Solved: A train started from rest and moved with constant acceleration. At one time it

Figure 2-16 gives the velocity of a particle moving on an x axis. What are (a) the initial and (b) the final directions of travel? (c) Does the particle stop momentarily? (d) Is the acceleration positive or negative? (e) Is it constant or varying?

Figure 2-17 gives the acceleration a(t) of a Chihuahua as it chases a German shepherd along an axis. In which of the time periods indicated does the Chihuahua move at constant speed?

Figure 2-18 shows four paths along which objects move from a starting point to a final point, all in the same time interval. The paths pass over a grid of equally spaced straight lines. Rank the paths according to (a) the average velocity of the objects and (b) the average speed of the objects, greatest first.

Figure 2-19 is a graph of a particle’s position along an x axis versus time. (a) At time t = 0, what is the sign of the particle’s position? Is the particle’s velocity positive, negative, or 0 at (b) t = 1 s, (c) t = 2 s, and (d) t = 3 s? (e) How many times does the particle go through the point x = 0?

Figure 2-20 gives the velocity of a particle moving along an axis. Point 1 is at the highest point on the curve; point 4 is at the lowest point; and points 2 and 6 are at the same height. What is the direction of travel at (a) time t = 0 and (b) point 4? (c) At which of the six numbered points does the particle reverse its direction of travel? (d) Rank the six points according to the magnitude of the acceleration, greatest first.

At t = 0, a particle moving along an x axis is at position \(x_{0}=-20 \mathrm{~m}\). The signs of the particle’s initial velocity \(v_{0}\) (at time \(t_{0}\)) and constant acceleration a are, respectively, for four situations: (1) +, +; (2) +, ?; (3) ?, +; (4) ?, ?. In which situations will the particle (a) stop momentarily, (b) pass through the origin, and (c) never pass through the origin? Text Transcription: x_0 =-20~m v_0 t_0

Hanging over the railing of a bridge, you drop an egg (no initial velocity) as you throw a second egg downward. Which curves in Fig. 2-21 give the velocity v(t) for (a) the dropped egg and (b) the thrown egg? (Curves A and B are parallel; so are C, D, and E; so are F and G.)

The following equations give the velocity v(t) of a particle in four situations: (a) v = 3; (b) \(v = 4t^{2}\) + 2t ? 6; (c) v = 3t ? 4; (d) \(v=5 t^{2}-3\). To which of these situations do the equations of Table 2-1 apply? Text Transcription: v=4t^2 v=5t^2 -3

In Fig. 2-22, a cream tangerine is thrown directly upward past three evenly spaced windows of equal heights. Rank the windows according to (a) the average speed of the cream tangerine while passing them, (b) the time the cream tangerine takes to pass them, (c) the magnitude of the acceleration of the cream tangerine while passing them, and (d) the change \(\Delta v\) in the speed of the cream tangerine during the passage, greatest first. Text Transcription: Delta v

Suppose that a passenger intent on lunch during his first ride in a hot-air balloon accidently drops an apple over the side during the balloon’s liftoff. At the moment of the apple’s release, the balloon is accelerating upward with a magnitude of \(4.0 \mathrm{~m} / \mathrm{s}^{2}\) and has an upward velocity of magnitude 2 m/s. What are the (a) magnitude and (b) direction of the acceleration of the apple just after it is released? (c) Just then, is the apple moving upward or downward, or is it stationary? (d) What is the magnitude of its velocity just then? (e) In the next few moments, does the speed of the apple increase, decrease, or remain constant? Text Transcription: 4.0 ~m/s^2

Figure 2-23 shows that a particle moving along an x axis undergoes three periods of acceleration. Without written computation, rank the acceleration periods according to the increases they produce in the particle’s velocity, greatest first.

