An object is projected with initial speed \(v_0\) from the edge of the roof of a building that has height H. The initial velocity of the object makes an angle \(\alpha_0\) with the horizontal. Neglect air resistance. (a) If \(\alpha_0\) is \(90^{\circ}\), so that the object is thrown straight up (but misses the roof on the way down), what is the speed v of the object just before it strikes the ground? (b) If \(\alpha_0 = -90^{\circ}\), so that the object is thrown straight down, what is its speed just before it strikes the ground? (c) Derive an expression for the speed v of the object just before it strikes the ground for general \(\alpha_0\). (d) The final speed v equals \(v_1\) when \(\alpha_0\) equals \(\alpha_1\). If \(\alpha_0\) is increased, does v increase, decrease, or stay the same?

A simple pendulum (a mass swinging at the end of a string) swings back and forth in a circular arc. What is the direction of the acceleration of the mass when it is at the ends of the swing? At the midpoint? In each case, explain how you obtained your answer.

Redraw Fig. 3.11a if \(\vec{a}\) is antiparallel to \(\vec{v}_1\). Does the particle move in a straight line? What happens to its speed?

A projectile moves in a parabolic path without air resistance. Is there any point at which \(\vec{a}\) is parallel to \(\vec{v}\)? Perpendicular to \(\vec{v}\)? Explain.

A book slides off a horizontal tabletop. As it leaves the table’s edge, the book has a horizontal velocity of magnitude \(v_0\). The book strikes the floor in time t. If the initial velocity of the book is doubled to \(2v_0\), what happens to (a) the time the book is in the air, (b) the horizontal distance the book travels while it is in the air, and (c) the speed of the book just before it reaches the floor? In particular, does each of these quantities stay the same, double, or change in another way? Explain.

At the instant that you fire a bullet horizontally from a rifle, you drop a bullet from the height of the gun barrel. If there is no air resistance, which bullet hits the level ground first? Explain.

A package falls out of an airplane that is flying in a straight line at a constant altitude and speed. If you ignore air resistance, what would be the path of the package as observed by the pilot? As observed by a person on the ground?

Sketch the six graphs of the x- and y-components of position, velocity, and acceleration versus time for projectile motion with \(x_0 = y_0 = 0\) and \(0 < \alpha_0 < 90^{\circ}\).

If a jumping frog can give itself the same initial speed regardless of the direction in which it jumps (forward or straight up), how is the maximum vertical height to which it can jump related to its maximum horizontal range \(R_{\max }=v_0^2 / \mathrm{g}\)?

A projectile is fired upward at an angle \(\theta\) above the horizontal with an initial speed \(v_0\). At its maximum height, what are its velocity vector, its speed, and its acceleration vector?

In uniform circular motion, what are the average velocity and average acceleration for one revolution? Explain.

In uniform circular motion, how does the acceleration change when the speed is increased by a factor of 3? When the radius is decreased by a factor of 2?

In uniform circular motion, the acceleration is perpendicular to the velocity at every instant. Is this true when the motion is not uniform—that is, when the speed is not constant?

Raindrops hitting the side windows of a car in motion often leave diagonal streaks even if there is no wind. Why? Is the explanation the same or different for diagonal streaks on the windshield?

In a rainstorm with a strong wind, what determines the best position in which to hold an umbrella?

You are on the west bank of a river that is flowing north with a speed of 1.2 m/s. Your swimming speed relative to the water is 1.5 m/s, and the river is 60 m wide. What is your path relative to the earth that allows you to cross the river in the shortest time? Explain your reasoning.

A stone is thrown into the air at an angle above the horizontal and feels negligible air resistance. Which graph in Fig. Q3.16 best depicts the stone’s speed v as a function of time t while it is in the air?

CP CALC A dog in an open field is at rest under a tree at time t = 0 and then runs with acceleration \(\vec{a} (t) = (0.400 \ \mathrm{m/s}^2) \hat{\imath} - (0.180 \ \mathrm{m/s}^3)t \hat{\jmath}\). How far is the dog from the tree 8.00 s after it starts to run?

