Chapter 3

A meteor streaking through the night sky is located with radar. At point \(A\) its coordinates are \((5.00 \mathrm{km}, 1.20 \mathrm{km}),\) and 1.14 s later it has moved to point \(B\) with coordinates \((6.24 \mathrm{km},\)0.925 \(\mathrm{km} ) .\) Find (a) the \(x\) and \(y\) components of its average velocity between \(A\) and \(B\) and (b) the magnitude and direction of its average velocity between these two points.

At an air show, a jet plane has velocity components \(v_{x}=\) 625 \(\mathrm{km} / \mathrm{h}\) and \(v_{y}=415 \mathrm{km} / \mathrm{h}\) at time 3.85 \(\mathrm{s}\) and \(v_{x}=838 \mathrm{km} / \mathrm{h}\) and \(v_{y}=365 \mathrm{km} / \mathrm{h}\) at time 6.52 s. For this time interval, find(a) the \(x\) and \(y\) components of the plane's average acceleration and (b) the magnitude and direction of its average acceleration.

\(\cdot\) A coyote chasing a rabbit is moving 8.00 \(\mathrm{m} / \mathrm{s}\) due east at one moment and 8.80 \(\mathrm{m} / \mathrm{s}\) due south 4.00 s later. Find (a) the \(x\) and \(y\) components of the coyote's average acceleration during that time and (b) the magnitude and direction of the coyote's average acceleration during that time.

A baseball pitcher throws a fastball horizontally at a speed of 42.0 \(\mathrm{m} / \mathrm{s} .\) Ignoring air resistance, how far does the ball drop between the pitcher's mound and home plate, 60 \(\mathrm{ft} 6\) in away?

A physics book slides off a horizontal tabletop with a speed of 1.10 \(\mathrm{m} / \mathrm{s}\) . It strikes the floor in 0.350 s. Ignore air resistance. Find (a) the height of the tabletop above the floor, (b) the horizontal distance from the edge of the table to the point where the book strikes the floor, and (c) the horizontal and vertical components of the book's velocity, and the magnitude and direction of its velocity, just before the book reaches the floor.

A tennis ball rolls off the edge of a tabletop 0.750 m above he floor and strikes the floor at a point 1.40 m horizontally from the edge of the table. (a) Find the time of flight of the ball. (b) Find the magnitude of the initial velocity of the ball.(c) Find the magnitude and direction of the velocity of the ball just before it strikes the floor.

A military helicopter on a training mission is flying horizontally at a speed of 60.0\(\mathrm{m} / \mathrm{s}\) when it accidentally drops a bomb(fortunately, not armed) at an elevation of 300 \(\mathrm{m.}\) You can ignore air resistance. (a) How much time is required for the bomb to reach the earth? (b) How far does it travel horizontally while falling? (c) Find the horizontal and vertical components of the bomb's velocity just before it strikes the earth. (d) Draw graphs of the horizontal distance vs. time and the vertical distance vs. time for the bomb's motion. (e) If the velocity of the helicopter remains constant, where is the helicopter when the bomb hits the ground?

Inside a star ship at rest on the earth, a ball rolls off the top of a horizontal table and lands a distance \(D\) from the foot of the table. This star ship now lands on the unexplored Planet \(X\) . The commander, Captain Curious, rolls the same ball off the same table with the same initial speed as on earth and finds that it lands a distance 2.76\(D\) from the foot of the table. What is the acceleration due to gravity on Planet \(\mathrm{X}\) ?

\(\bullet\) Leaping the river, A \(10,000 \mathrm{N}\) car comes to a bridge during a storm and finds the bridge washed out. The 650 \(\mathrm{N}\) driver must get to the other side, so he decides to try leaping it with his car. The side the car is on is 21.3 \(\mathrm{m}\) above the river, while the opposite side is a mere 1.80 \(\mathrm{m}\) above the river. The river itself is a raging torrent 61.0 \(\mathrm{m}\) wide. (a) How fast should the car be traveling just as it leaves the cliff in order to clear the river and land safely on the opposite side? (b) What is the speed of the car just before it lands safely on the other side?

football is thrown with an initial upward velocity component of 15.0 \(\mathrm{m} / \mathrm{s}\) and a horizontal velocity component of 18.0 \(\mathrm{m} / \mathrm{s}\) . (a) How much time is required for the football to reach the highest point in its trajectory? (b) How high does it get above its release point? (c) How much time after it is thrown does it take to return to its original height? How does this time compare with what you calculated in part (b)? Is your answer reasonable? How far has the football traveled horizontally from its original position?

