Chapter 2

While driving a car at \(90 \mathrm{~km} / \mathrm{h}\), how far do you move while your cyes shut for \(0.50 \mathrm{~s}\) during a hard sneeze?

Compute your averagc velocity in the following two cascs: (a) You walk \(73.2 \mathrm{~m}\) at a speed of \(1.22 \mathrm{~m} / \mathrm{s}\) and then run \(73.2 \mathrm{~m}\) at a speed of \(3.05 \mathrm{~m} / \mathrm{s}\) along a straight track. (b) You walk for \(1.00 \mathrm{~min}\) at a speed of \(1.22 \mathrm{~m} / \mathrm{s}\) and then run for \(1.00 \mathrm{~min}\) at \(3.05 \mathrm{~m} / \mathrm{s}\) along a straight track. (c) Graph \(x\) versus \(t\) for both cases and indicate how the average velocity is found on the graph.

A car moves uphill at \(40 \mathrm{~km} / \mathrm{h}\) and then back downhill at \(60 \mathrm{~km} / \mathrm{h}\). What is the average speed for the round trip?

The position of an object moving along an \(x\) axis is given by \(x=3 t-4 t^{2}+t^{3},\) where \(x\) is in meters and \(t\) in seconds. Find the position of the object at the following values of \(t:\) (a) \(1 \mathrm{~s}\). (b) \(2 \mathrm{~s}\). (c) \(3 \mathrm{~s}\), and (d) \(4 \mathrm{~s}\). (c) What is the object's dicplacement hetween \(t=0\) and \(t=4 \mathrm{~s} ?\) (f) What is its average velocity for the time interval from \(t=2 \mathrm{~s}\) to \(t=4 \mathrm{~s} ?\) (g) Graph \(x\) versus \(t\) for \(0 \leq t \leq 4 \mathrm{~s}\) and indicate how the answer for (f) can be found on the graph.

Two trains, each having a speed of \(30 \mathrm{~km} / \mathrm{h},\) are headed at each other on the same straight track. A bird that can fly \(60 \mathrm{~km} / \mathrm{h}\) flies off the front of one train when they are \(60 \mathrm{~km}\) apart and heads directly for the other train. On reaching the other train, the (crazy) bird flics directly hack to the first train, and so forth. What is the total distance the bird travcls before the trains collide?

In 1 km races, runner 1 on track 1 (with time \(2 \min .27 .95 \mathrm{~s}\) ) appears to be faster than runner 2 on track \(2(2 \min , 28.15 \mathrm{~s})\). Howcver, length \(L_{2}\) of track 2 might be slightly greater than length \(L_{1}\) of track 1. How large \(\operatorname{can} L_{2}-L_{1}\) be for us still to conclude that runner 1 is faster?

You drive on Interstate 10 from San Antonio to Houston, half the time at \(55 \mathrm{~km} / \mathrm{h}\) and the other half at \(90 \mathrm{~km} / \mathrm{h}\). On the way back you travel half the distance at \(55 \mathrm{~km} / \mathrm{h}\) and the other half at \(90 \mathrm{~km} / \mathrm{h}\). What is your average speed (a) from San Antonio to Houston, (b) from Houston back to San Antonio, and (c) for the entire trip? (d) What is your average velocity for the entire trip? (e) Sketch \(x\) versus \(t\) for (a), assuming the motion is all in the positive \(x\) direction. Indicate how the average velocity can be found on the sketch.

(a) If a particlc's position is given by \(x=4-12 t+3 t^{2}\) (where \(t\) is in seconds and \(x\) is in meters), what is its velocity at \(t=1 \mathrm{~s} ?(\mathrm{~b})\) Is it moving in the positive or negative direction of \(x\) just then? (c) What is its speed just then? (d) Is the speed increasing or decreasing just then? (Try answering the next two questions without further calculation.) (c) Is there cver an instant when the velocity is zero? If so. give the time \(r\), if not, answer no. (f) Is there a time after \(t=3 \mathrm{~s}\) when the particle is moving in the ncgative dircction of \(x ?\) If so, give the time \(t\), if not, answer no.

