College Physics 2013

Jay fills a wagon with sand (about 20 kg) and pulls it with a rope 30 $\mathrm{m}$ along the beach. He holds the rope $25^{\circ}$ above the horizontal. The rope exerts a $20-\mathrm{N}$ tension force on the wagon. How much work does the rope do on the wagon?

You have a 15 -kg suitcase and (a) slowly lift it 0.80 $\mathrm{m}$ upward, (b) hold it at rest to test whether you will be able to move the suitcase without help in the airport, and then (c) lower it 0.80 $\mathrm{m}$ . What work did you do in each case? What assumptions did you make to solve this problem?

You use a rope to slowly pull a sled and its passenger 50 $\mathrm{m}$ up a $20^{\circ}$ incline, exerting a $150-\mathrm{N}$ force on the rope. ( a ) How much work will you do if you pull parallel to the hill? (b) How much work will you do if you exert the same magnitude force while slowly lowering the sled back down the hill and pulling parallel to the hill? (c) How much work did Earth do on the sled for the trip in part (b)?

A rope attached to a truck pulls a 180 -kg motorcycle at 9.0 $\mathrm{m} / \mathrm{s}$ . The rope exerts a $400-\mathrm{N}$ force on the motorcycle at an angle of $15^{\circ}$ above the horizontal. (a) What is the work that the rope does in pulling the motorcycle 300 $\mathrm{m} ?$ (b) How will your answer change if the speed is 12 $\mathrm{m} / \mathrm{s} ?$ (c) How will your answer change if the truck accelerates?

You lift a 25-kg child 0.80 m, slowly carry him 10 m to the playroom, and finally set him back down 0.80 m onto the playroom floor. What work do you do on the child for each part of the trip and for the whole trip? List your assumptions.

A truck runs into a pile of sand, moving 0.80 $\mathrm{m}$ as it slows to a stop. The magnitude of the work that the sand does on the truck is $6.0 \times 10^{5} \mathrm{J} .$ (a) Determine the average force that the sand exerts on the truck. (b) Did the sand do positive or negative work? (c) How does the average force change if the stopping distance is doubled? Indicate any assumptions you made.

A 5.0 -kg rabbit and a 12 -kg Irish setter have the same kinetic energy. If the setter is running at speed 4.0 $\mathrm{m} / \mathrm{s}$ , how fast is the rabbit running?

Estimate your average kinetic energy when walking to physics class. What assumptions did you make?

A pickup truck (2268 kg) and a compact car (1100 kg) have the same momentum. (a) What is the ratio of their kinetic energies? (b) If the same horizontal net force were exerted on both vehicles, pushing them from rest over the same distance, what is the ratio of their final kinetic energies?

When does the kinetic energy of a car change more: when the car accelerates from 0 to 10 m/s or from 30 m/s to 40 m/s? Explain.

When exiting the highway, a 1100 -kg car is traveling at 22 $\mathrm{m} / \mathrm{s}$ . The car's kinetic energy decreases by $1.4 \times 10^{5} \mathrm{J}$ . The exit's speed limit is 35 $\mathrm{mi} / \mathrm{h}$ . Did the driver reduce the car's speed enough? Explain.

When exiting the highway, a 1100 -kg car is traveling at 22 $\mathrm{m} / \mathrm{s}$ . The car's kinetic energy decreases by $1.4 \times 10^{5} \mathrm{J} .$ The exit's speed limit is 35 $\mathrm{mi} / \mathrm{h}$ . Did the driver reduce the car's speed enough? Explain.

Flea jump $A$ 5.4 $\times 10^{-7}$ -kg flea pushes off a surface by extending its rear legs for a distance of slightly more than $2.0 \mathrm{mm},$ consequently jumping to a height of 40 $\mathrm{cm} .$ What physical quantities can you determine using this information? Make a list and determine three of them.

Roller coaster ride A roller coaster car drops a maximum vertical distance of 35.4 m. (a) Determine the maximum speed of the car at the bottom of that drop. (b) Describe any assumptions you made. (c) Will a car with twice the mass have more or less speed at the bottom? Explain.

