\) is in meters, determine the distance measured counterclockwise along the track from B to A when t = 1 s. What is the magnitude of the acceleration of each particle at this instant?

Particles A and B are traveling around a circular track at a speed of 8 m/s at the instant shown. If the speed of B is increasing by \(\left(a_t\right)_B=4\mathrm{\ m}/\mathrm{s}^2\), and at the same instant A has an increase in speed of \(\left(a_t\right)_A=0.8t\mathrm{\ m}/\mathrm{s}^2\), determine how long it takes for a collision to occur. What is the magnitude of the acceleration of each particle just before the collision occurs?

The jet plane is traveling with a speed of 120 m/s which is decreasing at \(40\mathrm{\ m}/\mathrm{s}^2\) when it reaches point A. Determine the magnitude of its acceleration when it is at this point. Also, specify the direction of flight, measured from the x axis.

The jet plane is traveling with a constant speed of 110 m/s along the curved path. Determine the magnitude of the acceleration of the plane at the instant it reaches point A(y = 0).

The train passes point B with a speed of 20 m/s which is decreasing at \(a_t=-0.5\mathrm{\ m}/\mathrm{s}^2\). Determine the magnitude of acceleration of the train at this point.

The train passes point A with a speed of 30 m/s and begins to decrease its speed at a constant rate of \(a_t=-0.25\mathrm{\ m}/\mathrm{s}^2\). Determine the magnitude of the acceleration of the train when it reaches point B, where \(s_{AB}=412\mathrm{\ m}\).

The particle travels with a constant speed of 300 mm/s along the curve. Determine the particle’s acceleration when it is located at point (200 mm, 100 mm) and sketch this vector on the curve.

A particle P travels along an elliptical spiral path such that its position vector r is defined by r = {2 cos(0.1t)i + 1.5 sin(0.1t)j + (2t)k} m, where t is in seconds and the arguments for the sine and cosine are given in radians. When t = 8 s, determine the coordinate direction angles \(\alpha\), \(beta\), and \(\gamma\), which the binormal axis to the osculating plane makes with the x, y, and z axes. Hint: Solve for the velocity \(\mathbf{v}_{P}\) and acceleration \(\mathbf{a}_{P}\) of the particle in terms of their i, j, k components. The binormal is parallel to \(\mathbf{v}_{P} \times \mathbf{a}_{P}\). Why?

The motion of a particle is defined by the equations \(x=\left(2t+t^2\right)\ \mathrm{m}\) and \(y=\left(t^2\right)\mathrm{\ m}\), where t is in seconds. Determine the normal and tangential components of the particle’s velocity and acceleration when t = 2 s.

If the speed of the crate at A is 15 ft/s, which is increasing at a rate \(\dot{\mathrm{v}}=3\mathrm{\ ft}/\mathrm{s}^2\), determine the magnitude of the acceleration of the crate at this instant.

A particle is moving along a circular path having a radius of 4 in. such that its position as a function of time is given by \(\theta=\cos 2 t\), where \(\theta\) is in radians and t is in seconds. Determine the magnitude of the acceleration of the particle when \(\theta=30^{\circ}\).

For a short time a rocket travels up and to the right at a constant speed of 800 m/s along the parabolic path \(y=600-35 x^{2}\). Determine the radial and transverse components of velocity of the rocket at the instant \(\theta=60^{\circ}\), where \(\theta\) is measured counterclockwise from the x axis.

A particle moves along a path defined by polar coordinates \(r=\left(2e^t\right)\ \mathrm{ft}\) and \(\theta=\left(8t^2\right)\ \mathrm{rad}\), where t is in seconds. Determine the components of its velocity and acceleration when t = 1 s.

An airplane is flying in a straight line with a velocity of 200 mi/h and an acceleration of \(3\ \mathrm{mi}/\mathrm{h}^2\). If the propeller has a diameter of 6 ft and is rotating at a constant angular rate of 120 rad/s, determine the magnitudes of velocity and acceleration of a particle located on the tip of the propeller.

The small washer is sliding down the cord OA. When it is at the midpoint, its speed is 28 m/s and its acceleration is \(7\mathrm{\ m}/\mathrm{s}^2\). Express the velocity and acceleration of the washer at this point in terms of its cylindrical components.

