Power and efficiency · printable packet

Power is how fast energy is transferred, in watts; efficiency is the share of the energy put in that comes out useful.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

By the end of this lesson you will be able to find a power from an energy and a time, and an efficiency from the energy in and out.

2. What you already have

You can find the work a force does, the energy it transfers, in joules. You know that energy is never used up but can end up as heat. This lesson adds time: how quickly energy is transferred, and how much of it ends up doing something useful.

3. Words for this lesson

4. How fast, and how well

$$\text{efficiency} = \dfrac{\text{useful energy out}}{\text{total energy in}}.$$

Term	What it means
Power	The rate of energy transfer, $P = E/t$, in watts.
Watt	One joule per second, W.
Kilowatt	A thousand watts, kW.
Kilowatt-hour	The energy used by one kilowatt for one hour, on electric bills.
Efficiency	Useful energy out divided by total energy in, often a percent.
Wasted energy	Energy that ends up in a form nobody wanted, usually heat.

Another way: picture

Picture two students carrying identical boxes up the same stairs. One jogs up in ten seconds; the other walks up in thirty. Both do the same work, but the jogger does it three times as fast, with three times the power, which is why the jogger is out of breath.

Another way: steps

Find the energy transferred or work done, in joules.
Find the time in seconds.
Divide energy by time for the power.
For efficiency, divide useful energy by total energy.
Multiply by a hundred for a percent.

5. The unit tells you what to do

A watt is a joule per second. A motor that transfers $1200$ joules in $60$ seconds has a power of $1200 \div 60 = 20$ watts. The unit, joules per second, says to divide the energy by the time.

Every electrical appliance is labeled with its power. A $60$ W light bulb uses sixty joules every second; a $1500$ W hair dryer uses fifteen hundred.

6. Power and energy are different

Energy is an amount; power is a rate. A small phone battery and a large car battery may deliver the same power for a moment, but the car battery stores far more energy and can keep going much longer.

A bigger power does not mean more energy overall. A powerful hair dryer used for a minute can use less energy than a small lamp left on all day.

7. Kilowatt-hours

Electric companies in the United States bill for energy in kilowatt-hours. One kilowatt-hour is the energy a one-kilowatt appliance uses in one hour, three million six hundred thousand joules.

To find kilowatt-hours, multiply power in kilowatts by time in hours. A two-kilowatt space heater running for three hours uses six kilowatt-hours.

8. Efficiency

No machine turns all its energy into the form we want. A car engine turns only about a quarter of its fuel's energy into motion; the rest becomes heat. An electric motor does much better, often ninety percent.

Efficiency is useful energy out divided by total energy in. A motor that takes in $500$ J and delivers $400$ J of useful work is $400 \div 500 = 0.8$ efficient, or eighty percent.

9. Where the wasted energy goes

Energy that is not useful does not disappear; it usually becomes heat. A warm laptop, a hot light bulb and a car's hot engine all show wasted energy leaving as heat.

Engineers work hard to reduce this waste. More efficient machines use less energy for the same job, saving money and reducing the fuel burned at power plants.

10. The method, step by step, and how to check it

Checking an answer. Power must be in watts, energy in joules. Efficiency must never exceed one hundred percent. A faster job with the same work must have more power.

11. Why each step is allowed

Power is defined as the energy transferred per second, so dividing energy by time is the definition itself. When the energy is transferred at a steady rate, every second carries the same share.

Efficiency can never be more than one, because energy is conserved: a machine cannot put out more energy than it takes in. Claims of machines that do are always wrong.

12. Light bulbs

An old incandescent bulb turns only about five percent of its electrical energy into light; the rest becomes heat, which is why these bulbs get too hot to touch. An LED bulb turns about forty percent into light.

That is why a ten-watt LED can light a room as brightly as a sixty-watt incandescent bulb. Many American states have phased out incandescent bulbs to save energy.

13. Human power

A person can sustain about a hundred watts of useful power for hours, such as when cycling steadily. A trained athlete can manage several hundred watts, and a sprinter over a thousand for a few seconds.

Running up a flight of stairs quickly, a student might produce three hundred watts, enough to light five sixty-watt bulbs, but only for a few seconds.

14. Power in cars

Car engines in the United States are usually rated in horsepower. One horsepower is about seven hundred forty-six watts. A car with two hundred horsepower can deliver about a hundred fifty kilowatts.

Cruising on a highway takes much less than full power, perhaps twenty kilowatts. Full power is needed only for quick acceleration or climbing steep hills.

15. Common slips

The most common error is confusing power and energy, or watts and watt-hours. A watt is a rate; a watt-hour or kilowatt-hour is an amount of energy.

Another is dividing time by energy, or forgetting to convert minutes and hours into seconds when the answer is wanted in watts. Writing units at each step catches both.

16. Estimating power

A few benchmarks help. A phone charges at about five to twenty watts. A laptop uses about fifty. A microwave oven, about a thousand. A car engine at full throttle, over a hundred thousand.

An answer that says a person climbing stairs produces a million watts, or a light bulb uses half a watt, almost certainly contains a unit error.

17. Saving energy at home

The average American home uses about thirty kilowatt-hours a day. The biggest users are heating, air conditioning, water heaters and clothes dryers, which all have high power and run for long times.