CALC If \(\vec{r} = bt^2 \hat{\imath} + ct^3 \hat{\jmath}\), where b and c are positive constants, when does the velocity vector make an angle of \(45.0^{\circ}\) with the x- and y-axes?

CALC A faulty model rocket moves in the xy-plane (the positive y-direction is vertically upward). The rocket’s acceleration has components \(a_{x} (t) = \alpha_{t^2}\) and \(a_{y} (t) = \beta - \gamma t\), where \(\alpha = 2.50 \ \mathrm{m/s}^4, \beta = 9.00 \ \mathrm{m/s}^2\), and \(\gamma = 1.40 \ \mathrm{m/s}^3\). At t = 0 the rocket is at the origin and has velocity \(\vec{v}_0 = v_{0x} \hat{\imath} + v_{0y} \hat{\jmath}\) with \(v0x = 1.00 \ \mathrm{m/s}\) and \(v_{0y} = 7.00 \ \mathrm{m/s}\). (a) Calculate the velocity and position vectors as functions of time. (b) What is the maximum height reached by the rocket? (c) What is the horizontal displacement of the rocket when it returns to y = 0?

CP A test rocket starting from rest at point A is launched by accelerating it along a 200.0 m incline at \(1.90 \ \mathrm{m/s}^2\) (Fig. P3.47). The incline rises at \(35.0^{\circ}\) above the horizontal, and at the instant the rocket leaves it, the engines turn off and the rocket is subject to gravity only (ignore air resistance). Find (a) the maximum height above the ground that the rocket reaches, and (b) the rocket’s greatest horizontal range beyond point A.

The position of a dragonfly that is flying parallel to the ground is given as a function of time by \(\vec{r}= [ 2.90 \ \mathrm{m} + (0.0900 \ \mathrm{m/s}^2) t^2 ] \hat{\imath} - (0.0150 \ \mathrm{m/s}^3) t^3 \hat{\jmath}\). (a) At what value of t does the velocity vector of the dragonfly make an angle of \(30.0^{\circ}\) clockwise from the +x-axis? (b) At the time calculated in part (a), what are the magnitude and direction of the dragonfly’s acceleration vector?

In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. For practice, a pilot drops 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 m above the ground and has a speed of 64.0 m/s (143 mi/h), at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

CALC A bird flies in the xy-plane with a velocity vector given by \(\vec{v}= (\alpha - \beta t^ 2) \hat{\imath}+ \gamma t \hat{\jmath}\), with \(\alpha = 2.4 \ \mathrm{m/s}, \beta = 1.6 \ \mathrm{m/s}^3\), and \(\gamma = 4.0 \ \mathrm{m/s}^2\). The positive y-direction is vertically upward. At t = 0 the bird is at the origin. (a) Calculate the position and acceleration vectors of the bird as functions of time. (b) What is the bird’s altitude (y-coordinate) as it flies over x = 0 for the first time after t = 0?

A movie stuntwoman drops from a helicopter that is 30.0 m above the ground and moving with a constant velocity whose components are 10.0 m/s upward and 15.0 m/s horizontal and toward the south. Ignore air resistance. (a) Where on the ground (relative to the position of the helicopter when she drops) should the stuntwoman have placed foam mats to break her fall? (b) Draw \(x-t, y-t, v_{x} -t\), and \(v_{y} -t\) graphs of her motion.

A cannon, located 60.0 m from the base of a vertical 25.0-m-tall cliff, shoots a 15 kg shell at \(43.0^{\circ}\) above the horizontal toward the cliff. (a) What must the minimum muzzle velocity be for the shell to clear the top of the cliff? (b) The ground at the top of the cliff is level, with a constant elevation of 25.0 m above the cannon. Under the conditions of part (a), how far does the shell land past the edge of the cliff?