The position of a particle moving along an \(x\) axis is given by \(x=12 t^{2}-2 t^{3}\), where \(x\) is in meters and \(t\) is in seconds. Determine (a) the position, (b) the velocity, and (c) the acceleration of the particle at \(t=3.0 \mathrm{~s}\). (d) What is the maximum positive coordinate reached by the particle and (e) at what time is it reached? (f) What is the maximum positive velocity reached by the particle and \(\underline{(g)}\) at what time is it reached? (h) What is the acceleration of the particle at the instant the particle is not moving (other than at \(t=0\) )? (i) Determine the average velocity of the particle between \(t=0\) and \(t=\overline{3} \mathrm{~s}\)

At a certain time a particle had a speed of \(18 \mathrm{~m} / \mathrm{s}\) in the positive \(x\) direction, and \(2.4 \mathrm{~s}\) later its speed was \(30 \mathrm{~m} / \mathrm{s}\) in the opposite direction. What is the average acceleration of the particle during this \(2.4 \mathrm{~s}\) interval?

(a) If the position of a particle is given by \(x=20 t-5 t^{3}\), where \(x\) is in meters and \(t\) is in seconds, when, if ever, is the particle's velocity zero? (b) When is its acceleration \(a\) zero? (c) For what time range (positive or negative) is a negative? (d) Positive? (e) Graph \(x(t), v(t),\) and \(a(t)\)

The position of a particle moving along the \(x\) axis depends on the time according to the equation \(x=c t^{2}-b t^{3},\) where \(x\) is in meters and \(t\) in seconds. What are the units of (a) constant \(c\) and (b) constant \(b\) ? Let their numerical valucs be 3.0 and \(2.0,\) respectivcly. (c) At what time does the particle reach its maximum positive \(x\) position? From \(t=0.0 \mathrm{~s}\) to \(t=4.0 \mathrm{~s},\) (d) what distance does the particle move and (e) what is its displacement? Find its velocity at times (f) \(1.0 \mathrm{~s},(\mathrm{~g}) 2.0 \mathrm{~s},(\mathrm{~h}) \overline{3.0 \mathrm{~s}, \text { and }}\) (i) \(4.0 \mathrm{~s}\). Find its acceleration at times (j) \(1.0 \mathrm{~s},(\mathrm{k}) 2.0 \mathrm{~s},(\mathrm{l}) 3.0 \mathrm{~s},\) and \((\mathrm{m}) 4.0 \mathrm{~s}\)

An electron with an initial velocity \(v_{0}=1.50 \times 10^{5} \mathrm{~m} / \mathrm{s}\) cnters a region of length \(L=1.00 \mathrm{~cm}\) where it is electrically accelerated (Fig. \(2-26\) ). It cmerges with \(v=5.70 \times 10^{6} \mathrm{~m} / \mathrm{s}\). What is its acceleration, assumed constant?

Catapulting mush- rooms. Certain mushrooms launch their spores by a catapult mechanism. As water condenses from the air onto a spore that is attached to the mushroom, a drop grows on onc side of the spore and a film grows on the other side. The spore is bent over by the drop's weight, but when the film reaches the drop, the drop's water suddenly spreads into the film and the spore springs upward so rapidly that it is slung off into the air. Typically, the spore reaches a speed of \(1.6 \mathrm{~m} / \mathrm{s}\) in a \(5.0 \mu \mathrm{m}\) launch; its speed is then reduced to zero in \(1.0 \mathrm{~mm}\) by the air. Using those data and assuming constant accelerations, find the acceleration in terms of \(g\) during (a) the launch and (b) the speed reduction.

an clectric vehicle starts from rest and accelerates at a rate of \(2.0 \mathrm{~m} / \mathrm{s}^{2}\) in a straight line until it reaches a speed of \(20 \mathrm{~m} / \mathrm{s} .\) The vchicle then slows at a constant rate of \(1.0 \mathrm{~m} / \mathrm{s}^{2}\) until it stops. (a) How much time elapses from start to stop? (b) How far does the vehicle travel from start to stop?

