Answer: A ball of mass m makes a head-on elastic collision

Problem 8PE Problem Suppose the ski patrol lowers a rescue sled and victim, having a total mass of 90.0 kg, down a 60.0º slope at constant speed, as shown in Figure 7.37. The coefficient of friction between the sled and the snow is 0.100. (a) How much work is done by friction as the sled moves 30.0 m along the hill? (b) How much work is done by the rope on the sled in this distance? (c) What is the work done by the gravitational force on the sled? (d) What is the total work done?

Problem 12Q At a hydroelectric power plant, water is directed at high speed against turbine blades on an axle that turns an electric generator. For maximum power generation, should the turbine blades be designed so that the water is brought to a dead stop, or so that the water rebounds?

Problem 15Q Why do you tend to lean backward when carrying a heavy load in your arms?

Problem 16CQ List four different forms or types of energy. Give one example of a conversion from each of these forms to another form.

Problem 16PE A hydroelectric power facility (see Figure 7.38) converts the gravitational potential energy of water behind a dam to electric energy. (a) What is the gravitational potential energy relative to the generators of a lake of volume 50.0 km3 (mass = 5.00×1013 kg), given that the lake has an average height of 40.0 m above the generators? (b) Compare this with the energy stored in a 9-megaton fusion bomb

Problem 16Q Why is the CM of a 1 -m length of pipe at its midpoint, whereas this is not true for your arm or leg?

Problem 17CQ List the energy conversions that occur when riding a bicycle.

Problem 17PE (a) How much gravitational potential energy (relative to the ground on which it is built) is stored in the Great Pyramid of Cheops, given that its mass is about 7 × 109 kg and its center of mass is 36.5 m above the surrounding ground? (b) How does this energy compare with the daily food intake of a person?

Problem 18PE Suppose a 350-g kookaburra (a large kingfisher bird) picks up a 75-g snake and raises it 2.5 m from the ground to a branch. (a) How much work did the bird do on the snake? (b) How much work did it do to raise its own center of mass to the branch?

Problem 18Q If only an external force can change the momentum of the center of mass of an object, how can the internal force of the engine accelerate a car?

Problem 18CQ Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, it is a zerowatt device.) Explain in terms of the definition of power.

A rocket following a parabolic path through the air suddenly explodes into many pieces. What can you say about the motion of this system of pieces?

Problem 19CQ Explain, in terms of the definition of power, why energy consumption is sometimes listed in kilowatt-hours rather than joules. What is the relationship between these two energy units?

Problem 19PE In Example 7.7, we found that the speed of a roller coaster that had descended 20.0 m was only slightly greater when it had an initial speed of 5.00 m/s than when it started from rest. This implies that ??PE >> KE?i . Confirm this statement by taking the ratio of ??PE to KE?i . (Note that mass cancels.) Example 7.7 Finding the Speed of a Roller Coaster from its Height (a) What is the final speed of the roller coaster shown in ?Figure 7.8 ?if it starts from rest at the top of the 20.0 m hill and work done by frictional forces is negligible? (b) What is its final speed (again assuming negligible friction) if its initial speed is 5.00 m/s?

Problem 20CQ A spark of static electricity, such as that you might receive from a doorknob on a cold dry day, may carry a few hundred watts of power. Explain why you are not injured by such a spark.

Suppose the force acting on a tennis ball (mass 0.060 kg) points in the +x direction and is given by the graph of Fig. 7-33 as a function of time, (a) Use graphical methods (count squares) to estimate the total impulse given the ball, (b) Estimate the velocity of the ball after being struck, assuming the ball is being served so it is nearly at rest initially.

Problem 20PE A 100-g toy car is propelled by a compressed spring that starts it moving. The car follows the curved track in Figure 7.39. Show that the final speed of the toy car is 0.687 m/s if its initial speed is 2.00 m/s and it coasts up the frictionless slope, gaining 0.180 m in altitude.

Problem 21CQ Problem Explain why it is easier to climb a mountain on a zigzag path rather than one straight up the side. Is your increase in gravitational potential energy the same in both cases? Is your energy consumption the same in both?

Problem 21PE Problem In a downhill ski race, surprisingly, little advantage is gained by getting a running start. (This is because the initial kinetic energy is small compared with the gain in gravitational potential energy on even small hills.) To demonstrate this, find the final speed and the time taken for a skier who skies 70.0 m along a 30º slope neglecting friction: (a) Starting from rest. (b) Starting with an initial speed of 2.50 m/s. (c) Does the answer surprise you? Discuss why it is still advantageous to get a running start in very competitive events.

Problem 22CQ Problem Do you do work on the outside world when you rub your hands together to warm them? What is the efficiency of this activity?

Problem 22PE Problem A 5.00×105 -kg subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the force constant k of the spring?

Problem 23CQ Problem Shivering is an involuntary response to lowered body temperature. What is the efficiency of the body when shivering, and is this a desirable value?

Problem 23PE Problem A pogo stick has a spring with a force constant of 2.50×104 N/m, which can be compressed 12.0 cm. To what maximum height can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 40.0 kg? Explicitly show how you follow the steps in the Problem-Solving Strategies for Energy.

Problem 24CQ Problem Discuss the relative effectiveness of dieting and exercise in losing weight, noting that most athletic activities consume food energy at a rate of 400 to 500 W, while a single cup of yogurt can contain 1360 kJ (325 kcal). Specifically, is it likely that exercise alone will be sufficient to lose weight? You may wish to consider that regular exercise may increase the metabolic rate, whereas protracted dieting may reduce it.

Problem 24PE Problem A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.50-m-high rise as shown in Figure 7.40. Find her final speed at the top, given that the coefficient of friction between her skis and the snow is 0.0800. (Hint: Find the distance traveled up the incline assuming a straight-line path as shown in the figure.)

Problem 25CQ Problem What is the difference between energy conservation and the law of conservation of energy? Give some examples of each.

