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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?
Integrated Concepts
(a) What force must be supplied by an elevator cable to produce an acceleration of $0\text{.}{\text{800 m/s}}^{2}$ 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?
(a) $\text{16}\text{.}1\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\mathrm{N}$
(b) $3\text{.}\text{22}\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\mathrm{J}$
(c) 5.66 m/s
(d) 4.00 kJ
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?
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?
(a) $4\text{.}\text{65}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{kcal}$
(b) 38.8 kcal/min
(c) This power output is higher than the highest value on [link] , which is about 35 kcal/min (corresponding to 2415 watts) for sprinting.
(d) It would be impossible to maintain this power output for 2 hours (imagine sprinting for 2 hours!).
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.)
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.
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?
(a) 4.32 m/s
(b) $3\text{.}\text{47}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\mathrm{N}$
(c) 8.93 kW
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