Concept #1: Power

Practice: A 350 g mass can undergo two different motions, propelled by gravity in each case: it can either fall a height of 10 cm, or it can slide down a frictionless incline at a 30° angle. In which case is the mass’ kinetic energy changing faster?

Air resistance at high speed is often given as Fair = bv2. If the terminal velocity of a 75 kg skydiver is 60 m/s, at what rate is the air resistance sapping the skydiver’s kinetic energy?

A car moves at a speed of 25 m/s. If the car were to experience 4.5 kN of air resistance at this speed, how much power would the engine need to produce to keep the car moving at a constant speed?

Your friend (mass 65.0 kg) is standing on the ice in the middle of a frozen pond. There is very little friction between her feet and the ice, so she is unable to walk. Fortunately, there is a light rope tied around her waist and you stand on the bank holding the other end. You pull on the rope for 3.00 s
and accelerate your friend from rest to a speed of 6.00 m/s
while you remain at rest.What is the average power supplied by the force you applied?

When the 210-kg Genesis Mission capsule crashed with a speed of 311 km/h, it buried itself 81.0 cm deep in the desert floor.Assuming constant acceleration during the crash, at what average rate did the capsule do work on the desert?

A 68 kg human sprinter can accelerate from rest to 10 m/s in 3.1 s . During the same time interval, a 31 kg greyhound can go from rest to 20 m/s.What is the average power output of the human? Average power over a time interval t is E/t.What is the average power output of the greyhound?

A 20.0-kg rock is sliding on a rough, horizontal surface at 8.00 m/s and eventually stops due to friction. The coefficient of kinetic friction between the rock and the surface is 0.200.What average thermal power is produced as the rock stops?

A pump is required to lift a mass of 810 kg of water per minute from a well of depth 14.1 m and eject it with a speed of 18.1 m/s .How much work is done per minute in lifting the water?How much in giving the water the kinetic energy it has when ejected?What must be the power output of the pump?

The Grand Coulee Dam is 1270 m long and 170 m high. The electrical power output from generators at its base is approximately 2010 MW .How many cubic meters of water must flow from the top of the dam per second to produce this amount of power if 91.0 % of the work done on the water by gravity is converted to electrical energy? (Each cubic meter of water has a mass of 1000 kg.)

An 80 kg football player traveling 5.0 m/s is stopped in 1.3 s by a tackler.What is the original kinetic energy of the player?What average power is required to stop him?

Six dogs pull a two-person sled with a total mass of
280
kg. The coefficient of kinetic friction between the sled and the snow is 0.080. The sled accelerates at
0.85 m/s2
until it reaches a cruising speed of
14
km/h.What is the teams maximum power output during the acceleration phase?What is the teams power output during the cruising phase?

A 1300 kg sports car accelerates from rest to 90 km/h in 7.3 s .What is the average power delivered by the engine?

You may want to review (Pages 189 - 191).
For related problemsolving
tips and strategies, you may
want to view a Video Tutor Solution of A "power climb".How many joules of energy does a 100-watt light bulb use per hour?How fast would a 71 kg person have to run to have that amount of energy?

For a physics lab experiment, four classmates run
up the stairs from the basement to the top floor of their physics
building - a vertical distance of
16.0 m . The classmates and their
masses are: Tatiana, 50.2 kg; Bill, 68.2 kg; Ricardo, 81.8 kg;
and Melanie, 59.1 kg. The time it takes each of them is shown in
.Considering only the work done against gravity,
which person had the largest average power output?Considering only the work done against gravity,
which person had the smallest average power output?Chang is very fit and has mass
64.7 kg . If his average power
output is 1.00 hp, how many seconds does it take him to run up the
stairs?

The hammer of a pile driver weighs 3800 N and must be lifted 2.80 m vertically at constant speed in 4.00 s.What horsepower must the engine provide to the hammer?

You may want to review (Pages 222 - 223).How much work does an elevator motor do to lift a
1600 kg
elevator a height of 160 m ?How much power must the motor supply to do this in 51 s at constant speed?

In a hydroelectric dam, water falls 30.0 m and then spins a turbine to generate electricity.
You may want to review (Pages 234 - 238).What is exttip{Delta U}{DeltaU} of 1.0 kg of water?Suppose the dam is 80% efficient at converting the waters potential energy to electrical energy. How many kilograms of water must pass through the turbines each second to generate 51.0 MW of electricity? This is a typical value for a small hydroelectric dam.