In an arcade video game, a spot is programmed to move across the screen according to \(x=9.00 t-0.750 t^{3}\), where x is distance in centimeters measured from the left edge of the screen and t is time in seconds. When the spot reaches a screen edge, at either x = 0 or x = 15.0 cm, t is reset to 0 and the spot starts moving again according to x(t). (a) At what time after starting is the spot instantaneously at rest? (b) At what value of x does this occur? (c) What is the spot’s acceleration (including sign) when this occurs? (d) Is it moving right or left just prior to coming to rest? (e) Just after? (f) At what time \(t>0\) does it first reach an edge of the screen? Text Transcription: x=9.00t-0.750t^3 t>0

At the instant the traffic light turns green, an automobile starts with a constant acceleration a of \(2.2 \mathrm{m} / \mathrm{s}^{2}\). At the same instant a truck, traveling with a constant speed of 9.5 m/s, overtakes and passes the automobile. (a) How far beyond the traffic signal will the automobile overtake the truck? (b) How fast will the automobile be traveling at that instant? Text Transcription: 2.2m/s^2

A pilot flies horizontally at 1300 km/h, at height h = 35 m above initially level ground. However, at time t = 0, the pilot begins to fly over ground sloping upward at angle \(\theta=4.3^{\circ}\) (Fig. 2-41). If the pilot does not change the airplane’s heading, at what time t does the plane strike the ground? Text Transcription: theta=4.3^circ

Figure 2-42 shows part of a street where traffic flow is to be controlled to allow a platoon of cars to move smoothly along the street. Suppose that the platoon leaders have just reached intersection 2, where the green appeared when they were distance d from the intersection. They continue to travel at a certain speed \(v_{p}\) (the speed limit) to reach intersection 3, where the green appears when they are distance d from it. The intersections are separated by distances \(D_{23}\) and \(D_{12}\). (a) What should be the time delay of the onset of green at intersection 3 relative to that at intersection 2 to keep the platoon moving smoothly? Suppose, instead, that the platoon had been stopped by a red light at intersection 1. When the green comes on there, the leaders require a certain time \(t_{r}\) to respond to the change and an additional time to accelerate at some rate a to the cruising speed \(v_{p}\). (b) If the green at intersection 2 is to appear when the leaders are distance d from that intersection, how long after the light at intersection 1 turns green should the light at intersection 2 turn green? Text Transcription: v_p D_23 D_12 t_r v_p

A hot rod can accelerate from 0 to 60 km/h in 5.4 s. (a) What is its average acceleration, in \(\mathrm{m} / \mathrm{s}^{2}\), during this time? (b) How far will it travel during the 5.4 s, assuming its acceleration is constant? (c) From rest, how much time would it require to go a distance of 0.25 km if its acceleration could be maintained at the value in (a)? Text Transcription: m/s^2

A red train traveling at 72 km/h and a green train traveling at 144 km/h are headed toward each other along a straight, level track. When they are 950 m apart, each engineer sees the other’s train and applies the brakes. The brakes slow each train at the rate of \(1.0 \mathrm{m} / \mathrm{s}^{2}\). Is there a collision? If so, answer yes and give the speed of the red train and the speed of the green train at impact, respectively. If not, answer no and give the separation between the trains when they stop. Text Transcription: 1.0m/s^2

At time t = 0, a rock climber accidentally allows a piton to fall freely from a high point on the rock wall to the valley below him. Then, after a short delay, his climbing partner, who is 10 m higher on the wall, throws a piton downward. The positions y of the pitons versus t during the falling are given in Fig. 2-43. With what speed is the second piton thrown?

Figure 2-44 gives the acceleration a versus time t for a particle moving along an x axis. The a-axis scale is set by \(a_{s}=12.0 \mathrm{m} / \mathrm{s}^{2}\). At t = ?2.0 s, the particle’s velocity is 7.0 m/s. What is its velocity at t = 6.0 s? Text Transcription: a_s =12.0m/s^2

Figure 2-45 shows a simple device for measuring your reaction time. It consists of a cardboard strip marked with a scale and two large dots. A friend holds the strip vertically, with thumb and forefinger at the dot on the right in Fig. 2-45. You then position your thumb and forefinger at the other dot (on the left in Fig. 2-45), being careful not to touch the strip. Your friend releases the strip, and you try to pinch it as soon as possible after you see it begin to fall. The mark at the place where you pinch the strip gives your reaction time. (a) How far from the lower dot should you place the 50.0 ms mark? How much higher should you place the marks for (b) 100, (c) 150, (d) 200, and (e) 250 ms? (For example, should the 100 ms marker be 2 times as far from the dot as the 50 ms marker? If so, give an answer of 2 times. Can you find any pattern in the answers?)