A radar gun at O rotates with the angular velocity of \(\dot{\theta}=0.1\mathrm{\ rad}/\mathrm{s}\) and angular acceleration of \(\ddot{\theta}=0.025 \mathrm{\ rad} / \mathrm{s}^{2}\), at the instant \(\theta=45^{\circ}\), as it follows the motion of the car traveling along the circular road having a radius of r = 200 m. Determine the magnitudes of velocity and acceleration of the car at this instant.

If a particle moves along a path such that r = (2 cos t) ft and \(\theta=(t/2)\mathrm{\ rad}\), where t is in seconds, plot the path \(r=f(\theta)\) and determine the particle’s radial and transverse components of velocity and acceleration.

If a particle moves along a path such that \(r=\left(e^{at}\right)\ \mathrm{m}\) and \(\theta=t\), where t is in seconds, plot the path \(r=f(\theta)\), and determine the particle’s radial and transverse components of velocity and acceleration.

The car travels along the circular curve having a radius r = 400 ft. At the instant shown, its angular rate of rotation is \(\dot{\theta}=0.025\mathrm{\ rad}/\mathrm{s}\), which is decreasing at the rate \(\ddot{\theta}=-0.008 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the radial and transverse components of the car’s velocity and acceleration at this instant and sketch these components on the curve.

The car travels along the circular curve of radius r = 400 ft with a constant speed of v = 30 ft/s. Determine the angular rate of rotation \(\dot{\theta}\) of the radial line r and the magnitude of the car’s acceleration.

The time rate of change of acceleration is referred to as the jerk, which is often used as a means of measuring passenger discomfort. Calculate this vector, \(\dot{\mathrm{a}}\), in terms of its cylindrical components, using Eq. 12–32.

A particle is moving along a circular path having a radius of 6 in. such that its position as a function of time is given by \(\theta=\sin 3 t\), where \(\theta\) and the argument for the sine are in radians, and t is in seconds. Determine the magnitude of the acceleration of the particle at \(\theta=30^{\circ}\). The particle starts from rest at \(\theta=0^{\circ}\).

The slotted link is pinned at O, and as a result of the constant angular velocity \(\dot{\theta}=3\ \mathrm{rad}/\mathrm{s}\) it drives the peg P for a short distance along the spiral guide \(r=(0.4\ \theta)\mathrm{\ m}\), where \(\theta\) is in radians. Determine the radial and transverse components of the velocity and acceleration of P at the instant \(\theta=\pi/3\mathrm{\ rad}\).

For a short time the bucket of the backhoe traces the path of the cardioid \(r=25(1-\cos\theta)\mathrm{\ ft}\). Determine the magnitudes of the velocity and acceleration of the bucket when \(\theta=120^{\circ}\) if the boom is rotating with an angular velocity of \(\dot{\theta}=2 \mathrm{rad} / \mathrm{s}\) and an angular acceleration of \(\ddot{\theta}=0.2 \mathrm{\ rad} / \mathrm{s}^{2}\) at the instant shown.

The slotted link is pinned at O, and as a result of the constant angular velocity \(\dot{\theta}=3\mathrm{\ rad}/\mathrm{s}\) it drives the peg P for a short distance along the spiral guide \(r=(0.4\theta)\mathrm{\ m}\), where \(\theta\) is in radians. Determine the velocity and acceleration of the particle at the instant it leaves the slot in the link, i.e., when r = 0.5 m.

A particle moves in the x-y plane such that its position is defined by \(r=\left\{2t\mathbf{i}+4t^2\mathbf{j}\right\}\ \mathrm{ft}\), where t is in seconds. Determine the radial and transverse components of the particle’s velocity and acceleration when t = 2 s.

At the instant shown, the man is twirling a hose over his head with an angular velocity \(\dot{\theta}=2\ \mathrm{rad}/\mathrm{s}\) and an angular acceleration \(\ddot{\theta}=3 \mathrm{\ rad} / \mathrm{s}^{2}\). If it is assumed that the hose lies in a horizontal plane, and water is flowing through it at a constant rate of 3 m/s, determine the magnitudes of the velocity and acceleration of a water particle as it exits the open end, r = 1.5 m.

The rod OA rotates clockwise with a constant angular velocity of 6 rad/s. Two pin-connected slider blocks, located at B, move freely on OA and the curved rod whose shape is a limaçon described by the equation \(r=200(2-\cos\theta)\mathrm{\ mm}\). Determine the speed of the slider blocks at the instant \(\theta=150^{\circ}\).