An electron has a constant acceleration of \(+3.2 \mathrm{~m} / \mathrm{s}^{2}\). At a certain instant its velocity is \(+9.6 \mathrm{~m} / \mathrm{s}\). What is its velocity (a) \(2.5 \mathrm{~s}\) earlier and (b) \(2.5 \mathrm{~s}\) later?

On a dry road, a car with good tires may be able to brake with a constant decclcration of \(4.92 \mathrm{~m} / \mathrm{s}^{2}\). (a) How long docs such a car, initially traveling at \(24.6 \mathrm{~m} / \mathrm{s}\), take to stop? (b) How far does it travel in this time? (c) Graph \(x\) versus \(t\) and \(v\) versus \(t\) for the deceleration.

A certain elevator cab has a total run of \(190 \mathrm{~m}\) and a maximum speed of \(305 \mathrm{~m} / \mathrm{min},\) and it accelerates from rest and then back to rest at \(1.22 \mathrm{~m} / \mathrm{s}^{2} .\) (a) How far docs the cab move whilc accelerating to full speed from rest? (b) How long does it take to make the nonstop \(190 \mathrm{~m}\) run, starting and ending at rest?

The brakes on your car can slow you at a rate of \(5.2 \mathrm{~m} / \mathrm{s}^{2}\). (a) If you are going \(137 \mathrm{~km} / \mathrm{h}\) and suddenly see a state trooper, what is the minimum time in which you can get your car under the \(90 \mathrm{~km} / \mathrm{h}\) speed limit? (The answer reveals the futility of braking to keep your high speed from being detected with a radar or laser gun.) (b) Graph \(x\) versus \(t\) and \(v\) versus \(t\) for such a slowing.

Suppose a rocket ship in deep space moves with con- stant acceleration equal to \(9.8 \mathrm{~m} / \mathrm{s}^{2}\), which gives the illusion of normal gravity during the flight. (a) If it starts from rest, how long will it take to acquire a speed onc-tenth that of light, which travels at \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s} ?\) (b) How far will it travel in so doing?

A world's land speed record was set by Colonel John P. Stapp when in March 1954 he rode a rocket-propelled sled that moved along a track at \(1020 \mathrm{~km} / \mathrm{h}\). He and the sled were brought to a stop in \(1.4 \mathrm{~s}\). (See Fig. \(2-7 .\) ) In terms of \(g\), what acceleration did he experience while stopping?

A car traveling \(56.0 \mathrm{~km} / \mathrm{h}\) is \(24.0 \mathrm{~m}\) from a barrier when the driver slams on the brakes. The car hits the barrier \(2.00 \mathrm{~s}\) later. (a) What is the magnitude of the car's constant acceleration before impact? (b) How fast is the car traveling at impact?

A car moves along an \(x\) axis through a distance of \(900 \mathrm{~m}\), starting at rest (at \(x=0\) ) and cnding at rest (at \(x=900 \mathrm{~m}\) ). Through the first \(\frac{1}{4}\) of that distance, its acceleration is \(+2.25 \mathrm{~m} / \mathrm{s}^{2}\). Through the rest of that distance, its acceleration is \(-0.750 \mathrm{~m} / \mathrm{s}^{2}\). What are (a) its travel time through the \(900 \mathrm{~m}\) and \((\mathrm{b})\) its maximum speed? (c) Graph position \(x,\) velocity \(v,\) and acceleration \(a\) versus time \(t\) for the trip.

(a) If the maximum acceleration that is tolerable for passengers in a subway train is \(1.34 \mathrm{~m} / \mathrm{s}^{2}\) and subway stations are located \(806 \mathrm{~m}\) apart, what is the maximum spced a subway train can attain between stations? (b) What is the travel time between stations? (c) If a subway train stops for \(20 \mathrm{~s}\) at each station, what is the maximum average speed of the train, from one start-up to the next? (d) Graph \(x\), \(v\), and \(a\) versus \(t\) for the interval from one start-up to the next.