Problem 25PE Problem (a) How high a hill can a car coast up (engine disengaged) if work done by friction is negligible and its initial speed is 110 km/h? (b) If, in actuality, a 750-kg car with an initial speed of 110 km/h is observed to coast up a hill to a height 22.0 m above its starting point, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a slope 2.5º above the horizontal?

Problem 26CQ Problem If the efficiency of a coal-fired electrical generating plant is 35%, then what do we mean when we say that energy is a conserved quantity?

Problem 26PE Problem Using values from Table 7.1, how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous x rays. Later model tube TVs had shielding that absorbed x rays before they escaped and exposed viewers.)

Problem 28PE Problem If the energy in fusion bombs were used to supply the energy needs of the world, how many of the 9-megaton variety would be needed for a year’s supply of energy (using data from Table 7.1)? This is not as far-fetched as it may sound—there are thousands of nuclear bombs, and their energy can be trapped in underground explosions and converted to electricity, as natural geothermal energy is.

Problem 27PE Problem Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown.

Problem 29PE Problem (a) Use of hydrogen fusion to supply energy is a dream that may be realized in the next century. Fusion would be a relatively clean and almost limitless supply of energy, as can be seen from Table 7.1. To illustrate this, calculate how many years the present energy needs of the world could be supplied by one millionth of the oceans’ hydrogen fusion energy. (b) How does this time compare with historically significant events, such as the duration of stable economic systems?

In a physics lab, a cube slides down a frictionless incline as shown in Fig. 7-50 and elastically strikes another cube at the bottom that is only one-half its mass. If the incline is 35 cm high and the table is 95 cm off the floor, where does each cube land? [Hint: Both leave the incline moving horizontally.) FIGURE 7-35 Problem 29.

(III) Take the general case of an object of mass \(m_{A}\) and velocity \(v_{A}\) elastically striking a stationary \(\left(v_{B}=0\right)\) object of mass \(m_{B}\) head-on. (a) Show that the final velocities \(v_{A}^{\prime} \text { and } v_{B}^{\prime}\) are given by \(v_{A}^{\prime}=\left(\frac{m_{A}-m_{B}}{m_{A}+m_{B}}\right) v_{A} v_{B}^{\prime}=\left(\frac{2 m_{A}}{m_{A}+m_{B}}\right) v_{A}\) (b) What happens in the extreme case when \(m_{A}\) is much smaller than \(m_{B}\)? Cite a common example of this. (c) What happens in the extreme case when \(m_{A}\) is much larger than \(m_{B}\)? Cite a common example of this. What happens in the case when \(m_{A}=m_{B}\)? Cite a common example. Equation Transcription: Text Transcription: mA vA (vB=0) mB v'A and v'B v_A^\prime=(\frac{m_A-m_B m_A+m_B) v_A v_B^\prime mA mB mA mB mA= mB

Problem 30PE Problem The Crab Nebula (see Figure 7.41) pulsar is the remnant of a supernova that occurred in A.D. 1054. Using data from Table 7.3, calculate the approximate factor by which the power output of this astronomical object has declined since its explosion.

Problem 31P (I) In a ballistic pendulum experiment, projectile 1 results in a maximum height f> of the pendulum equal to 2.6 cm. A second projectile (of the same mass) causes the pendulum to swing twice as high, h2 =5.2 cm. The second projectile was how many times faster than the first?

Problem 32PE Problem A person in good physical condition can put out 100 W of useful power for several hours at a stretch, perhaps by pedaling a mechanism that drives an electric generator. Neglecting any problems of generator efficiency and practical considerations such as resting time: (a) How many people would it take to run a 4.00-kW electric clothes dryer? (b) How many people would it take to replace a large electric power plant that generates 800 MW?

Problem 31PE Problem Suppose a star 1000 times brighter than our Sun (that is, emitting 1000 times the power) suddenly goes supernova. Using data from Table 7.3: (a) By what factor does its power output increase? (b) How many times brighter than our entire Milky Way galaxy is the supernova? (c) Based on your answers, discuss whether it should be possible to observe supernovas in distant galaxies. Note that there are on the order of 1011 observable galaxies, the average brightness of which is somewhat less than our own galaxy.

Problem 33PE Problem What is the cost of operating a 3.00-W electric clock for a year if the cost of electricity is $0.0900 per kW?h ?

(II) An internal explosion breaks an object, initially at rest, into two pieces, one of which has 1.5 times the mass of the other. If 7500 J were released in the explosion, how much kinetic energy did each piece acquire?

Problem 35PE Problem (a) What is the average power consumption in watts of an appliance that uses 5.00 kW ? h of energy per day? (b) How many joules of energy does this appliance consume in a year?

Problem 36PE Problem (a) What is the average useful power output of a person who does 6.00×106 J of useful work in 8.00 h? (b) Working at this rate, how long will it take this person to lift 2000 kg of bricks 1.50 m to a platform? (Work done to lift his body can be omitted because it is not considered useful output here.)

Problem 37PE Problem A 500-kg dragster accelerates from rest to a final speed of 110 m/s in 400 m (about a quarter of a mile) and encounters an average frictional force of 1200 N. What is its average power output in watts and horsepower if this takes 7.30 s?

Problem 38P (II) A wooden block is cut into two pieces, one with three times the mass of the other. A depression is made in both faces of the cut, so that a firecracker can be placed in it with the block reassembled. The reassembled block is set on a rough-surfaced table, and the fuse is lit .When the firecracker explodes inside, the two blocks separate and slide apart. What is the ratio of distances each block travels?

Problem 39PE Problem (a) Find the useful power output of an elevator motor that lifts a 2500-kg load a height of 35.0 m in 12.0 s, if it also increases the speed from rest to 4.00 m/s. Note that the total mass of the counterbalanced system is 10,000 kg—so that only 2500 kg is raised in height, but the full 10,000 kg is accelerated. (b) What does it cost, if electricity is $0.0900 per kW ? h ?

Problem 38PE Problem (a) How long will it take an 850-kg car with a useful power output of 40.0 hp (1 hp = 746 W) to reach a speed of 15.0 m/ s, neglecting friction? (b) How long will this acceleration take if the car also climbs a 3.00-m-high hill in the process?