When its engine of power 76 kW is generating full power, a small single-engine airplane with mass 710 kg gains altitude at a rate of 2.6 m/s .What fraction of the engine power is being used to make the airplane climb? (The remainder is used to overcome the effects of air resistance and of inefficiencies in the propeller and engine.)

An elevator has mass 600 kg, not including passengers. The elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 s, and it is driven by a motor that can provide up to 40 hp to the elevator.What is the maximum number of passengers that can ride in the elevator? Assume that an average passenger has mass 65.0 kg.

A hydroelectric power plant uses spinning turbines to transform the kinetic energy of moving water into electric energy with 80 % efficiency. That is, 80 % of the kinetic energy becomes electric energy. A small hydroelectric plant at the base of a dam generates 50 MW of electric power when the falling water has a speed of 18 m/s.What is the water flow rate - kilograms of water per second - through the turbines?

A
95 kg
firefighter needs to climb the stairs of a
30-m-tall building while carrying a
45 kg
backpack filled with gears.How much power does he need to reach the top in
60
s?

How long will it take a 1800 W motor to lift a 330 kg piano to a sixth-story window 16.5 m above?

A ski area claims that its lifts can move 44000 people per hour.If the average lift carries people about 210 m (vertically) higher, estimate the maximum total power needed. (Assume the mass of the average person to be 70 kg.)

What is the average power output of an elevator that lifts 880 kg a vertical height of 33.0 m in 13.0 s ?

Water flows over a dam at the rate of 560 kg/s and falls vertically 93 m before striking the turbine blades.Calculate the speed of the water just before striking the turbine blades (neglect air resistance).Calculate the rate at which mechanical energy is transferred to the turbine blades, assuming 55 % efficiency.

A 70 kg hiker climbs to the top of a 4300 m -high mountain. The climb is made in 5.5 h starting at an elevation of 2800 m .Calculate the work done by the hiker against gravity.Calculate the average power output in watts.Calculate assuming the body is 16 % efficient, what rate of energy input was required.Calculate the average power output in horsepower.

A fire hose for use in urban areas must be able to shoot a stream of water to a maximum height of 33 m . The water leaves the hose at ground level in a circular stream 3.5 cm in diameter.What minimum power is required to create such a stream of water? Every cubic meter of water has a mass of 1.00 103 kg.

Some electric power companies use water to store energy. Water is pumped by reversible turbine pumps from a low reservoir to a high reservoir.To store the energy produced in 1.0 hour by a 180 MW electric power plant, how many cubic meters of water will have to be pumped from the lower to the upper reservoir? Assume the upper reservoir is 360 m above the lower one, and we can neglect the small change in depths of each. Water has a mass of 1.00 103 kg for every 1.0 m3.

The motor of a crane uses power P to lift a steel beam.By what factor must the motors power increase to lift the beam twice as high in half the time?

If a volcano spews a 500 kg rock vertically upward a distance of 330 m , what was its velocity when it left the volcano?If the volcano spews 1300 rocks of this size every minute, estimate its power output.

Your job is to lift 30-kg
crates a vertical distance of 0.90 m
from the ground onto the bed of a truck.You may want to review (Pages 189 - 191).
For related problem-solving tips and strategies, you may want to view a Video Tutor Solution of Force and power.How many crates would you have to load onto the truck in one minute for the average power output you use to lift the crates to equal 0.50 hp?How many crates for an average power output of 100 W?

A typical flying insect applies an average force equal to twice its weight during each downward stroke while hovering. Take the mass of the insect to be 10 g, and assume the wings move an average downward distance of 1.0 cm during each stroke.You may want to review (Pages 189 - 191).
For related problem-solving tips and strategies, you may want to view a Video Tutor Solution of A "power climb".Assuming 100 downward strokes per second, estimate the average power output of the insect.

The aircraft carrier John F. Kennedy has mass 7.4107 kg. When its engines are developing their full power of 280000 hp, the John F. Kennedy travels at its top speed of 35 knots
( 65 km/h ).If 70% of the power output of the engines is applied to pushing the ship through the water, what is the magnitude of the force of water resistance that opposes the carriers motion at this speed?

When you ride a bicycle at constant speed, nearly all the energy you expend goes into the work you do against the drag force of the air. Model a cyclist as having cross-section area 0.46 m2 and, because the human body is not aerodynamically shaped, a drag coefficient of 0.90.What is the cyclists power output while riding at a steady 7.3 m/s (16 mph)? Metabolic power is the rate at which your body "burns" fuel to power your activities. For many activities, your body is roughly 25% efficient at converting the chemical energy of food into mechanical energy.
What is the cyclists metabolic power while cycling at 7.3 m/s? The food calorie is equivalent to 4190 J.
How many calories does the cyclist burn if he rides over level ground at 7.3 m/s for 1.6 h ?