You are driving toward a traffic signal when it turns yel- low. Your speed is the legal speed limit of \(v_{0}=55 \mathrm{~km} / \mathrm{h}\); your best deceleration rate has the magnitude \(a=5.18 \mathrm{~m} / \mathrm{s}^{2}\). Your best reaction time to begin braking is \(T=0.75 \mathrm{~s}\). To avoid having the front of your car enter the intersection after the light turns red, should you brake to a stop or continue to move at \(55 \mathrm{~km} / \mathrm{h}\) if the distance to the intersection and the duration of the yellow light are (a) \(40 \mathrm{~m}\) and \(2.8 \mathrm{~s}\), and (b) \(32 \mathrm{~m}\) and \(1.8 \mathrm{~s} ?\) Give an answer of brake, continuc, cither (if either strategy works), or neither (if neither strategy works and the yellow duration is inappropriate).

You are arguing over a cell phone while trailing an unmarked police car by \(25 \mathrm{~m} ;\) both your car and the police car are traveling at \(110 \mathrm{~km} / \mathrm{h}\). Your argument diverts your attention from the police car for \(2.0 \mathrm{~s}\) (long enough for you to look at the phone and yell, " 1 won't do that \(\left.\right|^{n}\) ). At the beginning of that \(2.0 \mathrm{~s}\). the police officer begins braking suddenly at \(5.0 \mathrm{~m} / \mathrm{s}^{2}\). (a) What is the separation between the two cars when your attention finally returns? Suppose that you take another \(0.40 \mathrm{~s}\) to realize your danger and begin braking. (b) If you too brake at \(5.0 \mathrm{~m} / \mathrm{s}^{2}\), what is your speed when you hit the police car?

When startled, an armadillo will leap upward. Suppose it riscs \(0.544 \mathrm{~m}\) in the first \(0.200 \mathrm{~s}\). (a) What is its initial speed as it leaves the ground? (b) What is its speed at the height of \(0.544 \mathrm{~m} ?\) (c) How much higher does it go?

(a) With what speed must a ball be thrown vertically from ground level to rise to a maximum height of \(50 \mathrm{~m} ?\) (b) How long will it be in the air? (c) Sketch graphs of \(y, v,\) and \(a\) versus \(t\) for the ball. On the first two graphs, indicate the time at which \(50 \mathrm{~m}\) is reached.

Raindrops fall \(1700 \mathrm{~m}\) from a cloud to the ground. (a) If they were not slowed by air resistance, how fast would the drops be moving when they struck the ground? (b) Would it be safe to walk outside during a rainstorm?

At a construction site a pipe wrench struck the ground with a speed of \(24 \mathrm{~m} / \mathrm{s}\). (a) From what height was it inadvertently dropped? (b) How long was it falling? (c) Sketch graphs of \(y, v\), and \(a\) versus \(t\) for the wrench.

A hoodlum throws a stone vertically downward with an initial spced of \(12.0 \mathrm{~m} / \mathrm{s}\) from the roof of a building, \(30.0 \mathrm{~m}\) above the ground. (a) How long does it take the stone to reach the ground? (b) What is the speed of the stone at impact?

A hot-air balloon is ascending at the rate of \(12 \mathrm{~m} / \mathrm{s}\) and is \(80 \mathrm{~m}\) above the ground when a package is dropped over the side. (a) How long does the package take to reach the ground? (b) With what spced docs it hit the ground?

A bolt is dropped from a bridec undering \(90 \mathrm{~m}\) construction, fall- to the valley below the bridge. (a) In how much time does it pass through the last \(20 \%\) of its fall? What is its speed (b) when it begins that last \(20 \%\) of its fall and (c) when it reaches the valley bencath the bridge?

A key falls from a bridge that is \(45 \mathrm{~m}\) above the watcr. It falls directly into a model boat, moving with constant velocity, that is \(12 \mathrm{~m}\) from the point of impact when the key is released. What is the speed of the boat?

A stone is dropped into a river from a bridge \(43.9 \mathrm{~m}\) above the water. Another stone is thrown vertically down \(1.00 \mathrm{~s}\) after the first is dropped. The stones strike the water at the same timc. (a) What is the initial speed of the scoond stonc? (b) Plot velocity versus time on a graph for each stone, taking zero time as the instant the first stone is released.