Heart pumps blood The heart does about 1 J of work while pumping blood into the aorta during each
heartbeat. (a) Estimate the work done by the heart in pumping blood during a lifetime. (b) If all of that work was used to lift a person, to what height could an average person be lifted? Indicate any assumptions you used for each part of the problem.

Wind energy Air circulates across Earth in regular patterns. A tropical air current called the Hadley cell carries about $2 \times 10^{11} \mathrm{kg}$ of air per second past a cross section of Earth's atmosphere while moving toward the equator. The average air speed is about 1.5 $\mathrm{m} / \mathrm{s}$ . (a) What is the kinetic energy of the air that passes the cross section each second? (b) About $1 \times 10^{20} \mathrm{J}$ of energy was consumed in the United States in 2005 . What is the ratio of the kinetic energy of the air that passes toward the equator each second and the energy consumed in the United States each second?

Bone break The tibia bone in the lower leg of an adult human will break if the compressive force on it exceeds about $4 \times 10^{5} \mathrm{N}$ (we assume that the ankle is pushing up). Suppose that you step off a chair that is 0.40 $\mathrm{m}$ above the floor. If landing stiff-legged on the surface below, what minimum stopping distance do you need to avoid breaking your tibias? Indicate any assumptions you made in your answer to this question.

Climbing Mt. Everest In 1953 Sir Edmund Hillary and Tenzing Norgay made the first successful ascent of Mt. Everest. How many slices of bread did each climber have to eat to compensate for the increase of the gravitational potential energy of the system climbers-Earth? (One piece of bread releases about $1.0 \times 10^{6} \mathrm{J}$ of energy in the body. Indicate all of the assumptions used. Note: The body is an inefficient energy converter see the reading passage at the end of this section.

A door spring is difficult to stretch. (a) What maximum force do you need to exert on a relaxed spring with a $1.2 \times 10^{4}-\mathrm{N} / \mathrm{m}$ spring constant to stretch it 6.0 $\mathrm{cm}$ from its equilibrium position? (b) How much does the elastic potential energy of the spring change? (c) Determine its change in elastic potential energy as it returns from the $6.0-\mathrm{cm}$ stretch
position to a 3.0 -cm stretch position. (d) Determine its elastic potential energy change as it moves from the $3.0-\mathrm{cm}$ stretch position back to its equilibrium position.

You compress a spring by a certain distance. Then you decide to compress it further so that its elastic potential energy increases by another 50%. What was the percent increase in the spring’s compression distance?

A moving car has 40,000 J of kinetic energy while moving at a speed of 7.0 m/s. A spring-loaded automobile bumper compresses 0.30 m when the car hits a wall and stops. What can you learn about the bumper’s spring using this information? Answer quantitatively and list the assumptions that you made.

The force required to stretch a slingshot by different amounts is shown in the graph in Figure P6.22. (a) What is the spring constant of the sling? (b) How much work does a child need to do to stretch the sling 15 cm from equilibrium?

Inverse bungee jump The Ejection Seat at Lake Biwa Amusement Park in Japan (Figure P6.23) is an inverse bungee system. A seat with passengers of total mass 160 kg is connected to elastic cables on the sides. The seat is pulled down 12 m, stretching the cables. When released, the stretched cables launch the passengers upward above the towers to a height of about 30 m above their starting position at the ground. What can you learn about the mechanical properties of the cables using this information?
Answer quantitatively. Assume that the cables are vertical.

Jim is driving a 2268-kg pickup truck at 20 m/s and releases his foot from the accelerator pedal. The car eventually stops due to an effective friction force that the road, air, and other things exert on the car. The friction force has an average magnitude of 800 N. (a) Make a list of the physical quantities you can determine using this information and determine three of them. Specify the system and the initial and final states. (b) Would a heavier car travel farther before stopping or stop sooner? Identify the assumptions in your answer.

A 1100 -kg car traveling at 24 $\mathrm{m} / \mathrm{s}$ coasts through some wet mud in which the net horizontal resistive force exerted on the car from all causes (mostly the force exerted by the mud) is $1.7 \times 10^{4} \mathrm{N}$ . Determine the car's speed as it leaves the 18 -m-long patch of mud.