A tennis player hits a ball at ground level, giving it an initial velocity of 24 \(\mathrm{m} / \mathrm{s}\) at \(57^{\circ}\) above the horizontal. (a) What are thehorizontal and vertical components of the ball's initial velocity? (b) How high above the ground does the ball go? (c) How long does it take the ball to reach its maximum height? (d) What are the ball's velocity and acceleration at its highest point? (e) For how long a time is the ball in the air? (f) When this ball lands on the court, how far is it from the place where it was hit?

\(\bullet(\) a) \(A\) pistol that fires a signal flare gives it an initial velocity (muzzle velocity) of 125 \(\mathrm{m} / \mathrm{s}\) at an angle of \(55.0^{\circ}\) above the horizontal. You can ignore air resistance. Find the flare's maximum height and the distance from its firing point to its landing point if it is fired (a) on the level salt flats of Utah, and (b) over the flat Sea of Tranquility on the moon, where \(g=1.67 \mathrm{m} / \mathrm{s}^{2}\) .

A major leaguer hits a baseball so that it leaves the bat at a speed of 30.0 \(\mathrm{m} / \mathrm{s}\) and at an angle of \(36.9^{\circ}\) above the horizontal. You can ignore air resistance. (a) At what two times is the base- ball at a height of 10.0 \(\mathrm{m}\) above the point at which it left the bat?(b) Calculate the horizontal and vertical components of the baseball's velocity at each of the two times you found in part (a). (c) What are the magnitude and direction of the base- ball's velocity when it returns to the level at which it left the bat?

A batted baseball leaves the bat at an angle of \(30.0^{\circ}\) above the horizontal and is caught by an outfielder 375 ft from home plate at the same height from which it left the bat. (a) What was the initial speed of the ball? (b) How high does the ball rise above the point where it struck the bat?

. A man stands on the roof of a 15.0 -m-tall building and throws a rock with a velocity of magnitude 30.0 \(\mathrm{m} / \mathrm{s}\) at an angle of \(33.0^{\circ}\) above the horizontal. You can ignore air resistance. Calculate (a) the maximum height above the roof reached by the rock, (b) the magnitude of the velocity of the rock just before it strikes the ground, and (c) the horizontal distance from the base of the building to the point where the rock strikes the ground.

\(\cdot\) The champion jumper of the insect world. The froghopper, Philaenus spumarius, holds the world record for insect jumps. When leaping at an angle of \(58.0^{\circ}\) above the horizontal, some of the tiny critters have reached a maximum height of 58.7 \(\mathrm{cm}\) above the level ground. (See Nature, Vol. \(424,31\) July \(2003, \mathrm{p} .509\) ) (a) What was the takeoff speed for such a leap? (b) What horizontal distance did the froghopper cover for this world-record leap?

Firemen are shooting a stream of water at a burning building. A high-pressure hose shoots out the water with a speed of 25.0 \(\mathrm{m} / \mathrm{s}\) as it leaves the hose nozzle. Once it leaves the hose, the water moves in projectile motion. The firemen adjust the angle of elevation of the hose until the water takes 3.00 s to reach a building 45.0 m away. You can ignore air resistance; assume that the end of the hose is at ground level.(a) Find the angle of elevation of the hose. (b) Find the speed and acceleration of the water at the highest point in its trajectory. (c) How high above the ground does the water strike the building, and how fast visit moving just before it hits the building?

. A water balloon slingshot launches its projectiles essentially from ground level at a speed of 25.0 \(\mathrm{m} / \mathrm{s}\) . (a) At what angle should the slingshot be aimed to achieve its maximum range? (b) If shot at the angle you calculated in part (a), how .far will a water balloon travel horizontally? (c) For how long will the balloon be in the air? (You can ignore air resistance.)

