Mechanics

A snowball rolls off a barn roof that slopes downward at an angle of $\mathbf{40}\mathbf{°}$ (Fig. P3.59) The edge of the roof is 14.0 m above the ground, and the snowball has a speed of 7.0m/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 $\mathit{x}\mathbf{-}\mathit{t}\mathbf{,}\mathit{y}\mathbf{-}\mathit{t}\mathbf{,}{\mathit{v}}_{\mathbf{x}}\mathbf{-}\mathit{t}\mathbf{,}{\mathit{v}}_{\mathbf{y}}\mathbf{-}\mathit{t}$ graphs for the motion in part (a). (c) A man 1.9 m tall is standing from the edge of the barn. Will the snowball hit him?

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 8.50m/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?

Suppose that the boy in Problem 3.60 throws the ball upward at 60^o above the horizontal, but all else is the same. Repeat parts (a) and  (b) of that problem.

Problem 3.60: 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 m/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 rock is thrown with a velocity ${\mathbf{V}}_{\mathbf{0}}$, at an angle of ${\mathbf{\alpha }}_{\mathbf{0}}$ from the horizontal, from the roof of a building of height h. Ignore air resistance. Calculate the speed of the rock just before it strikes the ground, and show that this speed is independent of ${\mathit{\alpha }}_{\mathbf{0}}$.

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 take off ramp was inclined at $\mathbf{53}\mathbf{°}$, the river was $\mathbf{40}\mathbf{.}\mathbf{0}\mathit{m}$ wide, and the far bank was $\mathbf{15}\mathbf{.}\mathbf{0}\mathit{m}$  lower than the top of the ramp. The river itself was $\mathbf{100}\mathit{m}$ 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?

A spring-gun projects a small rock from the ground with speed ${\mathit{v}}_{\mathbf{0}}$ at an angle  $\mathit{\theta }\mathbf{°}$above the ground. You have been asked to determine${\mathit{v}}_{\mathbf{0}}$  . From the way the spring-gun is constructed, you know that to a good approximation  ${\mathit{v}}_{\mathbf{0}}$ 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$\mathit{g}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{80}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$  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${\mathit{v}}_{\mathbf{0}}$ . (c) When the launch angle is$\mathbf{36}\mathbf{.}\mathbf{9}\mathbf{°}$  , what maximum height above the ground does the rock reach?

World-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude $15m/s^2$ . How much horizontal force must a 55-kg sprinter exert on the starting blocks to produce this acceleration? Which body exerts the force that propels the sprinter: the blocks

or the sprinter herself?

A 2.7kKg ball is thrown upward with an initial speed of 20m/s from the edge of a 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 60m/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.

A 76.0 Kg rock is rolling horizontally at the top of a vertical cliff that is above the surface of a lake (Fig. P3.65). The top of the vertical face of a dam is located 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 25m 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?

Henrietta is jogging on the sidewalk at 3.05m/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 38.0m above the street level and directly above the sidewalk, to throw the bag to her. He throws it horizontally 9.0s 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 cart carrying a vertical missile launcher moves horizontally at a constant velocity of 30.0m/s to the right. It launches a rocket vertically upward. The missile has an initial vertical velocity of 40.0m/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?

A fire fighting crew uses a water cannon that shoots water at 25.0m/s at a fixed angle of $\mathbf{53}\mathbf{.}\mathbf{0}\mathbf{°}$ above the horizontal. The firefighters want to direct the water at a blaze that is above ground level. How far from the building should they position their cannon? There are two possibilities; can you get them both? 

In the middle of the night you are standing a horizontal distance of 14.0m from the high fence that surrounds the estate of your rich uncle. The top of the fence is 5.00m 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 1.60m above the ground and at an angle of 56.$\mathbf{0}\mathbf{°}$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 student sits atop a platform a distance h above the ground. He throws a large firecracker horizontally with a speed v. However, a wind blowing parallel to the ground gives the firecracker a constant horizontal acceleration with magnitude a. As a result, the firecracker reaches the ground directly below the student. Determine the height h in terms of v, a, and g. Ignore the effect of air resistance on the vertical motion.

A student sits atop a platform a distance h above the ground. He throws a large firecracker horizontally with a speed v. However, a wind blowing parallel to the ground gives the firecracker a constant horizontal acceleration with magnitude a. As a result, the firecracker reaches the ground directly below the student. Determine the height h in terms of v, a, and g. Ignore the effect of air resistance on the vertical motion.

An airplane pilot sets a compass course due west and maintains an airspeed of $\mathbf{220}\mathbf{ }\mathit{K}\mathit{m}\mathbf{/}\mathit{h}\mathbf{ }$. After flying for $\mathbf{0}\mathbf{.}\mathbf{500}\mathit{h}$, she finds herself over a town $\mathbf{120}\mathbf{ }\mathit{K}\mathit{m}$ west and $\mathbf{20}\mathbf{ }\mathit{K}\mathit{m}$ south of her starting point. (a) Find the wind velocity (magnitude and direction). (b) If the wind velocity is $\mathbf{40}\mathbf{ }\mathit{K}\mathit{m}\mathbf{/}\mathit{h}$due south, in what direction should the pilot set her course to travel due west? Use the same airspeed of $\mathbf{220}\mathbf{ }\mathit{K}\mathit{m}\mathbf{/}\mathit{h}\mathbf{.}$

When a train’s velocity is 12.0m/s eastward, raindrops that are falling vertically with respect to the earth make traces that are inclined $\mathbf{30}\mathbf{°}\mathbf{0}$ 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 8.0 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 $\mathbf{37}\mathbf{.}\mathbf{0}\mathbf{°}$ east of north, relative to the ground. What are the magnitude and direction of the ball’s velocity relative to Juan?