Problem 40PE Problem (a) What is the available energy content, in joules, of a battery that operates a 2.00-W electric clock for 18 months? (b) How long can a battery that can supply 8.00×104 J run a pocket calculator that consumes energy at the rate of 1.00×10?3 W ?

Problem 41PE Problem (a) How long would it take a 1.50×105 -kg airplane with engines that produce 100 MW of power to reach a speed of 250 m/s and an altitude of 12.0 km if air resistance were negligible? (b) If it actually takes 900 s, what is the power? (c) Given this power, what is the average force of air resistance if the airplane takes 1200 s? (Hint: You must find the distance the plane travels in 1200 s assuming constant acceleration.)

The power output needed for a 950-kg car to climb a \(2.00^\circ\) slope at a constant 30.0 m/s while encountering wind resistance and friction totaling 600 N. Explicitly show how you follow the steps in the Strategies for Energy.

Problem 43PE Problem (a) Calculate the power per square meter reaching Earth’s upper atmosphere from the Sun. (Take the power output of the Sun to be 4.00×1026 W.) (b) Part of this is absorbed and reflected by the atmosphere, so that a maximum of 1.30 kW/m2 reaches Earth’s surface. Calculate the area in km2 of solar energy collectors needed to replace an electric power plant that generates 750 MW if the collectors convert an average of 2.00% of the maximum power into electricity. (This small conversion efficiency is due to the devices themselves, and the fact that the sun is directly overhead only briefly.) With the same assumptions, what area would be needed to meet the United States’ energy needs (1.05×1020 J)? Australia’s energy needs (5.4×1018 J)? China’s energy needs (6.3×1019 J)? (These energy consumption values are from 2006.)

(III) Two billiard balls of equal mass move at right angles and meet at the origin of an coordinate system. Ball A is moving upward along the axis at \(2.0 \mathrm{~m} / \mathrm{s}\), and ball is moving to the right along the axis with speed \(3.7 \mathrm{~m} / \mathrm{s}\). After the collision, assumed elastic, ball B is moving along the positive axis (Fig. ). What is the final direction of ball and what are their two speeds? FIGURE 7-37 Problem44. (BAll A after the collision is not shown.) Equation Transcription: Text Transcription: 2.0 m/s 3.7 m/s

Problem 44PE Problem (a) How long can you rapidly climb stairs (116/min) on the 93.0 kcal of energy in a 10.0-g pat of butter? (b) How many flights is this if each flight has 16 stairs?

Problem 45PE Problem (a) What is the power output in watts and horsepower of a 70.0-kg sprinter who accelerates from rest to 10.0 m/s in 3.00 s? (b) Considering the amount of power generated, do you think a well-trained athlete could do this repetitively for long periods of time?

(III) A neon atom (m =20.0 u) makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a 55.6° angle from its original direction and the unknown atom travels away at a -50.0° angle. What is the mass (in u) of the unknown atom? [Hint You could use the law of sines.)

(I) Find the center of mass of the three-mass system shown in Fig. 7-38. Specify relative to the left-hand 1.00-kg mass.

Problem 46PE Problem Calculate the power output in watts and horsepower of a shot-putter who takes 1.20 s to accelerate the 7.27-kg shot from rest to 14.0 m/s, while raising it 0.800 m. (Do not include the power produced to accelerate his body.)

(I) The distance between a carbon atom (m =12 u) and an oxygen atom (m =16 u) in the CO molecule is 1.3 X 10-10 m How far from the carbon atom is the center of mass of the molecule?

Problem 47PE Problem (a) What is the efficiency of an out-of-condition professor who does 2.10×105 J of useful work while metabolizing 500 kcal of food energy? (b) How many food calories would a well-conditioned athlete metabolize in doing the same work with an efficiency of 20%?

Problem 48PE Problem Energy that is not utilized for work or heat transfer is converted to the chemical energy of body fat containing about 39 kJ/g. How many grams of fat will you gain if you eat 10,000 kJ (about 2500 kcal) one day and do nothing but sit relaxed for 16.0 h and sleep for the other 8.00 h? Use data from Table 7.5 for the energy consumption rates of these activities.

Problem 49PE Problem Using data from Table 7.5, calculate the daily energy needs of a person who sleeps for 7.00 h, walks for 2.00 h, attends classes for 4.00 h, cycles for 2.00 h, sits relaxed for 3.00 h, and studies for 6.00 h. (Studying consumes energy at the same rate as sitting in class.)

(II) A (lightweight) pallet has a load of identical cases of tomato paste (see Fig. ), each of which is a cube of length . Find the center of gravity in the horizontal plane, so that the crane operator can pick up the load without tipping it. FIGURE 7-40 Problem 51

Problem 50PE Problem What is the efficiency of a subject on a treadmill who puts out work at the rate of 100 W while consuming oxygen at the rate of 2.00 L/min? (Hint: See Table 7.5.)

Problem 51PE Problem Shoveling snow can be extremely taxing because the arms have such a low efficiency in this activity. Suppose a person shoveling a footpath metabolizes food at the rate of 800 W. (a) What is her useful power output? (b) How long will it take her to lift 3000 kg of snow 1.20 m? (This could be the amount of heavy snow on 20 m of footpath.) (c) How much waste heat transfer in kilojoules will she generate in the process?

Problem 52PE Problem Very large forces are produced in joints when a person jumps from some height to the ground. (a) Calculate the magnitude of the force produced if an 80.0-kg person jumps from a 0.600–m-high ledge and lands stiffly, compressing joint material 1.50 cm as a result. (Be certain to include the weight of the person.) (b) In practice the knees bend almost involuntarily to help extend the distance over which you stop. Calculate the magnitude of the force produced if the stopping distance is 0.300 m. (c) Compare both forces with the weight of the person.

(I) Assume that your proportions are the same as those in Table , and calculate the mass of one of your legs.

