Chapter 2

You and your family take a trip to see your aunt who lives 100 miles away along a straight highway. The first 60 miles of the trip are driven at \(55 \mathrm{mi} / \mathrm{h}\) but then you get stuck in a standstill traffic jam for 20 minutes. In order to make up time, you then proceed at \(75 \mathrm{mi} / \mathrm{h}\) for the rest of the trip. What is the magnitude of your average velocity for the whole trip?

On March \(27,2004,\) the United States successfully tested the hypersonic X-43A scramjet, which flew at Mach 7 (seven times the speed of sound) for 11 s. (a) At this rate, how many minutes would it take such a scramjet to carry passengers the approximately \(5000 \mathrm{~km}\) from San Francisco to New York? (Use the speed of sound at \(\left.0^{\circ} \mathrm{C}, 331 \mathrm{~m} / \mathrm{s} .\right)\) (b) How many kilometers did the scramjet travel during its 11 s test?

The earth's crust is broken up into a series of more-or-less rigid plates that slide around due to motion of material in the mantle below. Although the speeds of these plates vary somewhat, they are typically about \(5 \mathrm{~cm} / \mathrm{y}\). Assume that this rate remains constant over time. (a) If you and your neighbor live on opposite sides of a plate boundary at which one plate is moving northward at \(5.0 \mathrm{~cm} / \mathrm{y}\) with respect to the other plate, how far apart do your houses move in a century? (b) Los Angeles is presently \(550 \mathrm{~km}\) south of San Francisco but is on a plate moving northward relative to San Francisco. If the \(5.0 \mathrm{~cm} / \mathrm{y}\) velocity continues, how many years will it take before Los Angeles has moved up to San Francisco?

A jogger covers one lap of a circular track \(40.0 \mathrm{~m}\) in diameter in \(62.5 \mathrm{~s}\). (a) For that lap, what were her average speed and average velocity? (b) If she covered the first half-lap in \(28.7 \mathrm{~s},\) what were her average speed and average velocity for that half-lap?

At room temperature, sound travels at a speed of about \(344 \mathrm{~m} / \mathrm{s}\) in air. You see a distant flash of lightning and hear the thunder arrive \(7.5 \mathrm{~s}\) later. How many miles away was the lightning strike? (Assume the light takes essentially no time to reach you.)

Nerve impulses travel at different speeds, depending on the type of fiber through which they move. The impulses for touch travel at \(76.2 \mathrm{~m} / \mathrm{s},\) while those registering pain move at \(0.610 \mathrm{~m} / \mathrm{s} .\) If a person stubs his toe, find (a) the time for each type of impulse to reach his brain, and (b) the time delay between the pain and touch impulses. Assume that his brain is \(1.85 \mathrm{~m}\) from his toe and that the impulses travel directly from toe to brain.

While riding on a bus traveling down the highway, you notice that it takes 2 min to travel from one roadside mile marker to the next one. (a) What is your speed in \(\mathrm{mi} / \mathrm{h} ?\) (b) How long does it take the bus to travel 100 yds?

A mouse travels along a straight line; its distance \(x\) from the origin at any time \(t\) is given by the equation \(x=\) \(\left(8.5 \mathrm{~cm} \cdot \mathrm{s}^{-1}\right) t-\left(2.5 \mathrm{~cm} \cdot \mathrm{s}^{-2}\right) t^{2} .\) Find the average velocity of the mouse in the interval from \(t=0\) to \(t=1.0 \mathrm{~s}\) and in the interval from \(t=0\) to \(t=4.0 \mathrm{~s}\)

When you normally drive the freeway between Sacramento and San Francisco at an average speed of \(105 \mathrm{~km} / \mathrm{h}\) (65 \(\mathrm{mi} / \mathrm{h}\) ), the trip takes \(1.0 \mathrm{~h}\) and \(20 \mathrm{~min}\). On a Friday afternoon, however, heavy traffic slows you down to an average of \(70 \mathrm{~km} / \mathrm{h}\) (43 \(\mathrm{mi} / \mathrm{h}\) ) for the same distance. How much longer does the trip take on Friday than on the other days?

Two runners start simultaneously at opposite ends of a \(200.0 \mathrm{~m}\) track and run toward each other. Runner \(A\) runs at a steady \(8.0 \mathrm{~m} / \mathrm{s}\) and runner \(B\) runs at a constant \(7.0 \mathrm{~m} / \mathrm{s}\). When and where will these runners meet?