An elevator is moving upward at a constant speed of $\mathbf{2}\mathbf{.}\mathbf{50}\mathit{m}\mathbf{/}\mathit{s}$ . A bolt in the elevator ceiling above the elevator floor works loose and falls. (a) How long does it take for the bolt to fall to the elevator floor? What is the speed of the bolt just as it hits the elevator floor (b) according to an observer in the elevator? (c) According to an observer standing on one of the floor landings of the building? (d) According to the observer in part (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

An elevator is moving upward at a constant speed of 2.50m/s . A bolt in the elevator ceiling 3.00m above the elevator floor works loose and falls. (a) How long does it take for the bolt to fall to the elevator floor? What is the speed of the bolt just as it hits the elevator floor (b) according to an observer in the elevator? (c) According to an observer standing on one of the floor landings of the building? (d) According to the observer in part (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

Two soccer players, Mia and Alice, are running as Alice passes the ball to Mia. Mia is running due north with a speed of 6.00m/s . The velocity of the ball relative to Mia is 5.00m/s in a direction $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{°}$ east of south. What are the magnitude and direction of the velocity of the ball relative to the ground?

You have constructed a hair-spray-powered potato gun and want to find the muzzle speed ${\mathit{v}}_{\mathbf{o}}$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 following data: 

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 ${\mathit{v}}_{\mathbf{o}}$. No wind is blowing, so you decide to ignore air resistance. You use$\mathit{g}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{8}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ 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${\mathit{v}}_{\mathbf{o}}$ . (c) What would be the horizontal range of a potato that is fired from ground level at an angle of $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{°}$above the horizontal? Use the value of ${\mathit{v}}_{\mathbf{o}}$that you calculated in part (b).

Question: 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  . Traveling due north across the river, you reach the opposite bank in  . For the return trip, you change the throttle setting so that the speed of the boat relative to the water is 9.00m/s  . You travel due south from one bank to the other and cross the river in  . (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?

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° 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° 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° to 75°, with a mean of 31°. Approximately 65% of the seeds are launched between 6° and 56°. (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.

Dancers experience large forces associated with the jumps they make. For example, when a dancer lands after a vertical jump, the force exerted on the head by the neck must exceed the head’s weight by enough to cause the head to slow down and come to rest. The head is about 9.4% of a typical person’s mass. Video analysis of a$\mathbf{65}\mathbf{-}\mathbf{kg}$ dancer landing after a vertical jump shows that her head decelerates from 4.0m/s to rest in a time of 0.20s.

Compared with the force her neck exerts on her head during the landing, the force her head exerts on her neck is (a) the same; (b) greater; (c) smaller; (d) greater during the first half of the landing and smaller during the second half of the landing.

Can a body be in equilibrium when only one force acts on it? Explain.

A ball thrown straight up has zero velocity at its highest point. Is the ball in equilibrium at this point? Why or why not?

A helium balloon hovers in mid-air, neither ascending nor descending. Is it in equilibrium? What forces act on it?

A manual for student pilots contains this passage: “When an airplane flies at a steady altitude, neither climbing nor descending, the upward lift force from the wings equals the plane’s weight. When the plane is climbing at a steady rate, the upward lift is greater than the weight; when the plane is descending at a steady rate, the upward lift is less than the weight.” Are these statements correct? Explain.

If your hands are wet and no towel is handy, you can remove some of the excess water by shaking them. Why does this work?

If you squat down (such as when you examine the books on a bottom shelf) and then suddenly get up, you may temporarily feel light-headed. What do Newton’s laws of motion have to say about why this happens?

When a car is hit from behind, the occupants may experience whiplash. Use Newton’s laws of motion to explain what causes this result.

When you fly in an airplane at night in smooth air, you have no sensation of motion, even though the plane may be moving at 800 km/h (500 mi/h). Why?

If the two ends of a rope in equilibrium are pulled with forces of equal magnitude and opposite directions, why isn’t the total tension in the rope zero?

You tie a brick to the end of a rope and whirl the brick around you in a horizontal circle. Describe the path of the brick after you suddenly let go of the rope.

When a car stops suddenly, the passengers tend to move forward relative to their seats. Why? When a car makes a sharp turn, the passengers tend to slide to one side of the car. Why?

Some people say that the “force of inertia” (or “force of momentum”) throws the passengers forward when a car brakes sharply. What is wrong with this explanation?

A passenger in a moving bus with no windows notices that a ball that has been at rest in the aisle suddenly starts to move toward the rear of the bus. Think of two possible explanations, and devise a way to decide which is correct

Suppose you chose the fundamental physical quantities to be force, length, and time instead of mass, length, and time. What would be the units of mass in terms of those fundamental quantities?

Why is the earth only approximately an inertial reference frame?

Does Newton’s second law hold true for an observer in a van as it speeds up, slows down, or rounds a corner? Explain.

Some students refer to the quantity ma S as “the force of acceleration.” Is it correct to refer to this quantity as a force? If so, what exerts this force? If not, what is a better description of this quantity?

The acceleration of a falling body is measured in an elevator that is traveling upward at a constant speed of 9.8 m/s. What value is obtained?

You can play catch with a softball in a bus moving with constant speed on a straight road, just as though the bus were at rest. Is this still possible when the bus is making a turn at constant speed on a level road? Why or why not?

Students sometimes say that the force of gravity on an object is 9.8 $\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. What is wrong with this view?

Why can it hurt your foot more to kick a big rock than a small pebble? Must the big rock hurt more? Explain.

“It’s not the fall that hurts you; it’s the sudden stop at the bottom.” Translate this saying into the language of Newton’s laws of motion.