A motorcyclist who is moving along an x axis directed toward the east has an acceleration given by \(a=(6.1-1.2 t) \mathrm{m} / \mathrm{s}^{2}\) for \(0 \leq t \leq 6.0 \mathrm{s}\). At t = 0, the velocity and position of the cyclist are 2.7 m/s and 7.3 m. (a) What is the maximum speed achieved by the cyclist? (b) What total distance does the cyclist travel between t = 0 and 6.0 s? Text Transcription: a=(6.1-1.2t)m/s^2 0 leq t leq 6.0s

A particle starts from the origin at t = 0 and moves along the positive x axis. A graph of the velocity of the particle as a function of the time is shown in Fig. 2-46; the v-axis scale is set by \(v_{s}=4.0 \mathrm{m} / \mathrm{s}\). (a) What is the coordinate of the particle at t = 5.0 s? (b) What is the velocity of the particle at t = 5.0 s? (c) What is the acceleration of the particle at t = 5.0 s? (d) What is the average velocity of the particle between t = 1.0 s and t = 5.0 s? (e) What is the average acceleration of the particle between t = 1.0 s and t = 5.0 s? Text Transcription: v_s =4.0m/s

Two subway stops are separated by 1100 m. If a subway train accelerates at \(+1.2 \mathrm{m} / \mathrm{s}^{2}\) from rest through the first half of the distance and decelerates at \(-1.2 \mathrm{m} / \mathrm{s}^{2}\) through the second half, what are (a) its travel time and (b) its maximum speed? (c) Graph x, v, and a versus t for the trip. Text Transcription: +1.2m/s^2 -1.2m/s^2

A stone is thrown vertically upward. On its way up it passes point A with speed v, and point B, 3.00 m higher than A, with speed \(\frac{1}{2} v\). Calculate (a) the speed v and (b) the maximum height reached by the stone above point B. Text Transcription: 1/2 v

An iceboat has a constant velocity toward the east when a sudden gust of wind causes the iceboat to have a constant acceleration toward the east for a period of 3.0 s. A plot of x versus t is shown in Fig. 2-47, where t = 0 is taken to be the instant the wind starts to blow and the positive x axis is toward the east. (a) What is the acceleration of the iceboat during the 3.0 s interval? (b) What is the velocity of the iceboat at the end of the 3.0 s interval? (c) If the acceleration remains constant for an additional 3.0 s, how far does the iceboat travel during this second 3.0 s interval?

A parachutist bails out and freely falls 50 m. Then the parachute opens, and thereafter she decelerates at \(2.0 \mathrm{m} / \mathrm{s}^{2}\). She reaches the ground with a speed of 3.0 m/s. (a) How long is the parachutist in the air? (b) At what height does the fall begin? Text Transcription: 2.0m/s^2

A ball is thrown down vertically with an initial speed of \(v_{0}\) from a height of h. (a) What is its speed just before it strikes the ground? (b) How long does the ball take to reach the ground? What would be the answers to (c) part a and (d) part b if the ball were thrown upward from the same height and with the same initial speed? Before solving any equations, decide whether the answers to (c) and (d) should be greater than, less than, or the same as in (a) and (b). Text Transcription: v_0

A proton moves along the x axis according to the equation \(x=50 t+10 t^{2}\), where x is in meters and t is in seconds. Calculate (a) the average velocity of the proton during the first 3.0 s of its motion, (b) the instantaneous velocity of the proton at t = 3.0 s, and (c) the instantaneous acceleration of the proton at t = 3.0 s. (d) Graph x versus t and indicate how the answer to (a) can be obtained from the plot. (e) Indicate the answer to (b) on the graph. (f) Plot v versus t and indicate on it the answer to (c). Text Transcription: x=50t+10t^2

A shuffleboard disk is accelerated at a constant rate from rest to a speed of 6.0 m/s over a 1.8 m distance by a player using a cue. At this point the disk loses contact with the cue and slows at a constant rate of \(2.5 \mathrm{m} / \mathrm{s}^{2}\) until it stops. (a) How much time elapses from when the disk begins to accelerate until it stops? (b) What total distance does the disk travel? Text Transcription: 2.5m/s^2

The head of a rattlesnake can accelerate at \(50 \mathrm{~m} / \mathrm{s}^{2}\) in striking a victim. If a car could do as well, how long would it take to reach a speed of 100 km/h from rest? Text Transcription: 50ms^2