A hall of moist clay falls \(15.0 \mathrm{~m}\) to the ground. It is in contact with the ground for \(20.0 \mathrm{~ms}\) before stopping. (a) What is the magnitude of the average acceleration of the ball during the time it is in contact with the ground? (Ireat the ball as a particle.) (b) Is the average acceleration up or down?

To test the quality of a tennis ball, you drop it onto the floor from a height of \(4.00 \mathrm{~m}\). It rebounds to a height of \(2.00 \mathrm{~m}\). If the ball is in contact with the floor for \(12.0 \mathrm{~ms}\), (a) what is the magnitude of its average acccleration during that contact and (b) is the average acceleration up or down?

An object falls a distance \(h\) from rest. If it travels 0.50 h in the last \(1.00 \mathrm{~s},\) find (a) the time and (b) the height of its fall. (c) Explain the physically unacceptable solution of the quadratic equation in \(t\) that you obtain.

Water drips from the nozzle of a shower onto the floor \(200 \mathrm{~cm}\) below. The drops fall at regular (equal) intervals of time. the first drop striking the floor at the instant the fourth drop begins to fall. When the first drop strikes the floor, how far below the nocrle are the (a) second and (b) third drops?

A steel ball is dropped from a building's roof and passes a window, taking \(0.125 \mathrm{~s}\) to fall from the top to the bottom of the window, a distance of \(1.20 \mathrm{~m}\). It then falls to a sidewalk and bounces back past the window, moving from bottom to top in \(0.125 \mathrm{~s}\). Assume that the upward flight is an exact reverse of the fall. The time the ball spends below the bottom of the window is \(2.00 \mathrm{~s}\). How tall is the building?

A basketball player grabbing a rebound jumps \(76.0 \mathrm{~cm}\) vertically. How much total time (ascent and descent) does the player spend (a) in the top \(15.0 \mathrm{~cm}\) of this jump and (b) in the bottom \(15.0 \mathrm{~cm} ?\) (The playcr secms to hang in the air at the top.)

A drowsy cat spots a flowerpot that sails first up and then down past an open window. The pot is in view for a total of \(0.50 \mathrm{~s}\), and the top-to- bottom height of the window is \(2.00 \mathrm{~m}\). How high above the window top docs the flowerpot go?

Two particles move along an \(x\) axis. The position of particle 1 is given by \(x=6.00 t^{2}+3.00 t+2.00\) (in meters and seconds); the acceleration of particle 2 is given by \(a=-8.00 t\) (in meters per second squared and seconds) and, at \(t=0,\) its velocity is \(20 \mathrm{~m} / \mathrm{s}\). When the velocities of the particles match, what is their velocity?

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 \(x=15.0 \mathrm{~cm}, t\) 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) \(A t\) what value of \(x\) does this occur? (c) What is the spot's acceleration (including sign) when this occurs? (d) 1st it moving right or left just prior to coming to rest? (e) Just after? (f) At what time \(t>0\) docs it first reach an edge of the screen?

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 \(1.60 \mathrm{~s}\) after being shot. Then, after barely missing the edge of the building as it falls downward, the rock strikes the ground \(6.00 \mathrm{~s}\) after it is launched. In \(\mathrm{SI}\) 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?

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 \(9.5 \mathrm{~m} / \mathrm{s}\), 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?

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

A red train traveling at \(72 \mathrm{~km} / \mathrm{h}\) and a green train traveling at \(144 \mathrm{~km} / \mathrm{h}\) are headed toward each other along a straight, level track. When they are \(950 \mathrm{~m}\) 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.

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

A particle's acceleration along an \(x\) axis is \(a=5.0 t,\) with \(t\) in seconds and \(a\) in meters per second squared. At \(t=2.0 \mathrm{~s}\) its velocity is \(+17 \mathrm{~m} / \mathrm{s}\). What is its velocity at \(t=4.0 \mathrm{~s} ?\)