If a car generates 21 hp when traveling at a steady 85 km/h , what must be the average force exerted on the car due to friction and air resistance?

A driver notices that her 1180 kg car slows down from 96 km/h to 64 km/h in about 7.0 s on the level when it is in neutral.Approximately what power (in watts) is needed to keep the car traveling at a constant 80 km/h?Approximately what power (in hp) is needed to keep the car traveling at a constant 80 km/h?

An outboard motor for a boat is rated at 65 hp .If it can move a particular boat at a steady speed of 45 km/h , what is the magnitude of the total force resisting the motion of the boat?What is the direction of the total force resisting the motion of the boat?

A tandem (two-person) bicycle team must overcome a force of 170 N to maintain a speed of 8.50 m/s .You may want to review (Pages 189 - 191).
For related problemsolving
tips and strategies, you may
want to view a Video Tutor Solution of A "power climb".Find the power required per rider, assuming that each contributes equally.Express your answer in watts.Express your answer in horsepower.

The total consumption of electrical energy in the United States is about 1.0 1019 joules per year.What is the average rate of electrical energy consumption in watts?If the population of the United States is 300 million, what is the average rate of electrical energy consumption per person?The sun transfers energy to the earth by radiation at a rate of approximately 1.0 kW
per square meter of surface. If this energy could be collected and converted to electrical energy with 40 %
efficiency, how great an area (in square kilometers) would be required to collect the electrical energy used by the United States?

It is not unusual for a 1000-kg car to get 30 mi/gal when traveling at 60 mi/h on a level road.If this car makes a 200-km trip, how many joules of energy does it consume?Note that 1.0 gal of gasoline yields 1.3 109 J (although this can vary).What is the average rate of energy consumption during the trip?

At midday, solar energy strikes the earth with an intensity of about 1 kW/m2.What is the area of a solar collector that could collect 150 MJ of energy in 1 hr? This is roughly the energy content of 1 gallon of gasoline.

Astronomers using a 2.0-m-diameter telescope observe a distant supernova-an exploding star. The telescopes detector records 8.3 -11 10J s of light energy during the first 10 37. Its known that this type of supernova has a visible-light power output of 5.5 -11 10s
m W for the first 10 37 of the explosion.How distant is the supernova? Give your answer in light years, where one light year is the distance light travels in one year. The speed of light is 3.0 8 10m/s
m m/s.

All birds, independent of their size, must maintain a power output of 10-25 watts per kilogram of body mass in order to fly by flapping their wings.The Andean giant hummingbird (Patagona gigas) has mass 70 g and flaps its wings 10 times per second while hovering. Estimate the amount of work done by such a hummingbird in each wingbeat.Explain.A 70-kg athlete can maintain a power output of 1.4 kW for no more than a few seconds; the steady power output of a typical athlete is only 500 W or so. Is it possible for a human-powered aircraft to fly for extended periods by flapping its wings?

A physics student spends part of her day walking between classes or for recreation, during which time she expends energy at an average rate of 280 W . The remainder of the day she is sitting in class, studying, or resting; during these activities, she expends energy at an average rate of 110 W .If she expends a total of 1.4×107 J of energy in a 24-hour day, how much of the day did she spend walking?

How much work can a 5.0 hp motor do in 2.0 h ?

On December 27, 2004, astronomers observed the greatest flash of light ever recorded from outside the solar system. It came from the highly magnetic neutron star SGR 180-20(a magnetar). During 0.20 s, this star released as much energy as our sun does in 250000 years.You may want to review (Pages 189 - 191).
For related problem-solving tips and strategies, you may want to view a Video Tutor Solution of A "power climb".If P is the average power output of our sun, what was the average power output (in terms of P) of this magnetar?

Six diesel units in series can provide 13.4 MW of power to the lead car of a freight train. The diesel units have total mass 1.10
106. The average car in the train has mass 8.2
104 and requires a horizontal pull of 2.8 kN to move at a constant 27 m/s on level tracks.How many cars can be in the train under these conditions?This would leave no power for accelerating or climbing hills. Show that the extra force needed to accelerate the train is about the same for a 0.10 m/s2 acceleration or a 1.0% slope (slope angle alpha=arctan 0.010).With the 1.0% slope, show that an extra 2.9 MW of power is needed to maintain the 27m/s speed of the diesel units.With 2.9 MW less power available, how many cars can the six diesel units pull up a 1.0% slope at a constant 27 m/s?