Problem 53PE Problem Jogging on hard surfaces with insufficiently padded shoes produces large forces in the feet and legs. (a) Calculate the magnitude of the force needed to stop the downward motion of a jogger’s leg, if his leg has a mass of 13.0 kg, a speed of 6.00 m/s, and stops in a distance of 1.50 cm. (Be certain to include the weight of the 75.0-kg jogger’s body.) (b) Compare this force with the weight of the jogger.

(III) A uniform circular plate of radius has a circular hole of radius cut out of it. The center \(C^{\prime}) of the smaller circle is a distance \(0.80 R\) from the center of the larger circle, Fig. . What is the position of the center of mass of the plate? [Hint: Try subtraction.] FIGURE 7-41 Problem 52 Equation Transcription: Text Transcription: C' 0.80R

Determine the of an outstretched arm using Table .

Problem 54PE Problem (a) Calculate the energy in kJ used by a 55.0-kg woman who does 50 deep knee bends in which her center of mass is lowered and raised 0.400 m. (She does work in both directions.) You may assume her efficiency is 20%. (b) What is the average power consumption rate in watts if she does this in 3.00 min?

(II) Use Table to calculate the position of the of an arm bent at a right angle. Assume that the person is tall.

Problem 56PE Problem The swimmer shown in Figure 7.44 exerts an average horizontal backward force of 80.0 N with his arm during each 1.80 m long stroke. (a) What is his work output in each stroke? (b) Calculate the power output of his arms if he does 120 strokes per minute.

Problem 55PE Problem Kanellos Kanellopoulos flew 119 km from Crete to Santorini, Greece, on April 23, 1988, in the Daedalus 88, an aircraft powered by a bicycle-type drive mechanism (see Figure 7.43). His useful power output for the 234-min trip was about 350 W. Using the efficiency for cycling from Table 7.2, calculate the food energy in kilojoules he metabolized during the flight.

(II) When a high jumper is in a position such that his arms and legs are hanging vertically, and his trunk and head are horizontal, calculate how far below the torso's median line the will be. Will this be outside the body? Use Table .

(II) The masses of the Earth and Moon are \(5.98 \times 10^{24} \mathrm{~kg}\) and \(7.35 \times 10^{22} \mathrm{~kg}\), respectively, and their centers are separated by \(3.84 \times 10^8 \mathrm{~m}\). (a) Where is the \(\mathrm{CM}\) of this system located? (b) What can you say about the motion of the Earth-Moon system about the Sun. and of the Earth and Moon separately about the Sun?

Problem 57PE Problem Mountain climbers carry bottled oxygen when at very high altitudes. (a) Assuming that a mountain climber uses oxygen at twice the rate for climbing 116 stairs per minute (because of low air temperature and winds), calculate how many liters of oxygen a climber would need for 10.0 h of climbing. (These are liters at sea level.) Note that only 40% of the inhaled oxygen is utilized; the rest is exhaled. (b) How much useful work does the climber do if he and his equipment have a mass of 90.0 kg and he gains 1000 m of altitude? (c) What is his efficiency for the 10.0-h climb? Step-by-step solution Step 1 of 6 The formula to find the efficiency is, Here is the efficiency, is the output work and is the input work. Step 2 of 6 (a) The amount of oxygen required in liters is, Therefore, the amount of oxygen required is . Step 3 of 6 (b) The useful work done by the climber, Substitute for , for and for . Therefore, useful work done by the climber is . Step 4 of 6 (c) Use the formula for efficiency, Step 5 of 6 Step 6 of 6 Substitute for and for . Therefore, the efficiency is .

Problem 58PE Problem The awe-inspiring Great Pyramid of Cheops was built more than 4500 years ago. Its square base, originally 230 m on a side, covered 13.1 acres, and it was 146 m high, with a mass of about 7×109 kg . (The pyramid’s dimensions are slightly different today due to quarrying and some sagging.) Historians estimate that 20,000 workers spent 20 years to construct it, working 12-hour days, 330 days per year. (a) Calculate the gravitational potential energy stored in the pyramid, given its center of mass is at one-fourth its height. (b) Only a fraction of the workers lifted blocks; most were involved in support services such as building ramps (see Figure 7.45), bringing food and water, and hauling blocks to the site. Calculate the efficiency of the workers who did the lifting, assuming there were 1000 of them and they consumed food energy at the rate of 300 kcal/h. What does your answer imply about how much of their work went into block-lifting, versus how much work went into friction and lifting and lowering their own bodies? (c) Calculate the mass of food that had to be supplied each day, assuming that the average worker required 3600 kcal per day and that their diet was 5% protein, 60% carbohydrate, and 35% fat. (These proportions neglect the mass of bulk and nondigestible materials consumed.)

Problem 59PE Problem (a) How long can you play tennis on the 800 kJ (about 200 kcal) of energy in a candy bar? (b) Does this seem like a long time? Discuss why exercise is necessary but may not be sufficient to cause a person to lose weight.

Problem 60PE Problem Integrated Concepts (a) Calculate the force the woman in Figure 7.46 exerts to do a push-up at constant speed, taking all data to be known to three digits. (b) How much work does she do if her center of mass rises 0.240 m? (c) What is her useful power output if she does 25 push-ups in 1 min? (Should work done lowering her body be included? See the discussion of useful work in Work, Energy, and Power in Humans.

Problem 61PE Problem Integrated Concepts A 75.0-kg cross-country skier is climbing a 3.0º slope at a constant speed of 2.00 m/s and encounters air resistance of 25.0 N. Find his power output for work done against the gravitational force and air resistance. (b) What average force does he exert backward on the snow to accomplish this? (c) If he continues to exert this force and to experience the same air resistance when he reaches a level area, how long will it take him to reach a velocity of 10.0 m/s?

Problem 63PE Problem Integrated Concepts A toy gun uses a spring with a force constant of 300 N/m to propel a 10.0-g steel ball. If the spring is compressed 7.00 cm and friction is negligible: (a) How much force is needed to compress the spring? (b) To what maximum height can the ball be shot? (c) At what angles above the horizontal may a child aim to hit a target 3.00 m away at the same height as the gun? (d) What is the gun’s maximum range on level ground?