A test driver at Incredible Motors, Inc., is testing a new model car having a speedometer calibrated to read \(\mathrm{m} / \mathrm{s}\) rather than \(\mathrm{mi} / \mathrm{h}\). The following series of speedometer readings was obtained during a test run: $$\begin{array}{l|lllllllll}\text { Time (s) } & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 \\\\\hline \text { Velocity (m/s) } & 0 & 0 & 2 & 5 & 10 & 15 & 20 & 22 & 22\end{array}$$ (a) Compute the average acceleration during each 2 s interval. Is the acceleration constant? Is it constant during any part of the test run? (b) Make a velocity-time graph of the data shown, using scales of \(1 \mathrm{~cm}=1\) s horizontally and \(1 \mathrm{~cm}=2 \mathrm{~m} / \mathrm{s}\) vertically. Draw a smooth curve through the plotted points. By measuring the slope of your curve, find the magnitude of the instantaneous acceleration at times \(t=9 \mathrm{~s}, 13 \mathrm{~s},\) and \(15 \mathrm{~s}\)

(a) The pilot of a jet fighter will black out at an acceleration greater than approximately \(5 g\) if it lasts more than a few seconds. Express this acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) and \(\mathrm{ft} / \mathrm{s}^{2} .\) (b) The acceleration of the passenger during a car crash with an air bag is about \(60 g\) for a very short time. What is this acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) and \(\mathrm{ft} / \mathrm{s}^{2} ?\) (c) The acceleration of a falling body on our moon is \(1.62 \mathrm{~m} / \mathrm{s}^{2}\). How many \(g^{\prime} \mathrm{s}\) is this? (d) If the acceleration of a test plane is \(24.3 \mathrm{~m} / \mathrm{s}^{2},\) how many \(g\) 's is it?

The driver of a car traveling on the highway suddenly slams on the brakes because of a slowdown in traffic ahead. If the car's speed decreases at a constant rate from \(60 \mathrm{mi} / \mathrm{h}\) to \(40 \mathrm{mi} / \mathrm{h}\) in \(3 \mathrm{~s},\) (a) what is the magnitude of its acceleration, assuming that it continues to move in a straight line? (b) What distance does the car travel during the braking period? Express your answers in feet.

Cheetahs, the fastest of the great cats, can reach \(45 \mathrm{mi} / \mathrm{h}\) in \(2.0 \mathrm{~s}\) starting from rest. Assuming that they have constant acceleration throughout that time, find (a) their acceleration (in \(\mathrm{ft} / \mathrm{s}^{2}\) and \(\left.\mathrm{m} / \mathrm{s}^{2}\right)\) and \((\mathrm{b})\) the distance (in \(\mathrm{m}\) and \(\mathrm{ft}\) ) they travel during that time.

A cat drops from a shelf \(4.0 \mathrm{ft}\) above the floor and lands on all four feet. His legs bring him to a stop in a distance of \(12 \mathrm{~cm}\). Calculate (a) his speed when he first touches the floor (ignore air resistance), (b) how long it takes him to stop, and (c) his acceleration (assumed constant) while he is stopping, in \(\mathrm{m} / \mathrm{s}^{2}\) and \(g\) 's.

A jet fighter pilot wishes to accelerate from rest at \(5 g\) to reach Mach 3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this acceleration lasts more than 5.0 s. Use \(331 \mathrm{~m} / \mathrm{s}\) for the speed of sound. (a) Will the period of acceleration last long enough to cause him to black out? (b) What is the greatest speed he can reach with an acceleration of \(5 g\) before blacking out?

A car is traveling at \(60 \mathrm{mi} / \mathrm{h}\) down a highway. (a) What magnitude of acceleration does it need to have to come to a complete stop in a distance of \(200 \mathrm{ft} ?\) (b) What acceleration does it need to stop in \(200 \mathrm{ft}\) if it is traveling at \(100 \mathrm{mi} / \mathrm{h} ?\)

If a pilot accelerates at more than \(4 g\), he begins to "gray out" but does not completely lose consciousness. (a) What is the shortest time that a jet pilot starting from rest can take to reach Mach 4 (four times the speed of sound) without graying out? (b) How far would the plane travel during this period of acceleration? (Use \(331 \mathrm{~m} / \mathrm{s}\) for the speed of sound.)