. A certain cannon with a fixed angle of projection has a range of 1500 \(\mathrm{m}\) . What will be its range if you add more powder so that the initial speed of the cannonball is tripled?

. The nozzle of a fountain jet sits in the center of a circular pool of radius 3.50 \(\mathrm{m}\) . If the nozzle shoots water at an angle of \(65^{\circ}\) , what is the maximum speed of the water at the nozzle that will allow it to land within the pool? (You can ignore air resistance.)

\(\bullet\) Two archers shoot arrows in the same direction from the same place with the same initial speeds but at different angles. One shoots at \(45^{\circ}\) above the horizontal, while the other shoots at \(60.0^{\circ} .\) If the arrow launched at \(45^{\circ}\) lands 225 \(\mathrm{m}\) from the archer, how far apart are the two arrows when they land? (You can assume that the arrows start at essentially ground level.)

A bottle rocket can shoot its projectile vertically to a height of 25.0 \(\mathrm{m}\) . At what angle should the bottle rocket be fired to reach its maximum horizontal range, and what is that range? (You can ignore air resistance.)

\cdots An airplane is flying with a velocity of 90.0 \(\mathrm{m} / \mathrm{s}\) at an angle of \(23.0^{\circ}\) above the horizontal. When the plane is 114 \(\mathrm{m}\) directly above a dog that is standing on level ground, a suitcase drops out of the lugage compartment. How far from the dog will the suitcase land? You can ignore air resistance.

\(\bullet\) You swing a 2.2 \(\mathrm{kg}\) stone in a circle of radius 75 \(\mathrm{cm} .\) At what speed should you swing it so its centripetal acceleration will be 9.8 \(\mathrm{m} / \mathrm{s}^{2}\) ?

\(\cdot\) A model of a helicopter rotor has four blades, each 3.40 \(\mathrm{m}\) in length from the central shaft to the tip of the blade. The model is rotated in a wind tunnel at 550 rev/min. (a) What is the linear speed, in \(\mathrm{m} / \mathrm{s}\) , of the blade tip? (b) What is the radial acceleration of the blade tip, expressed as a multiple of the acceleration \(g\) due to gravity?

\(\bullet\) A wall clock has a second hand 15.0 \(\mathrm{cm}\) long. What is the radial acceleration of the tip of this hand?

A curving freeway exit has a radius of 50.0 \(\mathrm{m}\) and a posted speed limit of 35 \(\mathrm{mi} / \mathrm{h} .\) What is your radial acceleration (in \(\mathrm{m} /\mathrm{s}^{2} )\) if you take this exit at the posted speed? What if you take the exit at a speed of 50 \(\mathrm{mi} / \mathrm{h} ?\)

Dizziness. Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose a dancer (or skater) is spinning at a very high 3. 0 revolutions per second about a vertical axis through the center of his head. Although the distance varies from person to person, the inner ear is approximately 7.0 \(\mathrm{cm}\) from the axis of spin. What is the radial acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime} s\) s of the endolymph fluid?

\(\bullet\) Pilot blackout in a power dive. A jet plane comes in for a downward dive as shown in Figure \(3.39 .\) The bottom part of the path is a quarter circle having a radius of curvature of 350 \(\mathrm{m} .\) According to medical tests, pilots lose consciousness at an acceleration of 5.5\(g .\) At what speed (in \(\mathrm{m} / \mathrm{s}\) and mph) will the pilot black out for this dive?

\(\bullet\) A canoe has a velocity of 0.40 \(\mathrm{m} / \mathrm{s}\) southeast relative to the earth. The canoe is on a river that is flowing 0.50 \(\mathrm{m} / \mathrm{s}\) east rela- tive to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

\(\bullet\) Crossing the river, I. A river flows due south with a speed of 2.0 \(\mathrm{m} / \mathrm{s} .\) A man steers a motorboat across the river; his velocity relative to the water is 4.2 \(\mathrm{m} / \mathrm{s}\) due east. The river is 800 \(\mathrm{m}\) wide. (a) What is his velocity (magnitude and direction) relative to the earth? (b) How much time is required for the man to cross the river? (c) How far south of his starting point will he reach the opposite bank?