CP CALC A toy rocket is launched with an initial velocity of 12.0 m/s in the horizontal direction from the roof of a 30.0-m-tall building. The rocket’s engine produces a horizontal acceleration of \((1.60 \ \mathrm{m/s}^3) t\), in the same direction as the initial velocity, but in the vertical direction the acceleration is g, downward. Ignore air resistance. What horizontal distance does the rocket travel before reaching the ground?

An important piece of landing equipment must be thrown to a ship, which is moving at 45.0 cm/s, before the ship can dock. This equipment is thrown at 15.0 m/s at \(60.0^{\circ}\) above the horizontal from the top of a tower at the edge of the water, 8.75 m above the ship’s deck (Fig. P3.54). For this equipment to land at the front of the ship, at what distance D from the dock should the ship be when the equipment is thrown? Ignore air resistance.

A baseball thrown at an angle of \(60.0^{\circ}\) above the horizontal strikes a building 18.0 m away at a point 8.00 m above the point from which it is thrown. Ignore air resistance. (a) Find the magnitude of the ball’s initial velocity (the velocity with which the ball is thrown). (b) Find the magnitude and direction of the velocity of the ball just before it strikes the building.

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 m above the level ground when the plane is flying at 75 m/s in a direction \(55^{\circ}\) above the horizontal. How far in front of the cattle should the pilot release the hay so that the bales land at the point where the cattle are stranded?

A grasshopper leaps into the air from the edge of a vertical cliff, as shown in Fig. P3.57. Find (a) the initial speed of the grasshopper and (b) the height of the cliff.

A water hose is used to fill a large cylindrical storage tank of diameter D and height 2D. 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 6D away (Fig. P3.58). For what range of launch speeds \((v_0)\) will the water enter the tank? Ignore air resistance, and express your answer in terms of D and g.

An object is projected with initial speed \(v_0\) from the edge of the roof of a building that has height H. The initial velocity of the object makes an angle \(\alpha_0\) with the horizontal. Neglect air resistance. (a) If \(\alpha_0\) is \(90^{\circ}\), so that the object is thrown straight up (but misses the roof on the way down), what is the speed v of the object just before it strikes the ground? (b) If \(\alpha_0 = -90^{\circ}\), so that the object is thrown straight down, what is its speed just before it strikes the ground? (c) Derive an expression for the speed v of the object just before it strikes the ground for general \(\alpha_0\). (d) The final speed v equals \(v_1\) when \(\alpha_0\) equals \(\alpha_1\). If \(\alpha_0\) is increased, does v increase, decrease, or stay the same?

Kicking an Extra Point. In Canadian football, after a touchdown the team has the opportunity to earn one more point by kicking the ball over the bar between the goal posts. The bar is 10.0 ft above the ground, and the ball is kicked from ground level, 36.0 ft horizontally from the bar (Fig. P3.60). Football regulations are stated in English units, but convert them to SI units for this problem. (a) There is a minimum angle above the ground such that if the ball is launched below this angle, it can never clear the bar, no matter how fast it is kicked. What is this angle? (b) If the ball is kicked at \(45.0^{\circ}\) above the horizontal, what must its initial speed be if it is just to clear the bar? Express your answer in m/s and in km/h.

Look Out! A snowball rolls off a barn roof that slopes downward at an angle of \(40^{\circ}\) (Fig. P3.61). The edge of the roof is 14.0 m above the ground, and the snowball has a speed of 7.00 m/s as it rolls off the roof. Ignore air resistance. (a) How far from the edge of the barn does the snowball strike the ground if it doesn’t strike anything else while falling? (b) Draw \(x-t, y-t, v_{x} -t\), and \(v_{y} -t\) graphs for the motion in part (a). (c) A man 1.9 m tall is standing 4.0 m from the edge of the barn. Will the snowball hit him?