During an auto accident, the vehicle's air bags deploy and slow down the passengers more gently than if they had hit the windshield or steering wheel. According to safety standards, the bags produce a maximum acceleration of \(60 g,\) but lasting only \(36 \mathrm{~ms}\) (or less). How far (in meters) does a person travel in coming to a complete stop in \(36 \mathrm{~ms}\) at a constant acceleration of \(60 \mathrm{~g} ?\)

Starting from rest, a boulder rolls down a hill with constant acceleration and travels \(2.00 \mathrm{~m}\) during the first second. (a) How far does it travel during the second second? (b) How fast is it moving at the end of the first second? at the end of the second second?

The Beretta Model \(92 \mathrm{~S}\) (the standard-issue U.S. army pistol) has a barrel \(127 \mathrm{~mm}\) long. The bullets leave this barrel with a muzzle velocity of \(335 \mathrm{~m} / \mathrm{s}\). (a) What is the acceleration of the bullet while it is in the barrel, assuming it to be constant? Express your answer in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime}\) s. (b) For how long is the bullet in the barrel?

An electric drag racer is much like its piston engine counterpart, but instead it is powered by an electric motor running off of onboard batteries. These vehicles are capable of covering a \(\frac{1}{4}\) mile straight-line track in \(10 \mathrm{~s}\). (a) Determine the acceleration of the drag racer in units of \(\mathrm{m} / \mathrm{s}\). (Assume that the acceleration is constant throughout the race.) (b) How does the value you get compare with the acceleration of gravity? (c) Calculate the final speed of the drag racer in \(\mathrm{mi} / \mathrm{h}\).

The "reaction time" of the average automobile driver is about \(0.7 \mathrm{~s}\). (The reaction time is the interval between the perception of a signal to stop and the application of the brakes.) If an automobile can slow down with an acceleration of \(12.0 \mathrm{ft} / \mathrm{s}^{2}\), compute the total distance covered in coming to a stop after a signal is observed (a) from an initial velocity of \(15.0 \mathrm{mi} / \mathrm{h}\) (in a school zone) and (b) from an initial velocity of \(55.0 \mathrm{mi} / \mathrm{h}\).

According to recent typical test data, a Ford Focus travels \(0.250 \mathrm{mi}\) in \(19.9 \mathrm{~s},\) starting from rest. The same car, when braking from \(60.0 \mathrm{mi} / \mathrm{h}\) on dry pavement, stops in \(146 \mathrm{ft}\). Assume constant acceleration in each part of its motion, but not necessarily the same acceleration when slowing down as when speeding up. (a) Find this car's acceleration while braking and while speeding up. (b) If its acceleration is constant while speeding up, how fast (in \(\mathrm{mi} / \mathrm{h}\) ) will the car be traveling after \(0.250 \mathrm{mi}\) of acceleration? (c) How long does it take the car to stop while braking from \(60.0 \mathrm{mi} / \mathrm{h} ?\)

If the radius of a circle of area \(A\) and circumference \(C\) is doubled, find the new area and circumference of the circle in terms of \(A\) and C. (Consult Chapter 0 if necessary.)

In the redesign of a machine, a metal cubical part has each of its dimensions tripled. By what factor do its surface area and volume change?

You have two cylindrical tanks. The tank with the greater volume is 1.20 times the height of the smaller tank. It takes 218 gallons of water to fill the larger tank and 150 gallons to fill the other. What is the ratio of the radius of the larger tank to the radius of the smaller one?

A spherical balloon has volume \(V\) and radius \(R .\) By what factor is its radius reduced if you let enough air out of the balloon to reduce its volume by a factor of \(8 ?\) (Consult Chapter 0 if necessary.)

Two rockets having the same acceleration start from rest, but rocket \(A\) travels for twice as much time as rocket \(B\). (a) If rocket \(A\) goes a distance of \(250 \mathrm{~km}\), how far will rocket \(B\) go? (b) If rocket \(A\) reaches a speed of \(350 \mathrm{~m} / \mathrm{s},\) what speed will rocket \(B\) reach?