Determine the magnitude of the acceleration of the slider blocks in Prob. 12–172 when \(\theta=150^{\circ}\).

A double collar C is pin connected together such that one collar slides over a fixed rod and the other slides over a rotating rod. If the geometry of the fixed rod for a short distance can be defined by a lemniscate, \(r^2=(4\cos2\theta)\mathrm{\ ft}^2\), determine the collar’s radial and transverse components of velocity and acceleration at the instant \(\theta=0^{\circ}\) as shown. Rod OA is rotating at a constant rate of \(\dot{\theta}=6\mathrm{\ rad}/\mathrm{s}\).

A block moves outward along the slot in the platform with a speed of \(\dot{r}=(4t)\ \mathrm{m}/\mathrm{s}\), where t is in seconds. The platform rotates at a constant rate of 6 rad/s. If the block starts from rest at the center, determine the magnitudes of its velocity and acceleration when t = 1 s.

The car travels around the circular track with a constant speed of 20 m/s. Determine the car’s radial and transverse components of velocity and acceleration at the instant \(\theta=\pi/4\mathrm{\ rad}\).

The car travels around the circular track such that its transverse component is \(\theta=\left(0.006t^2\right)\ \mathrm{rad}\), where t is in seconds. Determine the car’s radial and transverse components of velocity and acceleration at the instant t = 4 s.

The car travels along a road which for a short distance is defined by \(r=(200/\theta)\mathrm{\ ft}\), where \(\theta\) is in radians. If it maintains a constant speed of v = 35 ft/s, determine the radial and transverse components of its velocity when \(\theta=\pi/3\mathrm{\ rad}\).

A horse on the merry-go-round moves according to the equations \(r=8\mathrm{\ ft}\), \(\theta=(0.6t)\mathrm{\ rad}\), and \(z=(1.5\sin\theta)\mathrm{\ ft}\), where t is in seconds. Determine the cylindrical components of the velocity and acceleration of the horse when t = 4 s.

A horse on the merry-go-round moves according to the equations \(r=8\ \mathrm{ft}\), \(\dot{\theta}=2\mathrm{\ rad}/\mathrm{s}\) and \(z=(1.5\sin\theta)\mathrm{\ ft}\), where t is in seconds. Determine the maximum and minimum magnitudes of the velocity and acceleration of the horse during the motion.

If the slotted arm AB rotates counterclockwise with a constant angular velocity of \(\dot{\theta}=2\mathrm{\ rad}/\mathrm{s}\), determine the magnitudes of the velocity and acceleration of peg P at \(\theta=30^{\circ}\). The peg is constrained to move in the slots of the fixed bar CD and rotating bar AB.

The peg is constrained to move in the slots of the fixed bar CD and rotating bar AB. When \(\theta=30^{\circ}\), the angular velocity and angular acceleration of arm AB are \(\dot{\theta}=2\mathrm{\ rad}/\mathrm{s}\) and \(\ddot{\theta}=3 \mathrm{\ rad} / \mathrm{s}^{2}\), respectively. Determine the magnitudes of the velocity and acceleration of the peg P at this instant.

A truck is traveling along the horizontal circular curve of radius r = 60 m with a constant speed v = 20 m/s. Determine the angular rate of rotation \(\dot{\theta}\) of the radial line r and the magnitude of the truck’s acceleration.

A truck is traveling along the horizontal circular curve of radius r = 60 m with a speed of 20 m/s which is increasing at \(3\mathrm{\ m}/\mathrm{s}^2\). Determine the truck’s radial and transverse components of acceleration.

The rod OA rotates counterclockwise with a constant angular velocity of \(\dot{\theta}=5\ \mathrm{rad}/\mathrm{s}\). Two pin-connected slider blocks, located at B, move freely on OA and the curved rod whose shape is a limaçon described by the equation \(r=100(2-\cos\theta)\mathrm{\ mm}\). Determine the speed of the slider blocks at the instant \(\theta=120^{\circ}\).

Determine the magnitude of the acceleration of the slider blocks in Prob. 12–185 when \(\theta=120^{\circ}\).