After falling 18 m, a 0.057-kg tennis ball has a speed of 12 m/s (the ball’s initial speed is zero). Determine the average resistive force of the air in opposing the ball’s motion. Solve the problem for the ball as your system and then repeat for a ball-Earth system. Are the answers the same or different?

A water slide of length $l$ has a vertical drop of $h .$ Abby's mass is $m .$ An average friction force of magnitude $f$ opposes her motion. She starts down the slide at initial speed $v_{1}$ . Use work-energy ideas to develop an expression for her speed at the bottom of the slide. Then evaluate your result using unit analysis and limiting case analysis.

You are pulling a crate on a rug, exerting a constant force on the crate $\vec{F}_{Y \text { on } C}$ at an angle $\theta$ above the horizontal. The crate moves at constant speed. Represent this process using a motion diagram, a force diagram, a momentum bar chart, and an energy bar chart. Specify your choice of system for each representation. Make a list of physical quantities you can determine using this information.

A 900 -kg car initially at rest rolls 50 $\mathrm{m}$ down a hill inclined at an angle of $5.0^{\circ} .$ A 400 -N effective friction force opposes its motion. How fast is the car moving at the bottom? What distance will it travel on a similar horizontal surface at the bottom of the hill? Will the distance decrease or increase if the car's mass is 1800 $\mathrm{kg}$ ?

A car skids 18 m on a level road while trying to stop before hitting a stopped car in front of it. The two cars barely touch. The coefficient of kinetic friction between the first car and the road is 0.80. A policewoman gives the driver a ticket for exceeding the 35 mi/h speed limit. Can you defend the driver in court? Explain.

In a popular new hockey game, the players use small launchers with springs to move the 0.0030-kg puck. Each spring has a 120-N/m spring constant and can be compressed up to 0.020 m. What can you determine about the motion of the puck using this information? Make a list of quantities and determine their values.

A 500 -m-long ski slope drops at an angle of $6.4^{\circ}$ relative to the horizontal. (a) Determine the change in gravitational potential energy of a 60 -kg skier-Earth system when the skier goes down this slope. (b) If 20$\%$ of the gravitational potential energy change is converted into kinetic energy, how fast is the skier traveling at the bottom of the slope?

A Frisbee gets stuck in a tree. You want to get it out by throwing a 1.0-kg rock straight up at the Frisbee. If the rock’s speed as it reaches the Frisbee is 4.0 m/s, what was its speed as it left your hand 2.8 m below the Frisbee? Specify the system and the initial and final states.

A driver loses control of a car, drives off an embankment, and lands in a canyon 6.0 m below. What was the car’s speed just before touching the ground if it was traveling on the level surface at 12 m/s before the driver lost control?

You are pulling a box so it moves at increasing speed. Compare the work you need to do to accelerate it from 0 $\mathrm{m} / \mathrm{s}$ to speed $v$ to the work needed to accelerate it from speed $v$ to the speed of 2$v .$ Discuss whether your answer makes sense. How many different situations do you need to consider?

A cable lowers a 1200 -kg elevator so that the elevator's speed increases from zero to 4.0 $\mathrm{m} / \mathrm{s}$ in a vertical distance of 6.0 $\mathrm{m} .$ What is the force that the cable exerts on the elevator while lowering it? Specify the system, its initial and final states, and any assumptions you made. Then change the system and solve the problem again. Do the answers match?

Hit by a hailstone A 0.040-kg hailstone the size of a golf ball (4.3 cm in diameter) is falling at about 16 m/s when it reaches Earth’s surface. Estimate the force that the hailstone exerts on your head—a head-on collision. Indicate any assumptions used in your estimate. Note that the cheekbone will break if something exerts a 900-N or larger force on the bone for more than 6 ms. Is this hailstone likely to break a bone?

Froghopper jump Froghoppers may be the insect jumping champs. These 6-mm-long bugs can spring 70 cm into the air, about the same distance as the flea. But the froghopper is 60 times more massive than a flea, at 12 mg. The froghopper pushes off for about 4 mm. What average force does it exert on the surface? Compare this to the gravitational force that Earth exerts on the bug.