A car can be braked to a stop from the autobahn-like speed of 200 km/h in 170 m. Assuming the acceleration is constant, find its magnitude in (a) SI units and (b) in terms of g. (c) How much time \(T_{b}\) is required for the braking? Your reaction time \(T_{r}\) is the time you require to perceive an emergency, move your foot to the brake, and begin the braking. If \(T_{r}=400 \mathrm{ms}\), then (d) what is \(T_{b}\) in terms of \(T_{r}\), and (e) is most of the full time required to stop spent in reacting or braking? Dark sunglasses delay the visual signals sent from the eyes to the visual cortex in the brain, increasing \(T_{r}\). (f) In the extreme case in which \(T_{r}\) is increased by 100 ms, how much farther does the car travel during your reaction time? Text Transcription: T_b T_r T_r=400ms T_b T_r T_r T_r

The wings on a stonefly do not flap, and thus the insect cannot fly. However, when the insect is on a water surface, it can sail across the surface by lifting its wings into a breeze. Suppose that you time stoneflies as they move at constant speed along a straight path of a certain length. On average, the trips each take 7.1 s with the wings set as sails and 25.0 s with the wings tucked in. (a) What is the ratio of the sailing speed \(v_{s}\) to the nonsailing speed \(v_{n s}\)? (b) In terms of \(v_{s}\), what is the difference in the times the insects take to travel the first 2.0 m along the path with and without sailing? Text Transcription: v_s v_ns v_s

The position of a particle as it moves along a y axis is given by \(y=(2.0 \mathrm{cm}) \sin (\pi t / 4)\), with t in seconds and y in centimeters. (a) What is the average velocity of the particle between t = 0 and t = 2.0 s? (b) What is the instantaneous velocity of the particle at t = 0, 1.0, and 2.0 s? (c) What is the average acceleration of the particle between t = 0 and t = 2.0 s? (d) What is the instantaneous acceleration of the particle at t = 0, 1.0, and 2.0 s? Text Transcription: y=(2.0cm) sin (pi t/4)

A rock is shot vertically upward from the edge of the top of a tall building. The rock reaches its maximum height above the top of the building 1.60 s after being shot. Then, after barely missing the edge of the building as it falls downward, the rock strikes the ground 6.00 s after it is launched. In SI units: (a) with what upward velocity is the rock shot, (b) what maximum height above the top of the building is reached by the rock, and (c) how tall is the building?

To stop a car, first you require a certain reaction time to begin braking; then the car slows at a constant rate. Suppose that the total distance moved by your car during these two phases is 56.7 m when its initial speed is 80.5 km/h, and 24.4 m when its initial speed is 48.3 km/h. What are (a) your reaction time and (b) the magnitude of the acceleration?

A train started from rest and moved with constant acceleration. At one time it was traveling 30 m/s, and 160 m farther on it was traveling 50 m/s. Calculate (a) the acceleration, (b) the time required to travel the 160 m mentioned, (c) the time required to attain the speed of 30 m/s, and (d) the distance moved from rest to the time the train had a speed of 30 m/s. (e) Graph x versus t and v versus t for the train, from rest.

A particle’s acceleration along an x axis is a = 5.0t, with t in seconds and a in meters per second squared. At t = 2.0 s, its velocity is +17 m/s. What is its velocity at t = 4.0 s?

A rocket-driven sled running on a straight, level track is used to investigate the effects of large accelerations on humans. One such sled can attain a speed of 1600 km/h in 1.8 s, starting from rest. Find (a) the acceleration ( assumed constant) in terms of g and (b) the distance traveled.

A mining cart is pulled up a hill at 20 km/h and then pulled back down the hill at 35 km/h through its original level. (The time required for the cart’s reversal at the top of its climb is negligible.) What is the average speed of the cart for its round trip, from its original level back to its original level?

When the legal speed limit for the New York Thruway was increased from 55 mi/h to 65 mi/h, how much time was saved by a motorist who drove the 700 km between the Buffalo entrance and the New York City exit at the legal speed limit?

A car moving with constant acceleration covered the distance between two points 60.0 m apart in 6.00 s. Its speed as it passed the second point was 15.0 m/s. (a) What was the speed at the first point? (b) What was the magnitude of the acceleration? (c) At what prior distance from the first point was the car at rest? (d) Graph x versus t and v versus t for the car, from rest (t = 0).

A certain juggler usually tosses balls vertically to a height H. To what height must they be tossed if they are to spend twice as much time in the air?

A rock is dropped from a 100-m-high cliff. How long does it take to fall (a) the first 50 m and (b) the second 50 m?

A rock is dropped (from rest) from the top of a 60-m-tall building. How far above the ground is the rock 1.2 s before it reaches the ground?