A 2.7 kg ball is thrown upward with an initial speed of 20.0 m/s from the edge of a 45.0-m-high cliff. At the instant the ball is thrown, a woman starts running away from the base of the cliff with a constant speed of 6.00 m/s. The woman runs in a straight line on level ground. Ignore air resistance on the ball. (a) At what angle above the horizontal should the ball be thrown so that the runner will catch it just before it hits the ground, and how far does she run before she catches the ball? (b) Carefully sketch the ball’s trajectory as viewed by (i) a person at rest on the ground and (ii) the runner.

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 (Fig. P3.63). The takeoff ramp was inclined at \(53.0^{\circ}\), the river was 40.0 m wide, and the far bank was 15.0 m lower than the top of the ramp. The river itself was 100 m below the ramp. Ignore air resistance. (a) What should his speed have been at the top of the ramp to have just made it to the edge of the far bank? (b) If his speed was only half the value found in part (a), where did he land?

Tossing Your Lunch. Henrietta is jogging on the sidewalk at 3.05 m/s on the way to her physics class. Bruce realizes that she forgot her bag of bagels, so he runs to the window, which is 38.0 m above the street level and directly above the sidewalk, to throw the bag to her. He throws it horizontally 9.00 s after she has passed below the window, and she catches it on the run. Ignore air resistance. (a) With what initial speed must Bruce throw the bagels so that Henrietta can catch the bag just before it hits the ground? (b) Where is Henrietta when she catches the bagels?

A 76.0 kg rock is rolling horizontally at the top of a vertical cliff that is 20 m above the surface of a lake (Fig. P3.65). The top of the vertical face of a dam is located 100 m from the foot of the cliff, with the top of the dam level with the surface of the water in the lake. A level plain is 25 m below the top of the dam. (a) What must be the minimum speed of the rock just as it leaves the cliff so that it will reach the plain without striking the dam? (b) How far from the foot of the dam does the rock hit the plain?

A firefighting crew uses a water cannon that shoots water at 25.0 m/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 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.)

On a level football field a football is projected from ground level. It has speed 8.0 m/s when it is at its maximum height. It travels a horizontal distance of 50.0 m. Neglect air resistance. How long is the ball in the air?

You are standing on a loading dock at the top of a flat ramp that is at a constant angle \(\alpha_0\) below the horizontal. You slide a small box horizontally off the loading dock with speed \(v_0\) in a direction so that it lands on the ramp. How far vertically downward does the box travel before it strikes the ramp?

In the middle of the night you are standing a horizontal distance of 14.0 m from the high fence that surrounds the estate of your rich uncle. The top of the fence is 5.00 m above the ground. You have taped an important message to a rock that you want to throw over the fence. The ground is level, and the width of the fence is small enough to be ignored. You throw the rock from a height of 1.60 m above the ground and at an angle of \(56.0^{\circ}\) above the horizontal. (a) What minimum initial speed must the rock have as it leaves your hand to clear the top of the fence? (b) For the initial velocity calculated in part (a), what horizontal distance beyond the fence will the rock land on the ground?

A small object is projected from level ground with an initial velocity of magnitude 16.0 m/s and directed at an angle of \(60.0^{\circ}\) above the horizontal. (a) What is the horizontal displacement of the object when it is at its maximum height? How does your result compare to the horizontal range R of the object? (b) What is the vertical displacement of the object when its horizontal displacement is 80.0% of its horizontal range R? How does your result compare to the maximum height \(h_{\text{max}}\) reached by the object? (c) For when the object has horizontal displacement \(x - x_0 = \alpha R\), where \(\alpha\) is a positive constant, derive an expression (in terms of \(\alpha\)) for \((y - y_0)/h_{\text{max}}\). Your result should not depend on the initial velocity or the angle of projection. Show that your expression gives the correct result when \(\alpha = 0.80\), as is the case in part (b). Also show that your expression gives the correct result for \(\alpha = 0, \alpha = 0.50\), and \(\alpha = 1.0\).