The searchlight on the boat anchored 2000 ft from shore is turned on the automobile, which is traveling along the straight road at a constant speed 80 ft/s. Determine the angular rate of rotation of the light when the automobile is r = 3000 ft from the boat.

If the car in Prob. 12–187 is accelerating at \(15\ \mathrm{ft}/\mathrm{s}^2\) at the instant r = 3000 ft determine the required angular acceleration \(\ddot{\theta}\) of the light at this instant.

A particle moves along an Archimedean spiral \(r=(8\theta)\mathrm{\ ft}\), where \(\theta\) is given in radians. If \(\dot{\theta}=4\mathrm{\ rad}/\mathrm{s}\) (constant), determine the radial and transverse components of the particle’s velocity and acceleration at the instant \(\theta=\pi/2\ \mathrm{rad}\). Sketch the curve and show the components on the curve.

Solve Prob. 12–189 if the particle has an angular acceleration \(\ddot{\theta}=5 \mathrm{\ rad} / \mathrm{s}^{2}\) when \(\dot{\theta}=4 \mathrm{rad} / \mathrm{s}\) at \(\theta=\pi/2\ \mathrm{rad}\).

The arm of the robot moves so that r = 3 ft is constant, and its grip A moves along the path \(z=(3\sin4\theta)\ \mathrm{ft}\), where \(\theta\) is in radians. If \(\theta=(0.5t)\ \mathrm{rad}\), where t is in seconds, determine the magnitudes of the grip’s velocity and acceleration when t = 3 s.

For a short time the arm of the robot is extending such that \(\dot{r}=1.5\mathrm{\ ft}/\mathrm{s}\) when \(r=3\ \mathrm{ft}\), \(z=\left(4t^2\right)\ \mathrm{ft}\), and \(\theta=0.5t\mathrm{\ rad}), where t is in seconds. Determine the magnitudes of the velocity and acceleration of the grip A when t = 3 s.

The double collar C is pin connected together such that one collar slides over the fixed rod and the other slides over the rotating rod AB. If the angular velocity of AB is given as \(\dot{\theta}=\left(e^{0.5\ t^2}\right)\ \mathrm{rad}/\mathrm{s}\), where t is in seconds, and the path defined by the fixed rod is \(r=|(0.4\sin\theta+0.2)|\mathrm{\ m}\), determine the radial and transverse components of the collar’s velocity and acceleration when t = 1 s. When t = 0, \(\theta=0\). Use Simpson’s rule with n = 50 to determine \(\theta\) at t = 1 s.

The double collar C is pin connected together such that one collar slides over the fixed rod and the other slides over the rotating rod AB. If the mechanism is to be designed so that the largest speed given to the collar is 6 m/s, determine the required constant angular velocity \(\dot{\theta}\) of rod AB. The path defined by the fixed rod is \(r=(0.4\sin\theta+0.2)\mathrm{\ m}\).

If the end of the cable at A is pulled down with a speed of 2 m/s, determine the speed at which block B rises.

The motor at C pulls in the cable with an acceleration \(a_C=\left(3t^2\right)\mathrm{\ m}/\mathrm{s}^2\), where t is in seconds. The motor at D draws in its cable at \(a_D=5\mathrm{\ m}/\mathrm{s}^2\). If both motors start at the same instant from rest when d = 3 m, determine (a) the time needed for d = 0, and (b) the velocities of blocks A and B when this occurs.

The pulley arrangement shown is designed for hoisting materials. If BC remains fixed while the plunger P is pushed downward with a speed of 4 ft/s, determine the speed of the load at A.

If the end of the cable at A is pulled down with a speed of 5 m/s, determine the speed at which block B rises.

Determine the displacement of the log if the truck at C pulls the cable 4 ft to the right.

Determine the constant speed at which the cable at A must be drawn in by the motor in order to hoist the load 6 m in 1.5 s.

Starting from rest, the cable can be wound onto the drum of the motor at a rate of \(v_A=\left(3t^2\right)\ \mathrm{m}/\mathrm{s}\), where t is in seconds. Determine the time needed to lift the load 7 m.

If the end A of the cable is moving at \(v_A=3\mathrm{\ m}/\mathrm{s}\), determine the speed of block B.

Determine the time needed for the load at B to attain a speed of 10 m/s, starting from rest, if the cable is drawn into the motor with an acceleration of \(3\mathrm{\ m}/\mathrm{s}^2\).