A lead ball is dropped in a lake from a diving board 5.20 m above the water. It hits the water with a certain velocity and then sinks to the bottom with this same constant velocity. It reaches the bottom 4.80 s after it is dropped. (a) How deep is the lake? What are the (b) magnitude and (c) direction (up or down) of the average velocity of the ball for the entire fall? Suppose that all the water is drained from the lake. The ball is now thrown from the diving board so that it again reaches the bottom in 4.80 s. What are the (d) magnitude and (e) direction of the initial velocity of the ball?

The single cable supporting an unoccupied construction elevator breaks when the elevator is at rest at the top of a 120-m-high building. (a) With what speed does the elevator strike the ground? (b) How long is it falling? (c) What is its speed when it passes the halfway point on the way down? (d) How long has it been falling when it passes the halfway point?

Two diamonds begin a free fall from rest from the same height, 1.0 s apart. How long after the first diamond begins to fall will the two diamonds be 10 m apart?

A ball is thrown vertically downward from the top of a 36.6-m-tall building. The ball passes the top of a window that is 12.2 m above the ground 2.00 s after being thrown. What is the speed of the ball as it passes the top of the window?

The sport with the fastest moving ball is jai alai, where measured speeds have reached 303 km/h. If a professional jai alai player faces a ball at that speed and involuntarily blinks, he blacks out the scene for 100 ms. How far does the ball move during the blackout?

If a baseball pitcher throws a fastball at a horizontal speed of 160 km/h, how long does the ball take to reach home plate 18.4 m away?

A motorcycle is moving at 30 m/s when the rider applies the brakes, giving the motorcycle a constant deceleration. During the 3.0 s interval immediately after braking begins, the speed decreases to 15 m/s. What distance does the motorcycle travel from the instant braking begins until the motorcycle stops?

A jumbo jet must reach a speed of 360 km/h on the runway for takeoff. What is the lowest constant acceleration needed for takeoff from a 1.80 km runway?

An automobile driver increases the speed at a constant rate from 25 km/h to 55 km/h in 0.50 min. A bicycle rider speeds up at a constant rate from rest to 30 km/h in 0.50 min. What are the magnitudes of (a) the driver’s acceleration and (b) the rider’s acceleration?

On average, an eye blink lasts about 100 ms. How far does a MiG-25 “Foxbat” fighter travel during a pilot’s blink if the plane’s average velocity is 3400 km/h?

A certain sprinter has a top speed of 11.0 m/s. If the sprinter starts from rest and accelerates at a constant rate, he is able to reach his top speed in a distance of 12.0 m. He is then able to maintain this top speed for the remainder of a 100 m race. (a) What is his time for the 100 m race? (b) In order to improve his time, the sprinter tries to decrease the distance required for him to reach his top speed. What must this distance be if he is to achieve a time of 10.0 s for the race?

The speed of a bullet is measured to be 640 m/s as the bullet emerges from a barrel of length 1.20 m. Assuming constant acceleration, find the time that the bullet spends in the barrel after it is fired.

The Zero Gravity Research Facility at the NASA Glenn Research Center includes a 145 m drop tower. This is an evacuated vertical tower through which, among other possibilities, a 1-m diameter sphere containing an experimental package can be dropped. (a) How long is the sphere in free fall? (b) What is its speed just as it reaches a catching device at the bottom of the tower? (c) When caught, the sphere experiences an average deceleration of 25g as its speed is reduced to zero. Through what distance does it travel during the deceleration?

In 1889, at Jubbulpore, India, a tug-of-war was finally won after 2 h 41 min, with the winning team displacing the center of the rope 3.7 m. In centimeters per minute, what was the magnitude of the average velocity of that center point during the contest?

Most important in an investigation of an airplane crash by the U.S. National Transportation Safety Board is the data stored on the airplane’s flight-data recorder, commonly called the “black box” in spite of its orange coloring and reflective tape. The recorder is engineered to withstand a crash with an average deceleration of magnitude 3400g during a time interval of 6.50 ms. In such a crash, if the recorder and airplane have zero speed at the end of that time interval, what is their speed at the beginning of the interval?

From January 26, 1977, to September 18, 1983, George Meegan of Great Britain walked from Ushuaia, at the southern tip of South America, to Prudhoe Bay in Alaska, covering 30 600 km. In meters per second, what was the magnitude of his average velocity during that time period?