Problem 64PE Problem Integrated Concepts (a) What force must be supplied by an elevator cable to produce an acceleration of 0.800 m/s2 against a 200-N frictional force, if the mass of the loaded elevator is 1500 kg? (b) How much work is done by the cable in lifting the elevator 20.0 m? (c) What is the final speed of the elevator if it starts from rest? (d) How much work went into thermal energy?

Problem 65PE Problem Unreasonable Results A car advertisement claims that its 900-kg car accelerated from rest to 30.0 m/s and drove 100 km, gaining 3.00 km in altitude, on 1.0 gal of gasoline. The average force of friction including air resistance was 700 N. Assume all values are known to three significant figures. (a) Calculate the car’s efficiency. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?

Problem 66PE Problem Unreasonable Results Body fat is metabolized, supplying 9.30 kcal/g, when dietary intake is less than needed to fuel metabolism. The manufacturers of an exercise bicycle claim that you can lose 0.500 kg of fat per day by vigorously exercising for 2.00 h per day on their machine. (a) How many kcal are supplied by the metabolization of 0.500 kg of fat? (b) Calculate the kcal/min that you would have to utilize to metabolize fat at the rate of 0.500 kg in 2.00 h. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent?

Problem 67PE Problem Construct Your Own Problem Consider a person climbing and descending stairs. Construct a problem in which you calculate the long-term rate at which stairs can be climbed considering the mass of the person, his ability to generate power with his legs, and the height of a single stair step. Also consider why the same person can descend stairs at a faster rate for a nearly unlimited time in spite of the fact that very similar forces are exerted going down as going up. (This points to a fundamentally different process for descending versus climbing stairs.)

Problem 68PE Problem Construct Your Own Problem Consider humans generating electricity by pedaling a device similar to a stationary bicycle. Construct a problem in which you determine the number of people it would take to replace a large electrical generation facility. Among the things to consider are the power output that is reasonable using the legs, rest time, and the need for electricity 24 hours per day. Discuss the practical implications of your results.

Problem 69GP A golf ball rolls off the top of a flight of concrete steps of total vertical height 4.00 m. The ball hits four times on the way down, each time striking the horizontal part of a different step 1.00 m lower. If all collisions are perfectly elastic, what is the bounce height on the fourth bounce when the ball reaches the bottom of the stairs?

Problem 69PE Problem Integrated Concepts A 105-kg basketball player crouches down 0.400 m while waiting to jump. After exerting a force on the floor through this 0.400 m, his feet leave the floor and his center of gravity rises 0.950 m above its normal standing erect position. (a) Using energy considerations, calculate his velocity when he leaves the floor. (b) What average force did he exert on the floor? (Do not neglect the force to support his weight as well as that to accelerate him.) (c) What was his power output during the acceleration phase?

Problem 74GP (II) An object at rest is suddenly broken apart into two fragments by an explosion. One fragment acquires twice the kinetic energy of the other. What is the ratio of their masses?

The gravitational slingshot effect. Figure shows the planet Saturn moving in the negative direction at its orbital speed (with respect to the Sun) of \(9.6 \mathrm{~km} / \mathrm{s}\). The mass of Saturn is \(5.69 \times 10^{26} \mathrm{~kg}\). A spacecraft with mass approaches Saturn. When far from Saturn, it moves in the direction at \(10.4 \mathrm{~km} / \mathrm{s}\). The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Estimate the final speed of the spacecraft after it is far enough away to be considered free of Saturn's gravitational pull. FIGURE 7-47 Problem 81 Equation Transcription: Text Transcription: 9.6 km/s 5.69 x 10^26 kg 10.4 km/s

Problem 2P A constant friction force of 25 N acts on a 65-kg skier for 20 s. What is the skier’s change in velocity?

Problem 62PE Problem Integrated Concepts The 70.0-kg swimmer in Figure 7.44 starts a race with an initial velocity of 1.25 m/s and exerts an average force of 80.0 N backward with his arms during each 1.80 m long stroke. (a) What is his initial acceleration if water resistance is 45.0 N? (b) What is the subsequent average resistance force from the water during the 5.00 s it takes him to reach his top velocity of 2.50 m/s? (c) Discuss whether water resistance seems to increase linearly with velocity.

Problem 62GP A 0.145-kg baseball pitched horizontally at 35.0 m/s strikes a bat and is popped straight up to a height of 55.6 m. If the contact time is 1.4 ms, calculate the average force on the ball during the contact.

A rocket of mass m traveling with speed \(v_0\) along the x axis suddenly shoots out fuel, equal to one-third of its mass, parallel to the y axis (perpendicular to the rocket as seen from the ground) with speed \(2 v_0\). Give the components of the final velocity of the rocket.

A novice pool player is faced with the corner pocket shot shown in Fig. Relative dimensions are also shown. Should the player be worried about this being a "scratch shot," in which the cue ball will also fall into a pocket? Give details.

Problem 65GP A 140-kg astronaut (including space suit) acquires a speed of 2.50 m/s by pushing off with his legs from an 1800-kg space capsule. (a) What is the change in speed of the space capsule? (b) If the push lasts 0.40 s, what is the average force exerted on the astronaut by the space capsule? As the reference frame, use the position of the space capsule before the push.

(I) What is the magnitude of the momentum of a \(28-\mathrm{g}\) sparrow flying with a speed of \(8.4 \mathrm{~m} / \mathrm{s}\)?

Problem 1Q We claim that momentum is conserved. Yet most moving objects eventually slow down and stop. Explain.

Problem 2Q When a person jumps from a tree to the ground, what happens to the momentum of the person upon striking the ground?

Problem 3P A 0.145-kg baseball pitched at 39.0 m/s is hit on a horizontal line drive straight back toward the pitcher at 52.0 m/s. If the contact time between bat and ball is 3.00 × 10?3 s, calculate the average force between the ball and bat during contact .

Problem 3Q When you release an inflated but untied balloon, why does it fly across the room?

Problem 4Q It is said that in ancient times a rich man with a bag of gold coins was stranded on the surface of a frozen lake. Because the ice was frictionless, he could not push himself to shore and froze to death. What could he have done to save himself had he not been so miserly?