The cable at A is being drawn toward the motor at \(v_A=8\mathrm{\ m}/\mathrm{s}\). Determine the velocity of the block.

If block A of the pulley system is moving downward at 6 ft/s while block C is moving down at 18 ft/s, determine the relative velocity of block B with respect to C.

Determine the speed of the block at B.

Determine the speed of block A if the end of the rope is pulled down with a speed of 4 m/s.

The motor draws in the cable at C with a constant velocity of \(v_C=4\mathrm{\ m}/\mathrm{s}\). The motor draws in the cable at D with a constant acceleration of \(a_D=8\mathrm{\ m}/\mathrm{s}^2\). If \(v_{D}=0\) when t = 0, determine (a) the time needed for block A to rise 3 m, and (b) the relative velocity of block A with respect to block B when this occurs.

The cord is attached to the pin at C and passes over the two pulleys at A and D. The pulley at A is attached to the smooth collar that travels along the vertical rod. Determine the velocity and acceleration of the end of the cord at B if at the instant \(s_A=4\ \mathrm{ft}\) the collar is moving upward at 5 ft/s, which is decreasing at \(2\mathrm{\ ft}/\mathrm{s}^2\).

The 16-ft-long cord is attached to the pin at C and passes over the two pulleys at A and D. The pulley at A is attached to the smooth collar that travels along the vertical rod. When \(s_B=6\ \mathrm{ft}\), the end of the cord at B is pulled downward with a velocity of 4 ft/s and is given an acceleration of \(3\mathrm{\ ft}/\mathrm{s}^2\). Determine the velocity and acceleration of the collar at this instant.

The roller at A is moving with a velocity of \(v_{A}=4\mathrm{\ m}/\mathrm{s}\) and has an acceleration of \(a_A=2\mathrm{\ m}/\mathrm{s}^2\) when \(x_A=3\mathrm{\ m}\). Determine the velocity and acceleration of block B at this instant.

The girl at C stands near the edge of the pier and pulls in the rope horizontally at a constant speed of 6 ft/s. Determine how fast the boat approaches the pier at the instant the rope length AB is 50 ft.

If the hydraulic cylinder H draws in rod BC at 2 ft/s, determine the speed of slider A.

At the instant shown, the car at A is traveling at 10 m/s around the curve while increasing its speed at \(5\mathrm{\ m}/\mathrm{s}^2\). The car at B is traveling at 18.5 m/s along the straightaway and increasing its speed at \(2\mathrm{\ m}/\mathrm{s}^2\). Determine the relative velocity and relative acceleration of A with respect to B at this instant.

The motor draws in the cord at B with an acceleration of \(a_B=2\mathrm{\ m}/\mathrm{s}^2\). When \(s_A=1.5\mathrm{\ m}\), \(v_B=6\mathrm{\ m}/\mathrm{s}\). Determine the velocity and acceleration of the collar at this instant.

If block B is moving down with a velocity \(v_{B}\) and has an acceleration \(a_{B}\), determine the velocity and acceleration of block A in terms of the parameters shown.

The crate C is being lifted by moving the roller at A downward with a constant speed of \(v_A=2\mathrm{\ m}/\mathrm{s}\) along the guide. Determine the velocity and acceleration of the crate at the instant s = 1 m. When the roller is at B, the crate rests on the ground. Neglect the size of the pulley in the calculation. Hint: Relate the coordinates \(x_{C}\) and \(x_{A}\) using the problem geometry, then take the first and second time derivatives.

Two planes, A and B, are flying at the same altitude. If their velocities are \(v_A=500\mathrm{\ km}/\mathrm{h}\) and \(v_B=700\mathrm{\ km}/\mathrm{h}\) such that the angle between their straight-line courses is \(\theta=60^{\circ}\), determine the velocity of plane B with respect to plane A.

At the instant shown, cars A and B are traveling at speeds of 55 mi/h and 40 mi/h, respectively. If B is increasing its speed by \(1200\mathrm{\ mi}/\mathrm{h}^2\), while A maintains a constant speed, determine the velocity and acceleration of B with respect to A. Car B moves along a curve having a radius of curvature of 0.5 mi.

The boat can travel with a speed of 16 km/h in still water. The point of destination is located along the dashed line. If the water is moving at 4 km/h, determine the bearing angle \(\theta\) at which the boat must travel to stay on course.