Switching to LED bulbs, setting thermostats a few degrees higher in summer and lower in winter, and running full loads in the dishwasher all cut the energy used without changing daily life much.

18. Power plants

Power plants are rated by how many watts they can deliver. A large power plant may produce about a billion watts, a gigawatt, enough for hundreds of thousands of homes.

Even power plants are not perfectly efficient. A coal or gas plant turns only about a third to a half of its fuel's energy into electricity; the rest leaves as heat in cooling water and exhaust.

19. James Watt and horsepower

The watt is named after James Watt, the Scottish engineer whose improved steam engines helped power factories in the 1700s. To sell his engines, Watt compared them with the horses they replaced, measuring how much work a horse could do each minute and calling that rate one horsepower.

The name stuck. American car ads still list engine power in horsepower, and one horsepower is about seven hundred forty-six watts. So a lawn mower engine of five horsepower delivers about three and three-quarter kilowatts, and a pickup truck of four hundred horsepower, about three hundred kilowatts.

20. Power in sports

Cyclists in races like the Tour de France carry power meters on their bikes that measure how many watts they push into the pedals. A strong amateur might hold two hundred watts for an hour; a professional climbing a mountain can hold over four hundred.

Because climbing lifts the rider's own weight, coaches divide the power by the rider's mass. A lighter rider with the same power climbs faster, since less of the power goes into lifting body weight up the hill.

21. Batteries store energy, not power

A battery is labeled by how much energy it stores, often in watt-hours. A phone battery might hold fifteen watt-hours; an electric car's battery, seventy-five thousand. How fast that energy comes out depends on what the battery powers.

A phone playing video draws a few watts and lasts hours; a car accelerating hard draws hundreds of kilowatts. The same stored energy lasts a long time at low power and a short time at high power, since time is energy divided by power.

22. Checking efficiency answers

An efficiency must always lie between zero and one hundred percent. If a calculation gives more than one hundred percent, the useful and total energies have been swapped, or a unit is mixed up. Real machines usually fall between about twenty and ninety-five percent.

23. Time matters for power

The same job can take very different power depending on how quickly it is done. Lifting a heavy box slowly takes little power; snatching it up in an instant takes a great deal, although the energy is the same.

24. In the world: the electric bill

Every month, American households receive an electric bill measured in kilowatt-hours. A kilowatt-hour is the energy one kilowatt delivers in an hour, and a typical rate is about sixteen cents. The bill adds up the energy every appliance used.

Appliances with high power that run for long hours cost the most. A central air conditioner drawing three and a half kilowatts for eight hours on a hot day uses twenty-eight kilowatt-hours, over four dollars in a single day. A refrigerator, drawing much less power but running around the clock, uses about three and a half. Knowing power and time lets a family see where their energy goes and decide where to save.

25. In the world: efficient appliances

The U.S. Department of Energy and the Environmental Protection Agency run the Energy Star program, which labels appliances that are more efficient than standard models. An Energy Star refrigerator or washing machine does the same job while wasting less energy as heat.

Over years of use, the savings add up. Replacing ten incandescent bulbs, each sixty watts, with ten-watt LEDs saves five hundred watts whenever they are all on. Used four hours a day, that is two kilowatt-hours saved daily, over seven hundred kilowatt-hours a year, all from the higher efficiency of the new bulbs.

26. Power is not the same as energy

It is common to use power and energy as if they meant the same thing. But power is how fast energy is used. A powerful hair dryer used for one minute may use less energy than a dim lamp left on all night.

A related error is to think a machine could be more than one hundred percent efficient. Energy is conserved, so no machine can give out more energy than it takes in. Every real machine wastes some, usually as heat.

27. A crane lifting a beam

28. Comparing light bulbs

29. A day of television

30. Your turn: a motor does $3600$ J of work in $12$ s. What is its power?

31. Guided practice

32. Guided practice

Complete the worked solution: a $55$ kg student runs up stairs $5$ m high in $10$ s. With $g = 9.8$ N/kg, find the work done against gravity in J, the student's power in W, and how many sixty-watt light bulbs would use energy at the same rate.

33. Guided practice

34. Practice

An electric motor takes in $200$ W and is $0.9$ efficient, as a fraction. It runs for $300$ s. Fill in its useful power output in W, the energy it takes in over that time in kJ, and the energy wasted in kJ.

35. Practice

an old light bulb uses $60$ W while it runs. Write the energy it uses, in J, as a function of the time $t$ it runs, in seconds.

36. Practice

For every $200$ J of energy put into an old incandescent bulb, about $10$ J come out as the useful form. What is its efficiency, as a percent?

37. Somewhere new

In a typical American home, a home electric car charger uses about $7.2$ kW while running. If it runs for $6$ hours in a day, how much energy does it use, in kilowatt-hours?

38. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

39. Test question

an LED bulb uses $10$ W while it runs. Write the energy it uses, in J, as a function of the time $t$ it runs, in seconds.

40. What you can do now

You can find power and efficiency. Explain to someone why a ten-watt LED can replace a sixty-watt light bulb.

Working for the steps left to you

	value
useful power (W)
energy taken in (kJ)
energy wasted (kJ)

30. Your turn: a motor does $3600$ J of work in $12$ s. What is its power?, step 3