Problem 5Q How can a rocket change direction when it is far out in space and essentially in a vacuum?

Cars used to be built as rigid as possible to withstand collisions. Today, though, cars are designed to have "crumple zones" that collapse upon impact. What is the advantage of this new design?

Problem 8Q Why can a batter hit a pitched baseball farther than a ball he himself has tossed up in the air?

Problem 9Q Is it possible for an object to receive a larger impulse from a small force than from a large force? Explain.

Problem 11Q Problem Describe a collision in which all kinetic energy is lost.

Show on a diagram how your CM shifts when you move from a lying position to a sitting position.

According to Eq. , the longer the impact time of an impulse, the smaller the force can be for the same momentum change, and hence the smaller the deformation of the object on which the force acts. On this basis. explain the value of air bags, which are intended to inflate during an automobile collision and reduce the possibility of fracture or death.

(II) A child in a boat throws a package out horizontally with a speed of \(10.0 \mathrm{~m} / \mathrm{s}\), Fig. . Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is , and that of the boat is . Ignore water resistance. Equation Transcription: Text Transcription: 10.0 m/s

Problem 42P Billiard ball A of mass mA = 0.400 kg moving with speed vA = 1.80 m/s strikes ball B, initially at rest, of mass mB = 0.500 kg. As a result of the collision, ball A is deflected off at an angle of 30.0° with a speed v’A = 1.10 m/s. (a) Taking the x axis to be the original direction of motion of ball A, write down the equations expressing the conservation of momentum for the components in the x and y directions separately. (b) Solve these equations for the speed v'B and angle ?'B of ball B. Do not assume the collision is elastic.

Problem 5P Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of 1500 kg/s with a speed of 4.0 × 104m/s (at the moment of takeoff).

Problem 6P A 95-kg halfback moving at 4.1 m/s on an apparent breakaway for a touchdown is tackled from behind. When he was tackled by an 85-kg cornerback running at 5.5 m/s in the same direction, what was their mutual speed immediately after the tackle?

Problem 7P A 12,600-kg railroad car travels alone on a level frictionless track with a constant speed of 18.0 m/s. A 5350-kg load, initially at rest, is dropped onto the car. What will be the car’s new speed?

Problem 8P A 9300-kg boxcar traveling at 15.0 m/s strikes a second boxcar at rest. The two stick together and move off with a speed of 6.0 m/s. What is the mass of the second car?

(II) During a Chicago storm, winds can whip horizontally at speeds of \(100 \mathrm{~km} / \mathrm{h}\). If the air strikes a person at the rate of \(40 \mathrm{~kg} / \mathrm{s}\) per square meter and is brought to rest, estimate the force of the wind on a person. Assume the person is \(1.50 \mathrm{~m}\) high and \(0.50 \mathrm{~m}\) wide. Compare to the typical maximum force of friction \((\mu \approx 1.0)\) between the person and the ground, if the person has a mass of \(70 \mathrm{~kg}\).

Problem 10P A 3800-kg open railroad car coasts along with a constant speed of 8.60 m/s on a level track. Snow begins to fall vertically and fills the car at a rate of 3.50kg/min. Ignoring friction with the tracks, what is the speed of the car after 90.0 min?

Problem 10Q A 3800-kg open railroad car coasts along with a constant speed of 8.60 m/s on a level track. Snow begins to fall vertically and fills the car at a rate of 3.50kg/min. Ignoring friction with the tracks, what is the speed of the car after 90.0 min?

Problem 11P An atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction of its velocity, and the remaining nucleus slows to 350 m/s. If the alpha particle has a mass of 4.0 u and the original nucleus has a mass of 222 u, what speed does the alpha particle have when it is emitted?

(II) A 23-g bullet traveling 230 m/s penetrates a 2.0-kg block of wood and emerges cleanly at 170 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?

(III) A \(975-\mathrm{kg}\) two-stage rocket is traveling at a speed of \(5.80 \times 10^3 \mathrm{~m} / \mathrm{s}\) with respect to Earth when a pre-designed explosion separates the rocket into two sections of equal mass that then move at a speed of \(2.20 \times 10^3 \mathrm{~m} / \mathrm{s}\) relative to each other along the original line of motion. (a) What are the speed and direction of each section (relative to Earth) after the explosion? (b) How much energy was supplied by the explosion? [Hint: What is the change in KE as a result of the explosion?]

A squash ball hits a wall at a \(45^\circ\) angle as shown in Fig. 7-30. What is the direction (a) of the change in momentum of the ball, (b) of the force on the wall?

Problem 14P A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m/s. To alter its course by 35.0°, its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 m/s, how much mass must be expelled?

Problem 14Q A Superball is dropped from a height h onto a hard steel plate (fixed to the Earth), from which it rebounds at very nearly its original speed. (a) Is the momentum of the bah conserved during any part of this process? (b) If we consider the ball and Earth as our system, during what parts of the process is momentum conserved? (c) Answer part (b) for a piece of putty that falls and sticks to the steel plate.

Problem 15P A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. The golf club was in contact with the ball for 3.5 × 10?3 s. Find (a) the impulse imparted to the golf ball, and (b) the average force exerted on the ball by the golf club.

Problem 16P A 12-kg hammer strikes a nail at a velocity of 8.5 m/s and comes to rest in a time interval of 8.0 ms. (a) What is the impulse given to the nail? (b) What is the average force acting on the nail?

(II) A tennis ball of mass \(m=0.060 \mathrm{~kg}\) and speed \(v=25 \mathrm{~m} / \mathrm{s}\) strikes a wall at a \(45^{\circ}\) angle and rebounds with the same speed at \(45^{\circ}\) (Fig. ). What is the impulse (magnitude and direction) given to the ball? Equation Transcription: Text Transcription: m=0.060 kg v=25 m/s 45° 45°

(II) You are the design engineer in charge of the crashworthiness of new automobile models. Cars are tested by smashing them into fixed, massive barriers at 50 km/h (30mph). A new model of mass 1500 kg takes 0.15 s from the time of impact until it is brought to rest. (a) Calculate the average force exerted on the car by the barrier. (b) Calculate the average deceleration of the car.