It takes a force of 51 kN on the lead car of a 16-car passenger train with mass 9.1105 kg to pull it at a constant 46 m/s on level tracks.What power must the locomotive provide to the lead car?How much more power to the lead car than calculated in part A would be needed to give the train an acceleration of 1.6 m/s2 , at the instant that the train has a speed of 46 m/s on level tracks?How much more power to the lead car than that calculated in part A would be needed to move the train up a 1.5% grade (slope angle m/s=arctan 0.015) at a constant 46 {
m m/s} ?

If 12.0 hp are required to drive a 1800-kg automobile at 61.0 km/h on a level road, what is the total retarding force due to friction, air resistance, and so on?What power is necessary to drive the car at 61.0 km/h up a 10.0% grade (a hill rising 10.0 m vertically in 100.0 m horizontally)?What power is necessary to drive the car at 61.0 km/h
down a 1.00 % grade?Down what percent grade would the car coast at 61.0 km/h ?

A farmer uses a tractor to pull a
190 kg
bale of hay up a 15km/h
incline to the barn at a steady 5.0
m km/h. The coefficient of kinetic friction between the bale and the ramp is
0.50.What is the tractors power output?

A 80 kg skier grips a moving rope that is powered by an engine and is pulled at constant speed to the top of a 23 hill. The skier is pulled a distance exttip{x}{x} = 230 m along the incline and it takes 2.3 min to reach the top of the hill.If the coefficient of kinetic friction between the snow and skis is exttip{mu_{
m k}}{mu_k} = 0.13, what horsepower engine is required if 30 such skiers (max) are on the rope at one time?

A bicyclist of mass 75 kg (including the bicycle) can coast down a 4.0 hill at a steady speed of 12 km/h. Pumping hard, the cyclist can descend the hill at a speed of 32 km/h.Using the same power, at what speed can the cyclist climb the same hill? Assume the force of friction is proportional to the square of the speed v; that is, Ffr = bv2, where b is a constant.

In a common test for cardiac function (the "stress test"), the patient walks on an inclined treadmill. Estimate the power required from a 75 kg patient when the treadmill is sloping at an angle of 13 and the velocity is 3.7 km/h .How does this power compare to the power rating of a lightbulb? (Assume the power of a light bulb 100 W .)

A ski tow operates on a slope of angle 15.1 of length 310 m . The rope moves at a speed of 12.1 km/h and provides power for 51 riders at one time, with an average mass per rider of 71.0 kg .You may want to review (Pages 189 - 191).
For related problemsolving
tips and strategies, you may
want to view a Video Tutor Solution of A "power climb".Estimate the power required to operate the tow.

A 10 kg object slides along a surface with a coefficient of kinetic friction of 0.3, moving with an initial velocity of 20 m/s. What is the average thermal power generated as the object slows to a stop?

How much work must you do to push a 12.0 kg block of steel across a steel table (μk=0.6) at a steady speed of 1.20 m/s for 5.80 s? What is your power output while doing so?

It is 6.00 km from your home to the physics lab. As part of your physical fitness program, you could run that distance at 10.0 km/hr (which uses up energy at the rate of 700 W), or you could walk it leisurely at 3.00 km/hr (which uses energy at 290 W).(a) Which choice would burn up more energy?(b) Why is it that the more intense exercise actually burns up less energy than the less intense one?(c) How much energy (in joules) would it burn?

(a) How much energy is consumed by a 1.2 kW hair dryer used for 12 min?(b) How much energy is consumed by a 11 W night light left on for 18 hr?

An 820-N Marine in basic training climbs a 12.0-m vertical rope at a constant speed in 8.00 s. What is his power output?

During a workout, football players ran up the stadium stairs in 80 s. The stairs are 83 m long and inclined at an angle of 31 degrees. If a player has a mass of 87 kg, estimate his average power output on the way up. Ignore friction and air resistance.

A bicyclist coasts down a 6.5 degree hill at a steady speed of 4.5 m/s. Assuming a total mass of 75 kg (bicycle plus rider), what must be the cyclists power output to climb the same hill at the same speed?