Bar chart Jeopardy 1 Invent in words and with a sketch a process that is consistent with the qualitative work-energy bar chart shown in Figure P6.39. Then apply in symbols the generalized work-energy principle for that process.

Bar chart Jeopardy 2 Invent in words and with a sketch a process that is consistent with the qualitative work-energy bar chart shown in Figure P6.40. Then apply in symbols the generalized work-energy principle for that process.

Equation Jeopardy 1 Construct a qualitative work-energy bar chart for a process that is consistent with the equation below. Then invent in words and with a sketch a process that is consistent with both the equation and the bar chart.
$$
\begin{aligned}(1 / 2)(400 \mathrm{N} / \mathrm{m})(0.20 \mathrm{m})^{2} &=(1 / 2)(0.50 \mathrm{kg}) v^{2} \\ &+(0.50 \mathrm{kg})\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)(0.80 \mathrm{m}) \end{aligned}
$$

Equation Jeopardy 2 Construct a qualitative work-energy bar chart for a process that is consistent with the equation below. Then invent in words and with a sketch a process that is consistent with both the equation and the bar chart.
$$
\begin{array}{l}{(120 \mathrm{kg})\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)(100 \mathrm{m}) \sin 53^{\circ}} \\ {\quad=(1 / 2)(120 \mathrm{kg})(20 \mathrm{m} / \mathrm{s})^{2}+f_{\mathrm{k}}(100 \mathrm{m})}\end{array}
$$

Evaluation 1 Your friend provides a solution to the following problem. Evaluate his solution. Constructively identify any mistakes he made and correct the solution. Explain possible reasons for the mistakes. The problem: A 400-kg motorcycle, including the driver, travels up a $10-\mathrm{m}$ -long ramp inclined $30^{\circ}$ above the paved horizontal surface holding the ramp. The cycle leaves the ramp at speed 20 $\mathrm{m} / \mathrm{s}$ . Determine the cycle's speed just before it lands on the paved surface. Your friend's solution:
$$
\begin{aligned}(1 / 2)(400 \mathrm{kg})(20 \mathrm{m} / \mathrm{s})=&(400 \mathrm{kg})\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)(10 \mathrm{m}) \\ &+(1 / 2)(400 \mathrm{kg}) v^{2} \\=&-13.2 \mathrm{m} / \mathrm{s} \end{aligned}
$$

Evaluation 2 Your friend provides a solution to the following problem. Evaluate her solution. Constructively identify any mistakes she made and correct the solution. Explain possible reasons for the mistakes. The problem: Jim (mass 50 kg) steps off a ledge that is 2.0 m above a platform that sits on top of a relaxed spring of force constant 8000 N/m. How far will the spring compress while stopping Jim? Your friend’s solution:
$$
\begin{aligned}(50 \mathrm{kg})\left(9.8 \mathrm{m} / \mathrm{s}^{2}\right)(2.0 \mathrm{m}) &=(1 / 2)(8000 \mathrm{N} / \mathrm{m}) x \\ x &=0.25 \mathrm{m} \end{aligned}
$$

A puck of mass $m$ moving at speed $v_{i}$ on a horizontal, frictionless surface is stopped in a distance $\Delta x$ because a hockey stick exerts an opposing force of magnitude $F$ on it. (a) Using
the work-energy method, show that $F=m v_{i}^{2} / 2 \Delta x .(b)$ If the stopping distance $\Delta x$ increases by $50 \%,$ by what percent does the average force needed to stop the puck change, assuming that $m$ and $v_{\mathrm{i}}$ are unchanged? Justify your result.

A rope exerts an $18-\mathrm{N}$ force while lowering a 20 -kg crate down a plane inclined at $20^{\circ}$ (the rope is parallel to the plane). A $24-\mathrm{N}$ friction force opposes the motion. The crate starts at rest and moves 10 $\mathrm{m}$ down the plane. Make a list of the physical quantities you can determine using this information and determine three of them. Specify the system and the initial and final states of the process.

You fire an 80-g arrow so that it is moving at 80 m/s when it hits and embeds in a 10-kg block resting on ice. (a) What is the velocity of the block and arrow just after the collision? (b) How far will the block slide on the ice before stopping? A 7.2-N friction force opposes its motion. Specify the system and the initial and final states for (a) and (b).