Problem 19P A 95-kg fullback is running at 4.0 m/s to the east and is stopped in 0.75 s by a head-on tackle by a tackier running due west. Calculate (a) the original momentum of the fullback, (b) the impulse exerted on the fullback, (c) the impulse exerted on the tackier, and (d) the average force exerted on the tackler.

Problem 21P From what maximum height can a 75-kg person jump without breaking the lower leg bone of either leg? Ignore air resistance and assume the CM of the person moves a distance of 0.60 m from the standing to the seated position (that is, in breaking the fall). Assume the breaking strength (force per unit area) of bone is 170 × 106N/m2, and its smallest cross-sectional area is 2.5 × 10?4 m2. [Hint: Do not try this experimentally.]

Problem 22P A ball of mass 0.440 kg moving east (+ x direction) with a speed of 3.30 m/s collides head-on with a 0.220-kg ball at rest. If the collision is perfectly elastic, what will be the speed and direction of each ball after the collision?

Problem 23P A 0.450-kg ice puck, moving east with a speed of 3.00 m/s, has a head-on collision with a 0.900-kg puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of each object after the collision?

Problem 24P Two billiard balls of equal mass undergo a perfectly elastic head-on collision. If one ball’s initial speed was 2.00 m/s, and the other’s was 3.00 m/s in the opposite direction, what will be their speeds after the collision?

Problem 25P A 0.060-kg tennis ball, moving with a speed of 2.50 m/s, collides head-on with a 0.090-kg ball initially moving away from it at a speed of 1.15 m/s. Assuming a perfectly elastic collision, what arc the speed and direction of each ball after the collision?

Problem 26P A softball of mass 0.220 kg that is moving with a speed of 8.5 m/s collides head-on and elastically with another ball initially at rest. Afterward the incoming soft-ball bounces backward with a speed of 3.7 m/s. Calculate (a) the velocity of the target ball after the collision, and (b) the mass of the target ball.

(II) Two bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear (Fig. ). Car A has a mass of and car , owing to differences in passenger mass. If car approaches at \(4.50 \mathrm{~m} / \mathrm{s}\) and car is moving at \(3.70 \mathrm{~m} / \mathrm{s}\), calculate ( ) their velocities after the collision, and the change in momentum of each. Equation Transcription: Text Transcription: 4.50 m/s 3.70 m/s

(II) A \(0.280-\mathrm{kg}\) croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball. (a) What is the mass of the second ball? (b) What fraction of the original kinetic energy \((\Delta \mathrm{KE} / \mathrm{KE})\) gets transferred to the second ball?

Problem 32P A 28-g rifle bullet traveling 230 m/s buries itself in a 3.6-kg pendulum hanging on a 2.8-m-long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulum’s displacement.

(II) (a) Derive a formula for the fraction of kinetic energy lost,\(\Delta K E / K E\), for the ballistic pendulum collision of Example Evaluate for and Equation Transcription: Text Transcription: \Delta K E / K E

An internal explosion breaks an object, initially at rest, into two pieces, one of which has 1.5 times the mass of the other. If 7500 J were released in the explosion, how much kinetic energy did each piece acquire?

Problem 35P A 920-kg sports car collides into the rear end of a 2300-kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.8 m before stopping. The police officer, knowing that the coefficient of kinetic friction between tires and road is 0.80, calculates the speed of the sports car at impact. What was that speed?

Problem 36P A ball is dropped from a height of 1.50 m and rebounds to a height of 1.20 m. Approximately how many rebounds will the ball make before losing 90% of its energy?

(II) A measure of inelasticity in a head-on collision of two objects is the coefficient of restitution, e, defined as \(e=\frac{v_{\mathrm{A}}^{\prime}-v_{\mathrm{B}}^{\prime}}{v_{\mathrm{B}}-v_{\mathrm{A}}}\), where \(v_{\mathrm{A}}^{\prime}-v_{\mathrm{B}}^{\prime}\) is the relative velocity of the two objects after the collision and \(v_{\mathrm{B}}-v_{\mathrm{A}}\) is their relative velocity before it. (a) Show that e=1 for a perfectly elastic collision, and e=0 for a completely inelastic collision. (b) A simple method for measuring the coefficient of restitution for an object colliding with a very hard surface like steel is to drop the object onto a heavy steel plate, as shown in Fig. 7-36. Determine a formula for e in terms of the original height h and the maximum height \(h^{\prime}\) reached after one collision.

Problem 39P A 15.0-kg object moving in the +x direction at 5.5 m/s collides head-on with a 10.0-kg object moving in the ?x direction at 4.0 m/s. Find the final velocity of each mass if: (a) the objects stick together; (b) the collision is elastic; (c) the 15.0-kg object is at rest after the collision; (d) the 10.0-kg object is at rest after the collision; (e) the 15.0-kg object has a velocity of 4.0 m/s in the ?x direction after the collision. Are the results in (c), (d), and (e) “reasonable”? Explain.

(II) A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino are emitted at right angles and have momenta of \(9.30 \times 10^{-23} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) and \(5.40 \times 10^{-23} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\), respectively. What are the magnitude and direction of the momentum of the second (recoiling) nucleus?

Problem 41P An eagle (mA = 4.3 kg) moving with speed vA = 7.8 m/s is on a collision course with a second eagle (mB = 5.6 kg) moving at vB = 10.2 m/s in a direction perpendicular to the first. After they collide, they hold onto one another. In what direction, and with what speed, are they moving after the collision?

Problem 43P After a completely inelastic collision between two objects of equal mass, each having initial speed v, the two move off together with speed v/3. What was the angle between their initial directions?

Problem 48P The CM of an empty 1050-kg car is 2.50 m behind the front of the car. How far from the front of the car will the CM be when two people sit in the front seat 2.80 m from the front of the car, and three people sit in the back seat 3.90 m from the front? Assume that each person has a mass of 70.0 kg.