A test car of mass 740 kg is moving at a speed of 7.3 m/s when it crashes into a wall to test its bumper. If the car comes to rest in 0.36 s, how much average power is expended in the process?
1. 42820.9
2. 47940.6
3. 59118.6
4. 49662.2
5. 50789.2
6. 54770.3
7. 45825.2
8. 44990.3
9. 46540.4
10. 51904.1

Four students run up the stairs as shown in the figure: Adam (mass of 80 kg) goes up 10 m in 10 s, Bob (80 kg) 10 m in 8 seconds, Claire (64 kg) 10 m in 8 s, and David (80 kg) 20 m in 25 s.
Who has the smallest power output?
A. Bob
B. Claire
C. David
D. Adam

The electric motor of a model train accelerates the train from rest to 0.620 m/s in 21.0 ms. The total mass of the train is 875 g.(a) Find the minimum power delivered to the train by electrical transmission from the metal rails during the acceleration.(b) Why is it the minimum power?

Sewage at a certain pumping station is raised vertically by 5.49 m at the rate of 1 890 000 liters each day. The sewage, of density 1 050 kg/m3, enters and leaves the pump at atmospheric pressure and through pipes of equal diameter.(a) Find the output mechanical power of the lift station.(b) Assume an electric motor continuously operating with average power 5.90 kW runs the pump. Find its efficiency.

A certain rain cloud at an altitude of 1.75 km contains 3.20 x 107 kg of water vapor. How long would it take a 2.70-kW pump to raise the same amount of water from the Earth’s surface to the cloud’s position?

An electric scooter has a battery capable of supplying 120 Wh of energy. If friction forces and other losses account for 60.0% of the energy usage, what altitude change can a rider achieve when driving in hilly terrain if the rider and scooter have a combined weight of 890 N?

A 3.50-kN piano is lifted by three workers at constant speed to an apartment 25.0 m above the street using a pulley system fastened to the roof of the building. Each worker is able to deliver 165 W of power, and the pulley system is 75.0% efficient (so that 25.0% of the mechanical energy is transformed to other forms due to friction in the pulley). Neglecting the mass of the pulley, find the time required to lift the piano from the street to the apartment.

A pump lifts 23.0 kg of water per minute through a height of 3.60 m .What minimum power output rating (watts) must the pump motor have?

While running, a person transforms about 0.600 J of chemical energy to mechanical energy per step per kilogram of body mass. If a 60.0-kg runner transforms energy at a rate of 70.0 W during a race, how fast is the person running? Assume that a running step is 1.50 m long.

When an automobile moves with constant speed down a highway, most of the power developed by the engine is used to compensate for the energy transformations due to friction forces exerted on the car by the air and the road. If the power developed by an engine is 175 hp, estimate the total friction force acting on the car when it is moving at a speed of 29 m/s. One horsepower equals 746 W.

An energy-efficient lightbulb, taking in 28.0 W of power, can produce the same level of brightness as a conventional lightbulb operating at power 100 W. The lifetime of the energy-efficient bulb is 10 000 h and its purchase price is $4.50, whereas the conventional bulb has a lifetime of 750 h and costs $0.42. Determine the total savings obtained by using one energy-efficient bulb over its lifetime as opposed to using conventional bulbs over the same time interval. Assume an energy cost of $0.200 per kilowatt-hour.

For saving energy, bicycling and walking are far more efficient means of transportation than is travel by automobile. For example, when riding at 10.0 mi/h, a cyclist uses food energy at a rate of about 400 kcal/h above what he would use if merely sitting still. (In exercise physiology, power is often measured in kcal/h rather than in watts. Here 1 kcal = 1 nutritionist’s Calorie = 4 186 J.) Walking at 3.00 mi/h requires about 220 kcal/h. It is interesting to compare these values with the energy consumption required for travel by car. Gasoline yields about 1.30 x 108 J/gal. Find the fuel economy in equivalent miles per gallon for a person (a) walking and (b) bicycling.

The mass of a car is 1 500 kg. The shape of the car’s body is such that its aerodynamic drag coefficient is D = 0.330 and its frontal area is 2.50 m2. Assuming the drag force is proportional to v2 and ignoring other sources of friction, calculate the power required to maintain a speed of 100 km/h as the car climbs a long hill sloping at 3.20°.

A loaded ore car has a mass of 950 kg and rolls on rails with negligible friction. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at 30.08 above the horizontal. The car accelerates uniformly to a speed of 2.20 m/s in 12.0 s and then continues at constant speed. (a) What power must the winch motor provide when the car is moving at constant speed? (b) What maximum power must the winch motor provide? (c) What total energy has transferred out of the motor by work by the time the car moves off the end of the track, which is of length 1 250 m?