You're standing outside on a windless day when raindrops begin to fall straight down. You run for shelter at a speed of \(5.0 \mathrm{m} / \mathrm{s},\) and you notice while you're running that the raindrops appear to be falling at an angle of about \(30^{\circ}\) from the vertical. What's the vertical speed of the raindrops?

\(\bullet\) Bird migration. Canadian geese migrate essentially along a north- south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 \(\mathrm{km} / \mathrm{h}\) . If one such bird is flying at 100 \(\mathrm{km} / \mathrm{h}\) relative to the air, but there is a 40 \(\mathrm{km} / \mathrm{h}\) wind blowing from west to east, (a) at what angle relative to the north-south direction should this bird head so that it will be traveling directly southward relative to the ground? (b) How long will it take the bird to cover a ground distance of 500 \(\mathrm{km}\) from north to south? (Note: Even on cloudy nights, many birds can navigate using the earth's magnetic field to fix the north-south direction.)

\bulletA test rocket is launched by accelerating it along a \(200.0-\mathrm{m}\) incline at 1.25 \(\mathrm{m} / \mathrm{s}^{2}\) starting from rest at point \(A\) (Figure \(3.40 . )\) The incline rises at \(35.0^{\circ}\) above the horizontal,and at the instant the rocket leaves it, its engines turn off and it is subject only to gravity (air resistance can be ignored). Find (a) the maximum height above the ground that the rocket reaches, and (b) the greatest horizontal range of the rocket beyond point \(A .\)

\(\bullet\) A player kicks a football at an angle of \(40.0^{\circ}\) from the horizontal, with an initial speed of 12.0 \(\mathrm{m} / \mathrm{s} .\) A second player standing at a distance of 30.0 \(\mathrm{m}\) from the first (in the direction of the kick) starts running to meet the ball at the instant it is kicked. How fast must he run in order to catch the ball just before it hits the ground?

\(\bullet\) Dynamite! A demolition crew uses dynamite to blow an old building apart. Debris from the explosion flies off in all directions and is later found at distances as far as 50 \(\mathrm{m}\) from the explosion. Estimate the maximum speed at which debris was blown outward by the explosion. Describe any assumptions that you make.

Fighting forest fires. When fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing by dropping a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path 90.0 \(\mathrm{m}\) above the ground and with a speed of \(64.0 \mathrm{m} / \mathrm{s}(143 \mathrm{mi} / \mathrm{h}),\) at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

An errand of mercy. An airplane is dropping bales of hay to cattle stranded in a blizzard on the Great Plains. The pilot releases the bales at 150 \(\mathrm{m}\) above the level ground when the plane is flying at 75 \(\mathrm{m} / \mathrm{s} 55^{\circ}\) above the horizontal. How far in front of the cattle should the pilot release the hay so that the bales will land at the point where the cattle are stranded?

\bullet A cart carrying a vertical missile launcher moves horizontally at a constant velocity of 30.0 \(\mathrm{m} / \mathrm{s}\) to the right. It launches a rocket vertically upward. The missile has an initial vertical velocity of 40.0 \(\mathrm{m} / \mathrm{s}\) relative to the cart. (a) How high does the rocket go? (b) How far does the cart travel while the rocket is in the air? (c) Where does the rocket land relative to the cart?

\(\bullet\) The longest home run. According to the Guinness Book of World Records, the longest home run ever measured was hit by Roy "Dizzy" Carlyle in a minor-league game. The ball traveled 188 \(\mathrm{m}(618 \mathrm{ft})\) before landing on the ground outside the ball- park. (a) Assuming that the ball's initial velocity was 45 above the horizontal, and ignoring air resistance, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 \(\mathrm{m}(3.0 \mathrm{ft})\) above ground level? Assume that the ground was perfectly flat. (b) How far would the ball be above a fence 3.0 \(\mathrm{m}(10 \mathrm{ft})\) in height if the fence were 116 \(\mathrm{m}\) \((380 \mathrm{ft})\) from home plate?