(II) A square uniform raft, \(18 \mathrm{~m}\) by \(18 \mathrm{~m}\), of mass \(6800 \mathrm{~kg}\), is used as a ferryboat. If three cars, each of mass \(1200 \mathrm{~kg}\), occupy its NE, SE, and SW corners, determine the \(\mathrm{CM}\) of the loaded ferryboat.

Problem 58P A 55-kg woman and an 80-kg man stand 10.0 m apart on frictionless ice. (a) How far from the woman is their CM? (b) If each holds one end of a rope, and the man pulls on the rope so that he moves 2.5 m, how far from the woman will he be now? (c) How far will the man have moved when he collides with the woman?

(II) Three cubes, of sides \(l_{0}, 2 l_{0}, \text { and } 3 l_{0}\), are placed next to one another (in contact) with their centers along a straight line and the \(l=2 l_{0}\) cube in the center (Fig. 7-39). What is the position, along this line, of the of this system? Assume the cubes are made of the same uniform material. Equation Transcription: Text Transcription: l_0, 2l_0, and 3l_0 l=2l_0

(II) A mallet consists of a uniform cylindrical head of mass 2.00 kg and a diameter 0.0800 m mounted on a uniform cylindrical handle of mass 0.500 kg and length 0.240 m, as shown in Fig. 7-42. If this mallet is tossed, spinning, into the air, how far above the bottom of the handle is the point that will follow a parabolic trajectory?

(II) Suppose that in Example (Fig. 7-29), \(m_{I I}=3 m_{I}\). Where then would \(m_{\Pi I}\) land? (b) What if \(m_{I}=3 m_{I I}\)? Equation Transcription: Text Transcription: m_II=3m_I M \Pi _I m_I=3m_II

Problem 61P A helium balloon and its gondola, of mass M, are in the air and stationary with respect to the ground. A passenger, of mass m, then climbs out and slides down a rope with speed v, measured with respect to the balloon. With what speed and direction (relative to Earth) does the balloon then move? What happens if the passenger stops?

Problem 66GP Two astronauts, one of mass 60 kg and the other 80 kg, are initially at rest in outer space. They then push each other apart. How far apart are they when the lighter astronaut has moved 12 m?

Problem 67GP A ball of mass m makes a head-on elastic collision with a second ball (at rest) and rebounds in the opposite direction with a speed equal to one-fourth its original speed. What is the mass of the second ball?

Problem 68GP You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car A of mass 1900 kg which crashed into stationary car B of mass 1100 kg. The driver of car A applied his brakes 15 m before he crashed into car B. After the collision, car A slid 18 m while car B slid 30 m. The coefficient of kinetic friction between the locked wheels and the road was measured to be 0.60. Show that the driver of car A was exceeding the 55-mph (90 km/h) speed limit before applying the brakes.

A bullet is fired vertically into a 1.40-kg block of wood at rest directly above it. If the bullet has a mass of 29.0 g and a speed of 510 m/s, how high will the block rise after the bullet becomes embedded in it?

Problem 71GP A 25-g bullet strikes and becomes embedded in a 1.35-kg block of wood placed on a horizontal surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface is 0.25, and the impact drives the block a distance of 9.5 m before it comes to rest, what was the muzzle speed of the bullet?

Problem 72GP Two people, one of mass 75 kg and the other of mass 60 kg, sit in a rowboat of mass 80 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat 3.2 m apart from each other, now exchange seats. How far and in what direction will the boat move?

A meteor whose mass was about \(1.0 \times 10^8 \mathrm{~kg}\) struck the Earth \(\left(m_{\mathrm{E}}=6.0 \times 10^{24} \mathrm{~kg}\right)\) with a speed of about \(15 \mathrm{~km} / \mathrm{s}\) and came to rest in the Earth. (a) What was the Earth's recoil speed? (b) What fraction of the meteor's kinetic energy was transformed to kinetic energy of the Earth? (c) By how much did the Earth's kinetic energy change as a result of this collision?

Problem 75GP The force on a bullet is given by the formula F = 580 ? (l.8 × 105)t over the time interval t = 0 to t = 3.0 × 10?3 s. In this formula, t is in seconds and F is in newtons. (a) Plot a graph of F vs. t for t = 0 to t = 3.0 ms. (b) Estimate, using graphical methods, the impulse given the bullet. (c) If the bullet achieves a speed of 220 m/s as a result of this impulse, given to it in the barrel of a gun, what must its mass be?

Two balls, of masses \(m_{A}=40 \mathrm{~g} \text { and } m_{B}=60 \mathrm{~g}\), are suspended as shown in Fig. . The lighter ball is pulled away to a angle with the vertical and released. (a) What is the velocity of the lighter ball before impact? (b) What is the velocity of each ball after the elastic collision? (c) What will be the maximum height of each ball after the elastic collision? Equation Transcription: Text Transcription: m_A=40 g and m_B=60 g

Problem 77GP An atomic nucleus at rest decays radioactively into an alpha particle and a smaller nucleus. What will be the speed of this recoiling nucleus if the speed of the alpha particle is 3.8 × 105m/s? Assume the recoiling nucleus has a mass 57 times greater than that of the alpha particle.

A -kg skeet (clay target) is fired at an angle of to the horizon with a speed of \(25 \mathrm{~m} / \mathrm{s}\) (Fig. ). When it reaches the maximum height, it is hit from below by a pellet traveling vertically upward at a speed of \(200 \mathrm{~m} / \mathrm{s}\) The pellet is embedded in the skeet. (a) How much higher did the skeet go up? (b) How much extra distance, \(\Delta x\). does the skeet travel because of the collision? Equation Transcription: Text Transcription: 25 m/s 200 m/s \Delta x

A block of mass m = 2.20 kg slides down a \(30.0^\circ\) incline which is 3.60 m high. At the bottom, it strikes a block of mass m=7.00 kg which is at rest on a horizontal surface, Fig. 7-46. (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine (a) the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go.

In Problem 79 (Fig. 7-46), what is the upper limit on mass if it is to rebound from , slide up the incline, stop, slide down the incline, and collide with again?