You fire a 50-g arrow that moves at an unknown speed. It hits and embeds in a 350-g block that slides on an air track. At the end, the block runs into and compresses a 4000-N/m spring 0.10 m. How fast was the arrow traveling? Indicate the assumptions that you made and discuss how they affect the result.

To confirm the results of Problem 48, you try a new experiment. The 50-g arrow is launched in an identical manner so that it hits and embeds in a 3.50-kg block. The block hangs from strings. After the arrow joins the block, they swing up so that they are 0.50 m higher than the block’s starting point. How fast was the arrow moving before it joined the block?

A 1060 -kg car moving west at 16 $\mathrm{m} / \mathrm{s}$ collides with and locks onto a 1830 -kg stationary car. (a) Determine the velocity of the cars just after the collision. (b) After the collision, the road exerts a $1.2 \times 10^{4}-\mathrm{N}$ friction force on the car tires. How far do the cars skid before stopping? Specify the system and the initial and final states of the process.

Jay rides his 2.0-kg skateboard. He is moving at speed 5.8 m/s when he pushes off the board and continues to move forward in the air at 5.4 m/s. The board now goes forward at 13 m/s. Determine Jay’s mass and the change in the internal energy of the system during this process.

A 36-kg child is moving on a 2.0-kg skateboard at speed 6.0 m/s when she comes to a ledge that is 1.2 m above the surface below. Just before reaching the ledge, she pushes off the board. The board leaves the ledge moving horizontally and lands 8.0 m horizontally from the edge of the ledge. Make a list of the physical quantities describing the motion of the child after leaving the ledge and determine two of them.
Describe any assumptions you made.

Falcons While perched on an elevated site, a peregrine falcon spots a flying pigeon. The falcon dives, reaching a speed of 90 m/s (200 mi/h). The falcon hits its prey with its feet, stunning or killing it, then swoops back around to catch it in mid-air. Assume that the falcon has a mass of 0.60 kg and hits a 0.20-kg pigeon almost head-on. The falcon’s speed after the collision is 60 m/s in the same direction. (a) Determine the final speed of the pigeon immediately after the hit.
(b) Determine the internal energy produced by the collision.
(c) Why does the falcon strike its prey with its feet and not head-on?

When you play billiards, can you predict the velocities of the billiard balls after a collision if you know the velocity of a moving ball before the collision? Assume that the collision is head-on and elastic and that rotational motion can be ignored.

A block of mass $m_{1}$ moving at speed $v$ toward the west on a frictionless surface has an elastic head-on collision with a second, stationary block of mass $m_{2} .$ Determine expressions for the final velocity of each block.

A 4.0-kg block moving at 2.0 m/s toward the west on a frictionless surface has an elastic head-on collision with a second 1.0-kg block traveling east at 3.0 m/s. Determine the final velocity of each block. (b) Determine the kinetic energy of each block before and after the collision. Note: The block with the least initial kinetic energy actually gains energy and the one with the most loses an equal amount. This is analogous to what happens when cool air comes into contact with warm air. The cool air warms (its molecules speed up) and the warm air cools (its molecules slow down).

(a) What is the power involved in lifting a 1.0 -kg object 1.0 $\mathrm{m}$ in 1.0 $\mathrm{s} ?(\mathrm{b})$ While lifting a 10 -kg object 1.0 $\mathrm{m}$ in 0.50$?(\mathrm{c})$ While lifting the 10 -kg object 2.0 $\mathrm{m}$ in 1.0 $\mathrm{s} ?$ (d) While lifting a 20 -kg object 1.0 $\mathrm{m}$ in 1.0 $\mathrm{s}$ ?

A fire engine must lift 30 kg of water a vertical distance of 20 m each second. What is the amount of power needed for the water pump for this fire hose?

Internal energy change while biking You set your stationary bike on a high 80-N friction-like resistive force and cycle for 30 min at a speed of 8.0 m/s. Your body is 10% efficient at converting chemical energy in your body into mechanical work. (a) What is your internal chemical energy change? (b) How long must you bike to convert $3.0 \times 10^{5} \mathrm{J}$ of chemical potential while staying at this speed? (This amount of energy equals the energy released by the body after eating three slices of bread.)