A professional golfer can hit a ball with a speed of 70.0 \(\mathrm{m} / \mathrm{s}\) . What is the maximum distance a golf ball hit with this speed could travel on Mars, where the value of \(g\) is 3.71 \(\mathrm{m} / \mathrm{s}^{2}\) ? (The distances golf balls travel on earth are greatly shortened by air resistance and spin, as well as by the stronger force of gravity.)

\(\bullet\) A boy 12.0 m above the ground in a tree throws a ball for his dog, who is standing right below the tree and starts running the instant the ball is thrown. If the boy throws the ball horizontally at 8.50 \(\mathrm{m} / \mathrm{s}\) , (a) how fast must the dog run to catch the ball just as it reaches the ground, and (b) how far from the tree will the dog catch the ball?

A firefighting crew uses a water cannon that shoots water at 25.0 \(\mathrm{m} / \mathrm{s}\) at a fixed angle of \(53.0^{\circ}\) above the horizontal. The firefighters want to direct the water at a blaze that is 10.0 \(\mathrm{m}\) above ground level. How far from the building should they position their cannon? There are two possibilities; can you get them both? (Hint: Start with a sketch showing the trajectory of the water.)

A gun shoots a shell into the air with an initial velocity of \(100.0 \mathrm{m} / \mathrm{s}, 60.0^{\circ}\) above the horizontal on level ground. Sketch quantitative graphs of the shell's horizontal and vertical velocity components as functions of time for the complete motion.

Spiraling up. It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8.00 \(\mathrm{m}\) every 5.00 \(\mathrm{s}\) and rises vertically at a rate of 3.00 \(\mathrm{m} / \mathrm{s} .\) Determine: (a) the speed of the birc relative to the ground; (b) the bird's acceleration (magnitude) and direction); and (c) the angle between the bird's velocity vector and the horizontal.

. A water hose is used to fill a large cylindrical storage tank of diameter \(D\) and height 2\(D\) The hose shoots the water at \(45^{\circ}\) above the horizontal from the same level as the base of the tank and is a distance 6\(D\) away (Fig. \(3.43 ) .\) For what range of launch speeds \(\left(v_{0}\right)\) will the water enter the tank? Ignore air resistance, and express your answer in terms of \(D\) and \(g .\)

A \(\mathbf{A}\) world record. In the shot put, a standard track-and- field event, a 7.3 \(\mathrm{kg}\) object (the shot) is thrown by releasing it at approximately \(40^{\circ}\) over a straight left leg. The world record for distance, set by Randy Barnes in \(1990,\) is 23.11 \(\mathrm{m} .\) Assuming that Barnes released the shot put at \(40.0^{\circ}\) from a height of 2.00 \(\mathrm{m}\) above the ground, with what speed, in \(\mathrm{m} / \mathrm{s}\) and mph, did he release it?

\cdotse: Leaping the river, II. A physics professor did daredevil stunts in his spare time. His last stunt was an attempt to jump across a river on a motorcycle. (See Figure \(3.45 .\) ) The takeoff ramp was inclined at \(53.0^{\circ},\) the river was 40.0 \(\mathrm{m}\) wide, and the far bank was 15.0 \(\mathrm{m}\) lower than the top of the ramp. The river itself was 100 \(\mathrm{m}\) below the ramp. You can ignore air resistance. (a) What should his speed have been at the top of the ramp for him to have just made it to the edge of the far bank? (b) If his speed was only half the value found in \((a)\) where did he land?

A batter hits a baseball at a speed of 35.0 \(\mathrm{m} / \mathrm{s}\) and an angle of \(65.0^{\circ}\) above the horizontal. At the same instant, an outfielder 70.0 \(\mathrm{m}\) away begins running away from the batter in the line of the ball's flight, hoping to catch it. How fast must the out fielder run to catch the ball? ( (ignore air resistance, and assume the fielder catches the ball at the same height at which it left the bat.)

If the time of flight of the ball is \(t\) seconds, at what point in time will the ball have zero vertical velocity? A. \(t / 4 \mathrm{s}\) B. \(t / 4 \mathrm{s}\) C. \(t / 2 \mathrm{s}\) D. \(t / 2 \mathrm{s}\) E. There is no place on the path where the vertical velocity is zero.