Two boats leave the pier P at the same time and travel in the directions shown. If \(v_A=40\ \mathrm{ft}/\mathrm{s}\) and \(v_B=30\ \mathrm{ft}/\mathrm{s}\), determine the velocity of boat A relative to boat B. How long after leaving the pier will the boats be 1500 ft apart?

A car is traveling north along a straight road at 50 km/h. An instrument in the car indicates that the wind is coming from the east. If the car’s speed is 80 km/h, the instrument indicates that the wind is coming from the northeast. Determine the speed and direction of the wind.

Two boats leave the shore at the same time and travel in the directions shown. If \(v_A=10\mathrm{\ m}/\mathrm{s}\) and \(v_B=15\mathrm{\ m}/\mathrm{s}\), determine the velocity of boat A with respect to boat B. How long after leaving the shore will the boats be 600 m apart?

At the instant shown, car A has a speed of 20 km/h, which is being increased at the rate of \(300\mathrm{\ km}/\mathrm{h}^2\) as the car enters the expressway. At the same instant, car B is decelerating at \(250\mathrm{\ km}/\mathrm{h}^2\) while traveling forward at 100 km/h. Determine the velocity and acceleration of A with respect to B.

Cars A and B are traveling around the circular race track. At the instant shown, A has a speed of 90 ft/s and is increasing its speed at the rate of \(15\mathrm{\ ft}/\mathrm{s}^2\), whereas B has a speed of 105 ft/s and is decreasing its speed at \(25\mathrm{\ ft}/\mathrm{s}^2\). Determine the relative velocity and relative acceleration of car A with respect to car B at this instant.

A man walks at 5 km/h in the direction of a 20 km/h wind. If raindrops fall vertically at 7 km/h in still air, determine the direction in which the drops appear to fall with respect to the man.

At the instant shown, cars A and B are traveling at velocities of 40 m/s and 30 m/s, respectively. If B is increasing its velocity by \(2\mathrm{\ m}/\mathrm{s}^2\), while A maintains a constant velocity, determine the velocity and acceleration of B with respect to A. The radius of curvature at B is \(\rho_B=200\mathrm{\ m}\).

At the instant shown, cars A and B are traveling at velocities of 40 m/s and 30 m/s, respectively. If A is increasing its velocity by \(4\mathrm{\ m}/\mathrm{s}^2\), whereas the speed of B is decreasing at \(3\mathrm{\ m}/\mathrm{s}^2\), determine the velocity and acceleration of B with respect to A. The radius of curvature at B is \(\rho_B=200\mathrm{\ m}\).

A passenger in an automobile observes that raindrops make an angle of \(30^{\circ}\) with the horizontal as the auto travels forward with a speed of 60 km/h. Compute the terminal (constant) velocity \(\mathbf{v}_{r}\) of the rain if it is assumed to fall vertically.

A man can swim at 4 ft/s in still water. He wishes to cross the 40-ft-wide river to point B, 30 ft downstream. If the river flows with a velocity of 2 ft/s, determine the speed of the man and the time needed to make the crossing. Note: While in the water he must not direct himself toward point B to reach this point. Why?

The ship travels at a constant speed of \(v_s=20\mathrm{\ m}/\mathrm{s}\) and the wind is blowing at a speed of \(v_w=10\mathrm{\ m}/\mathrm{s}\), as shown. Determine the magnitude and direction of the horizontal component of velocity of the smoke coming from the smoke stack as it appears to a passenger on the ship.

The football player at A throws the ball in the y–z plane at a speed \(v_A=50\ \mathrm{ft}/\mathrm{s}\) and an angle \(\theta_{A}=60^{\circ}\) with the horizontal. At the instant the ball is thrown, the player is at B and is running with constant speed along the line BC in order to catch it. Determine this speed, \(v_B\), so that he makes the catch at the same elevation from which the ball was thrown.

The football player at A throws the ball in the y–z plane at a speed \(v_A=50\ \mathrm{ft}/\mathrm{s}\) and an angle \(\theta_{A}=60^{\circ}\) with the horizontal. At the instant the ball is thrown, the player is at B and is running with constant speed of \(v_B=23\ \mathrm{ft}/\mathrm{s}\) along the line BC. Determine if he can reach point C, which has the same elevation as A, before the ball gets there.