Tree evaporation A large tree can lose 500 kg of water a day. (a) How much work does the tree need to do to lift the water 8.0 m? (b) If the loss of water occurs over a 12-h period, what is the average power in watts needed to provide this increase in gravitational energy in the water-Earth system?

Climbing Mt. Mitchell An 82-kg hiker climbs to the summit of Mount Mitchell in western North Carolina. During one 2.0-h period, the climber’s vertical elevation increases 540 m. Determine (a) the change in gravitational potential energy of the climber-Earth system and (b) the power of the process needed to increase the gravitational energy.

Sears stair climb The fastest time for the Sears Tower (now Willis Tower) stair climb (103 flights, or 2232 steps) is about 20 min. (a) Estimate the mechanical power in watts for a top climber. Indicate any assumptions you made. (b) If the body is 20% efficient at converting chemical energy into mechanical energy, approximately how many joules and kilocalories of chemical energy does the body expend during the stair climb? Note: 1 food calorie = 1 kilocalorie = 4186 J.

ES T Exercising so you can eat ice cream You curl a 5.5-kg (12 lb) barbell that is hanging straight down in your hand up to your shoulder. (a) Estimate the work that your hand does in lifting the barbell. (b) Estimate the average mechanical power of the lifting process. Indicate any assumptions used in making the estimate. (c) Assuming the efficiency described at the end of Problem 62, how many times would you have to lift the barbell in order to burn enough calories to use up the energy absorbed by eating a 300-food-calorie dish of ice cream? (Problem 62 provides the joule equivalent of a food calorie.) List the assumptions that you made.

Salmon move upstream In the past, salmon would swim more than 1130 km (700 mi) to spawn at the headwaters of the Salmon River in central Idaho. The trip took about 22 days, and the fish consumed energy at a rate of 2.0 W for each kilogram of body mass. (a) What is the total energy used by a 3.0-kg salmon while making this 22@day trip? (b) About 80% of this energy is released by burning fat and the other 20% by burning protein. How many grams of fat are burned? One gram of fat releases $3.8 \times 10^{4} \mathrm{J}$ of energy. (c) If the salmon is about 15$\%$ fat at the beginning of the trip, how many grams of fat does it have at the end of the trip?

Estimate the maximum horsepower of the process of raising your body mass as fast as possible up a flight of 20 stair steps. Justify any numbers used in your estimate. The only energy change you should consider is the change in gravitational potential energy of the system you-Earth.

A 1600-kg car smashes into a shed and stops. The force that the shed exerts on the car as a function of a position at the car’s center is shown in Figure P6.66. How fast was the car traveling just before hitting the shed?

Suppose the car in Problem 66 was moving at twice the speed and was stopped in the same manner and in the same distance, only now by a more solidly constructed shed. Draw a new graph for the force that this new shed exerted on the car as a function of the car’s position. Include the appropriate numbers on the force axis and on the distance axis.

At what distance from Earth is the gravitational potential energy of a spaceship-Earth system reduced to half the energy of the system before the launch?

Possible escape of different air molecule types (a) Determine the ratio of escape speeds from Earth for a hydrogen molecule $\left(\mathrm{H}_{2}\right)$ and for an oxygen molecule $\left(\mathrm{O}_{2}\right) .$ The mass of the oxygen is approximately 16 times that of the hydrogen.
(b) In the atmosphere, the average random kinetic energy of hydrogen molecules and oxygen molecules is the same. Determine the ratio of the average speeds of the hydrogen and the oxygen molecules. (c) Based on these two results, give one reason why our atmosphere lacks hydrogen but retains oxygen.

Determine the escape speed for a rocket to leave Earth’s Moon.

Determine the escape speed for a rocket to leave the solar system.

If the Sun were to become a black hole, how much would it increase the gravitational potential energy of the Sun-Earth system?

A satellite moves in elliptical orbit around Earth, which is one of the foci of the elliptical orbit. (a) The satellite is moving faster when it is closer to Earth. Explain why. (b) If the satellite moves faster when it is closer to Earth, is the energy of the satellite-Earth system constant? Explain.