An airplane pilot sets a compass course due west and maintains an airspeed of 220 km/h. After flying for 0.500 h, she finds herself over a town 120 km west and 20 km south of her starting point. (a) Find the wind velocity (magnitude and direction). (b) If the wind velocity is 40 km/h due south, in what direction should the pilot set her course to travel due west? Use the same airspeed of 220 km/h.

Raindrops. When a train’s velocity is 12.0 m/s eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined \(30.0^{\circ}\) to the vertical on the windows of the train. (a) What is the horizontal component of a drop’s velocity with respect to the earth? With respect to the train? (b) What is the magnitude of the velocity of the raindrop with respect to the earth? With respect to the train?

In a World Cup soccer match, Juan is running due north toward the goal with a speed of 8.00 m/s relative to the ground. A teammate passes the ball to him. The ball has a speed of 12.0 m/s and is moving in a direction \(37.0^{\circ}\) east of north, relative to the ground. What are the magnitude and direction of the ball’s velocity relative to Juan?

A shortstop is running due east as he throws a baseball to the catcher, who is standing at home plate. The velocity of the baseball relative to the shortstop is 6.00 m/s in the direction due south, and the speed of the baseball relative to the catcher is 9.00 m/s. What is the speed of the shortstop relative to the ground when he throws the ball?

Two soccer players, Mia and Alice, are running as Alice passes the ball to Mia. Mia is running due north with a speed of 6.00 m/s. The velocity of the ball relative to Mia is 5.00 m/s in a direction \(30.0^{\circ}\) east of south. What are the magnitude and direction of the velocity of the ball relative to the ground?

DATA A spring-gun projects a small rock from the ground with speed \(v_0\) at an angle \(\theta_0\) above the ground. You have been asked to determine \(v_0\). From the way the spring-gun is constructed, you know that to a good approximation \(v_0\) is independent of the launch angle. You go to a level, open field, select a launch angle, and measure the horizontal distance the rock travels. You use \(g = 9.80 \ \mathrm{m/s}^2\) and ignore the small height of the end of the spring-gun’s barrel above the ground. Since your measurement includes some uncertainty in values measured for the launch angle and for the horizontal range, you repeat the measurement for several launch angles and obtain the results given in Fig. 3.76. You ignore air resistance because there is no wind and the rock is small and heavy. (a) Select a way to represent the data well as a straight line. (b) Use the slope of the best straight line fit to your data from part (a) to calculate \(v_0\). (c) When the launch angle is \(36.9^{\circ}\), what maximum height above the ground does the rock reach?

DATA You have constructed a hairspray-powered potato gun and want to find the muzzle speed \(v_0\) of the potatoes, the speed they have as they leave the end of the gun barrel. You use the same amount of hair spray each time you fire the gun, and you have confirmed by repeated firings at the same height that the muzzle speed is approximately the same for each firing. You climb on a microwave relay tower (with permission, of course) to launch the potatoes horizontally from different heights above the ground. Your friend measures the height of the gun barrel above the ground and the range R of each potato. You obtain the data in the table. Each of the values of h and R has some measurement error: The muzzle speed is not precisely the same each time, and the barrel isn’t precisely horizontal. So you use all of the measurements to get the best estimate of \(v_0\). No wind is blowing, so you decide to ignore air resistance. You use \(g = 9.80 \ \mathrm{m/s}^2\) in your analysis. (a) Select a way to represent the data well as a straight line. (b) Use the slope of the best-fit line from part (a) to calculate the average value of \(v_0\). (c) What would be the horizontal range of a potato that is fired from ground level at an angle of \(30.0^{\circ}\) above the horizontal? Use the value of \(v_0\) that you calculated in part (b).