The drivers of two cars having equal speeds hit their brakes at the same time, but car \(A\) has three times the acceleration of car \(B\). (a) If car \(A\) travels a distance \(D\) before stopping, how far (in terms of \(D)\) will car \(B\) go before stopping? (b) If car \(B\) stops in time \(T\), how long (in terms of \(T\) ) will it take for car \(A\) to stop?

Two bicyclists start a sprint from rest, each riding with a constant acceleration. Bicyclist \(A\) has twice the acceleration of bicyclist \(B ;\) however, bicyclist \(B\) rides for twice as long as bicyclist \(A\). What is the ratio of the distance traveled by bicyclist \(A\) to that traveled by bicyclist \(B\) ? What is the ratio of the speed of bicyclist \(A\) to that of bicyclist \(B\) at the end of their sprint?

(a) If a flea can jump straight up to a height of \(22.0 \mathrm{~cm}\), what is its initial speed (in \(\mathrm{m} / \mathrm{s}\) ) as it leaves the ground, neglecting air resistance? (b) How long is it in the air? (c) What are the magnitude and direction of its acceleration while it is (i) moving upward? (ii) moving downward? (iii) at the highest point?

A brick is released with no initial speed from the roof of a building and strikes the ground in \(2.50 \mathrm{~s},\) encountering no appreciable air drag. (a) How tall, in meters, is the building? (b) How fast is the brick moving just before it reaches the ground? (c) Sketch graphs of this falling brick's acceleration, velocity, and vertical position as functions of time.

Suppose that you drop a marble from the top of the Burj Khalifa building in Dubai, which is about \(830 \mathrm{~m}\) tall. If you ignore air resistance, (a) how long will it take for the marble to hit the ground? (b) How fast will it be moving just before it hits?

A tennis ball on Mars, where the acceleration due to gravity is \(0.379 g\) and air resistance is negligible, is hit directly upward and returns to the same level \(8.5 \mathrm{~s}\) later. (a) How high above its original point did the ball go? (b) How fast was it moving just after being hit? (c) Sketch clear graphs of the ball's vertical position, vertical velocity, and vertical acceleration as functions of time while it's in the Martian air.

One way to measure \(g\) on another planet or moon by remote sensing is to measure how long it takes an object to fall a given distance. A lander vehicle on a distant planet records the fact that it takes \(3.17 \mathrm{~s}\) for a ball to fall freely \(11.26 \mathrm{~m},\) starting from rest. (a) What is the acceleration due to gravity on that planet? Express your answer in \(\mathrm{m} / \mathrm{s}^{2}\) and in earth \(g^{\prime} \mathrm{s}\). (b) How fast is the ball moving just as it lands?

A hot-air balloonist, rising vertically with a constant speed of \(5.00 \mathrm{~m} / \mathrm{s},\) releases a sandbag at the instant the balloon is \(40.0 \mathrm{~m}\) above the ground. (See Figure \(2.54 .)\) After it is released, the sandbag encounters no appreciable air drag. (a) Compute the position and velocity of the sandbag at \(0.250 \mathrm{~s}\) and \(1.00 \mathrm{~s}\) after its release. (b) How many seconds after its release will the bag strike the ground? (c) How fast is it moving as it strikes the ground? (d) What is the greatest height above the ground that the sandbag reaches? (e) Sketch graphs of this bag's acceleration, velocity, and vertical position as functions of time.

Astronauts on our moon must function with an acceleration due to gravity of \(1.62 g .\) (a) If an astronaut can throw a certain wrench \(12.0 \mathrm{~m}\) vertically upward on earth, how high could he throw it on our moon if he gives it the same starting speed in both places? (b) How much longer would it be in motion (going up and coming down) on the moon than on earth?

A student throws a water balloon vertically downward from the top of a building. The balloon leaves the thrower's hand with a speed of \(15.0 \mathrm{~m} / \mathrm{s}\). (a) What is its speed after falling freely for \(2.00 \mathrm{~s}\) ? (b) How far does it fall in \(2.00 \mathrm{~s} ?\) (c) What is the magnitude of its velocity after falling \(10.0 \mathrm{~m} ?\)

A rock is thrown vertically upward with a speed of \(12.0 \mathrm{~m} / \mathrm{s}\) from the roof of a building that is \(60.0 \mathrm{~m}\) above the ground. (a) In how many seconds after being thrown does the rock strike the ground? (b) What is the speed of the rock just before it strikes the ground? Assume free fall.