Determine the maximum radius Earth’s Moon would have to have in order for it to be a black hole.

You wish to try bungee jumping, but want to make sure it is safe. The brochure provided at the ticket office says that the cord holding the jumper is initially 12 m long and has a spring constant of 160 N/m. The tower from which you plan to jump appears to be 10 floors high. Should you try the bungee jump? Explain your answer.

Pose a problem involving work-energy ideas with real numbers. Then solve the problem choosing two different systems and discuss whether the answers were different.

Your dormitory has a nice balcony that looks over a pond in the grass below. You attach a 16-m-long rope to the limb of a tall tree beside the pond and pull it to the balcony so that it makes a $53^{\circ}$ angle from the vertical. You hold the rope while standing on the balcony and then swing down. (a) How fast are you moving at the lowest point in your swing? Specify the system and the initial and final states. List the assumptions that you made. (b) How strong should the rope be to withstand your adventure (use your own mass for calculations)?

Bungee jump at Squaw Valley At the Squaw Valley ski area, 2500 m above sea level (Figure P6.78), a bungee jumper falls over a 152-m cliff above Lake Tahoe. Choose numbers for quantities, and make and solve a problem related to this bungee system.

Six Flags roller coaster A loop-the-loop on the Six Flags Shockwave roller coaster has a 10-m radius (Figure P6.79). The car is moving at 24 m/s at the bottom of the loop. Determine the force exerted by the
seat of the car on an 80-kg rider when passing inverted at the top of the loop.

Designing a ride You are asked to help design a new type of loop-the-loop ride. Instead of rolling down a long hill to generate the speed to go around the loop, the 300-kg cart starts at rest (with two
passengers) on a track at the same level as the bottom of the 10-m radius loop. The cart is pressed against a compressed spring that, when released, launches the cart along the track around the loop. Choose a spring of the appropriate spring constant to launch the cart so that the downward force exerted by the track on the cart as it passes the top of the loop is 0.2 times the force that Earth exerts on
the cart. The spring is initially compressed 6.0 m.

Impact extinction 65 million years ago over 50% of all species became extinct, ending the reign of dinosaurs and opening the way for mammals to become the dominant land vertebrates. A theory for this extinction, with considerable supporting evidence, is that a $10-\mathrm{km}$ -wide $1.8 \times 10^{15}-\mathrm{kg}$ asteroid traveling at speed 11 $\mathrm{km} / \mathrm{s}$ crashed into Earth. Use this information and any other information or assumptions of your choosing to (a) estimate the change in velocity of Earth due to the impact; (b) estimate the average force that Earth exerted on the asteroid while stopping it; and (c) estimate the internal energy produced by the collision (a bar chart for the process might help. By comparison, the atomic bombs dropped on Japan during World War II were each equivalent to $15,000$ tons of TNT $(1 \text { ton of TNT releases }$ $4.2 \times 10^{9} \mathrm{J}$ of energy).

Newton’s cradle is a toy that consists of several metal balls touching each other and suspended on strings (Figure P6.82). When you pull one ball to the side and let it strike the next ball, only one ball swings out on the other side. When you use two balls to hit the others, two balls swing out. Can you account for this effect using your knowledge about elastic collisions?

Design of looping roller coaster You are an engineer helping to design a roller coaster that carries passengers down a steep track and around a vertical loop. The coaster’s speed must be great enough when at the top of the loop so that the rider stays in contact with the cart and the cart stays in contact with the track. Riders can withstand acceleration no more than a few "g's", where one "g" is 9.8 $\mathrm{m} / \mathrm{s}^{2}$ . What are some reasonable values for the physical quantities you can use in the design of the ride? For example, one consideration is the height at which the cart and rider should start so that they can safely make it around the loop and the radius of the loop.