DATA You are a member of a geological team in Central Africa. Your team comes upon a wide river that is flowing east. You must determine the width of the river and the current speed (the speed of the water relative to the earth). You have a small boat with an outboard motor. By measuring the time it takes to cross a pond where the water isn’t flowing, you have calibrated the throttle settings to the speed of the boat in still water. You set the throttle so that the speed of the boat relative to the river is a constant 6.00 m/s. Traveling due north across the river, you reach the opposite bank in 20.1 s. For the return trip, you change the throttle setting so that the speed of the boat relative to the water is 9.00 m/s. You travel due south from one bank to the other and cross the river in 11.2 s. (a) How wide is the river, and what is the current speed? (b) With the throttle set so that the speed of the boat relative to the water is 6.00 m/s, what is the shortest time in which you could cross the river, and where on the far bank would you land?

CALC A projectile thrown from a point P moves in such a way that its distance from P is always increasing. Find the maximum angle above the horizontal with which the projectile could have been thrown. Ignore air resistance.

Two students are canoeing on a river. While heading upstream, they accidentally drop an empty bottle overboard. They then continue paddling for 60 minutes, reaching a point 2.0 km farther upstream. At this point they realize that the bottle is missing and, driven by ecological awareness, they turn around and head downstream. They catch up with and retrieve the bottle (which has been moving along with the current) 5.0 km downstream from the turnaround point. (a) Assuming a constant paddling effort throughout, how fast is the river flowing? (b) What would the canoe speed in a still lake be for the same paddling effort?

CP A rocket designed to place small payloads into orbit is carried to an altitude of 12.0 km above sea level by a converted airliner. When the airliner is flying in a straight line at a constant speed of 850 km/h, the rocket is dropped. After the drop, the airliner maintains the same altitude and speed and continues to fly in a straight line. The rocket falls for a brief time, after which its rocket motor turns on. Once that motor is on, the combined effects of thrust and gravity give the rocket a constant acceleration of magnitude 3.00g directed at an angle of \(30.0^{\circ}\) above the horizontal. For safety, the rocket should be at least 1.00 km in front of the airliner when it climbs through the airliner’s altitude. Your job is to determine the minimum time that the rocket must fall before its engine starts. Ignore air resistance. Your answer should include (i) a diagram showing the flight paths of both the rocket and the airliner, labeled at several points with vectors for their velocities and accelerations; (ii) an x-t graph showing the motions of both the rocket and the airliner; and (iii) a y-t graph showing the motions of both the rocket and the airliner. In the diagram and the graphs, indicate when the rocket is dropped, when the rocket motor turns on, and when the rocket climbs through the altitude of the airliner.

The experiment is designed so that the seeds move no more than 0.20 mm between photographic frames. What minimum frame rate for the high-speed camera is needed to achieve this? (a) 250 frames/s; (b) 2500 frames/s; (c) 25,000 frames/s; (d) 250,000 frames/s.

About how long does it take a seed launched at \(90^{\circ}\) at the highest possible initial speed to reach its maximum height? Ignore air resistance. (a) 0.23 s; (b) 0.47 s; (c) 1.0 s; (d) 2.3 s.

If a seed is launched at an angle of \(0^{\circ}\) with the maximum initial speed, how far from the plant will it land? Ignore air resistance, and assume that the ground is flat. (a) 20 cm; (b) 93 cm; (c) 2.2 m; (d) 4.6 m.

A large number of seeds are observed, and their initial launch angles are recorded. The range of projection angles is found to be \(-51^{\circ}\) to \(75^{\circ}\), with a mean of \(31^{\circ}\). Approximately 65% of the seeds are launched between \(6^{\circ}\) and \(56^{\circ}\). (See W. J. Garrison et al., “Ballistic seed projection in two herbaceous species,” Amer. J. Bot., Sept. 2000, 87:9, 1257–64.) Which of these hypotheses is best supported by the data? Seeds are preferentially launched (a) at angles that maximize the height they travel above the plant; (b) at angles below the horizontal in order to drive the seeds into the ground with more force; (c) at angles that maximize the horizontal distance the seeds travel from the plant; (d) at angles that minimize the time the seeds spend exposed to the air.