At a given instant the football player at A throws a football C with a velocity of 20 m/s in the direction shown. Determine the constant speed at which the player at B must run so that he can catch the football at the same elevation at which it was thrown. Also calculate the relative velocity and relative acceleration of the football with respect to B at the instant the catch is made. Player B is 15 m away from A when A starts to throw the football.

At the instant shown, car A travels along the straight portion of the road with a speed of 25 m/s. At this same instant car B travels along the circular portion of the road with a speed of 15 m/s. Determine the velocity of car B relative to car A.

If you measured the time it takes for the construction elevator to go from A to B, then B to C, and then C to D, and you also know the distance between each of the points, how could you determine the average velocity and average acceleration of the elevator as it ascends from A to D? Use numerical values to explain how this can be done.

If the sprinkler at A is 1 m from the ground, then scale the necessary measurements from the photo to determine the approximate velocity of the water jet as it flows from the nozzle of the sprinkler.

The basketball was thrown at an angle measured from the horizontal to the man’s outstretched arm. If the basket is 3 m from the ground, make appropriate measurements in the photo and determine if the ball located as shown will pass through the basket.

The pilot tells you the wingspan of her plane and her constant airspeed. How would you determine the acceleration of the plane at the moment shown? Use numerical values and take any necessary measurements from the photo.

The position of a particle along a straight line is given by \(s=\left(t^3-9t^2+15t\right)\ \mathrm{ft}\), where t is in seconds. Determine its maximum acceleration and maximum velocity during the time interval \(0\le t\le10\mathrm{\ s}\).

If a particle has an initial velocity \(v_0=12\ \mathrm{ft}/\mathrm{s}\) to the right, and a constant acceleration of \(2\mathrm{\ ft}/\mathrm{s}^2\) to the left, determine the particle’s displacement in 10 s. Originally \(s_0=0\).

A projectile, initially at the origin, moves along a straight-line path through a fluid medium such that its velocity is \(v=1800\left(1-e^{-0.3t}\right)\mathrm{\ mm}/\mathrm{s}\) where t is in seconds. Determine the displacement of the projectile during the first 3 s.

The v–t graph of a car while traveling along a road is shown. Determine the acceleration when t = 2.5 s, 10 s, and 25 s. Also if s = 0 when t = 0, find the position when t = 5 s, 20 s, and 30 s.

A car traveling along the straight portions of the road has the velocities indicated in the figure when it arrives at points A, B, and C. If it takes 3 s to go from A to B, and then 5 s to go from B to C, determine the average acceleration between points A and B and between points A and C.

From a videotape, it was observed that a player kicked a football 126 ft during a measured time of 3.6 seconds. Determine the initial speed of the ball and the angle \(\theta\) at which it was kicked.

The truck travels in a circular path having a radius of 50 m at a speed of v = 4 m/s. For a short distance from s = 0, its speed is increased by \(\dot{v}=(0.05s)\mathrm{\ m}/\mathrm{s}^2\), where s is in meters. Determine its speed and the magnitude of its acceleration when it has moved s = 10 m.

Car B turns such that its speed is increased by \(\left(a_t\right)_B=\left(0.5e^t\right)\ \mathrm{m}/\mathrm{s}^2\), where t is in seconds. If the car starts from rest when \(\theta=0^{\circ}\), determine the magnitudes of its velocity and acceleration when t = 2 s. Neglect the size of the car.

A particle is moving along a circular path of 2-m radius such that its position as a function of time is given by \(\theta=\left(5t^2\right)\ \mathrm{rad}\), where t is in seconds. Determine the magnitude of the particle’s acceleration when \(\theta=30^{\circ}\). The particle starts from rest when \(\theta=0^{\circ}\).

Determine the time needed for the load at B to attain a speed of 8 m/s, starting from rest, if the cable is drawn into the motor with an acceleration of \(0.2\mathrm{\ m}/\mathrm{s}^2\).

Two planes, A and B, are flying at the same altitude. If their velocities are \(v_A=600\mathrm{\ km}/\mathrm{h}\) and \(v_B=500\mathrm{\ km}/\mathrm{h}\) such that the angle between their straight-line courses is \(\theta=75^{\circ}\), determine the velocity of plane B with respect to plane A.