The rocket driven sled Sonic Wind No. 2, used for investigating the physiological effects of large accelerations, runs on a straight, level track that is \(1080 \mathrm{~m}\) long. Starting from rest, it can reach a speed of \(1610 \mathrm{~km} / \mathrm{h}\) in \(1.80 \mathrm{~s}\). (a) Compute the acceleration in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g\) 's. (b) What is the distance covered in 1.80 s? (c) A magazine article states that, at the end of a certain run, the speed of the sled decreased from \(1020 \mathrm{~km} / \mathrm{h}\) to zero in \(1.40 \mathrm{~s}\) and that, during this time, its passenger was subjected to more than \(40 g .\) Are these figures consistent?

Two stones are thrown vertically upward from the ground, one with three times the initial speed of the other. (a) If the faster stone takes \(10 \mathrm{~s}\) to return to the ground, how long will it take the slower stone to return? (b) If the slower stone reaches a maximum height of \(H,\) how high (in terms of \(H\) ) will the faster stone go? Assume free fall.

Two coconuts fall freely from rest at the same time, one from a tree twice as high as the other. (a) If the coconut from the taller tree reaches the ground with a speed \(V,\) what will be the speed (in terms of \(V\) ) of the coconut from the other tree when it reaches the ground? (b) If the coconut from the shorter tree takes time \(T\) to reach the ground, how long (in terms of \(T\) ) will it take the other coconut to reach the ground?

A Toyota Prius driving north at \(65 \mathrm{mi} / \mathrm{h}\) and a VW Passat driving south at \(42 \mathrm{mi} / \mathrm{h}\) are on the same road heading toward each other (but in different lanes). What is the velocity of each car relative to the other (a) when they are \(250 \mathrm{ft}\) apart, just before they meet, and (b) when they are \(525 \mathrm{ft}\) apart, after they have passed each other?

You are driving eastbound on the interstate at \(70 \mathrm{mi} / \mathrm{h}\). You observe that you are approaching a truck in your lane at a relative speed of \(20 \mathrm{mi} / \mathrm{h}\). (a) How fast is the truck moving relative to the highway? (b) If the truck were instead traveling at this speed in the westbound lane, what would be the relative velocity between you and the truck?

A helicopter 8.50 m above the ground and descending at \(3.50 \mathrm{~m} / \mathrm{s}\) drops a package from rest (relative to the helicopter). Just as it hits the ground, find (a) the velocity of the package relative to the helicopter and (b) the velocity of the helicopter relative to the package. The package falls freely.

A jetliner has a cruising air speed of \(600 \mathrm{mi} / \mathrm{h}\) relative to the air. How long does it take this plane to fly round trip from San Francisco to Chicago, an east-west flight of \(2000 \mathrm{mi}\) each way, (a) if there is no wind blowing and (b) if the wind is blowing at \(150 \mathrm{mi} / \mathrm{h}\) from the west to the east?

At the instant the traffic light turns green, an automobile that has been waiting at an intersection starts ahead with a constant acceleration of \(2.50 \mathrm{~m} / \mathrm{s}^{2}\). At the same instant, a truck, traveling with a constant speed of \(15.0 \mathrm{~m} / \mathrm{s}\), overtakes and passes the automobile. (a) How far beyond its starting point does the automobile overtake the truck? (b) How fast is the automobile traveling when it overtakes the truck?

A state trooper is traveling down the interstate at \(30 \mathrm{~m} / \mathrm{s}\). He sees a speeder traveling at \(50 \mathrm{~m} / \mathrm{s}\) approaching from behind. At the moment the speeder passes the trooper, the trooper hits the gas and gives chase at a constant acceleration of \(2.5 \mathrm{~m} / \mathrm{s}^{2}\). Assuming that the speeder continues at \(50 \mathrm{~m} / \mathrm{s},\) (a) how long will it take the trooper to catch up to the speeder? (b) How far down the highway will the trooper travel before catching up to the speeder?

Two rocks are thrown directly upward with the same initial speeds, one on earth and one on our moon, where the acceleration due to gravity is one-sixth what it is on earth. (a) If the rock on the moon rises to a height \(H\), how high, in terms of \(H\), will the rock rise on the earth? (b) If the earth rock takes \(4.0 \mathrm{~s}\) to reach its highest point, how long will it take the moon rock to do so?