Why is the metabolic rate different for different people?
(a) They have different masses.
(b) They have different body function efficiencies.
(c) They have different levels of physical activity.
(d) All of the above

A 50-kg mountain climber moves 30 m up a vertical slope. If the muscles in her body convert chemical energy into gravitational potential energy with an efficiency of no more than 5%, what is the chemical energy used to climb the slope?
(a) 7 kcal (b) 3000 J
(c) 70 kcal (d) 300,000 J

If 10% of a 50-kg rock climber’s total energy expenditure goes into the gravitational energy change when climbing a 100-m vertical slope, what is the climber’s average metabolic rate during the climb if it takes her 10 min to complete the climb?
(a) 8 W (b) 80 W (c) 500 W
(d) 700 W (e) 800 W

A 68-kg person wishes to lose 4.5 kg in 2 months. Estimate the time that this person should spend in moderate exercise each day to achieve this goal (without altering her food consumption).
(a) 0.4 h (b) 0.9 h (c) 1.4 h
(d) 1.9 h (e) 2.4 h

A 68-kg person walks at 5 km per hour for 1 hour a day for 1 year. Estimate the extra number of kilocalories of energy used because of the walking.
(a) 40,000 kcal (b) 47,000 kcal
(c) 88,000 kcal (d) 150,000 kcal

Suppose that a 90-kg person walks for 1 hour each day for a year, expending 50,000 extra kilocalories of metabolic energy (in addition to his normal resting metabolic energy use). What approximately is the person’s mass at the end of the year, assuming his food consumption does not change?
(a) 57 kg (b) 61 kg (c) 64 kg
(d) 66 kg (e) 67 kg

Why is hopping an energy-efficient mode of transportation for a kangaroo?
(a) There is less resistance since there is less contact with the ground.
(b) The elastic energy stored in muscles and tendons when landing is returned to help with the next hop.
(c) The kangaroo has long feet that cushion the landing.
(d) The kangaroo’s long feet help launch the kangaroo.

Why does the horizontal force exerted by the ground on the kangaroo change direction as the kangaroo lands and then hops forward?
(a) The backward force when it lands prevents it from slipping, and the forward force when taking off helps propel it forward.
(b) One horizontal force is needed to help stop the kangaroo’s fall and the other to help launch its upward vertical hop.
(c) Both forces oppose the kangaroo’s motion, but one looks like it is forward because the kangaroo is moving fast.
(d) The kangaroo is not an inertial reference frame, and the forward force is not real.
(e) All of the above

Which answer below is closest to the vertical impulse that the ground exerts on the kangaroo while it takes off?
(a) Zero (b)$+50 N \cdot s$ (c)$+150 N \cdot s$
(d) $-50 \mathrm{N} \cdot \mathrm{s}$ (e) $-150 \mathrm{N} \cdot \mathrm{s}$

Which answer below is closest to the vertical impulse due to the gravitational force exerted on the kangaroo by Earth during the short time interval while it takes off?
$\begin{array}{llll}{\text { (a) } 0} & {\text { (b) }+50 \mathrm{N} \cdot \mathrm{s}} & {\text { (c) }+150 \mathrm{N} \cdot \mathrm{s}}\end{array}$ (d) $-50 \mathrm{N} \cdot \mathrm{s} \quad(\mathrm{e})-150 \mathrm{N} \cdot \mathrm{s}$

Suppose the net vertical impulse on the 50 -kg kangaroo due to all external forces was $+100 \mathrm{N} \cdot$ s. Which answer below is closest to its vertical component of velocity when it leaves the ground?
$\begin{array}{llll}{\text { (a) }+2.0 \mathrm{m} / \mathrm{s}} & {\text { (b) }+3.0 \mathrm{m} / \mathrm{s}} & {\text { (c) }+4.0 \mathrm{m} / \mathrm{s}}\end{array}$ $(\mathrm{d})+8.0 \mathrm{m} / \mathrm{s} \quad(\mathrm{e})+10 \mathrm{m} / \mathrm{s}$

Which answer below is closest to the vertical height above the ground that the kangaroo reaches if it leaves the ground traveling with a vertical component of velocity of 2.5 m/s?
$\begin{array}{llll}{\text { (a) } 0.2 \mathrm{m}} & {\text { (b) } 0.3 \mathrm{m}} & {\text { (c) } 0.4 \mathrm{m}}\end{array}$ (d) 0.6 $\mathrm{m} \quad$ (e) 0.8 $\mathrm{m}$