Power

Power is the rate at which work is done or energy is transferred.

Power tells us how quickly energy is transferred.

Two machines can transfer exactly the same amount of energy but have very different power outputs if one transfers the energy in less time.

For example:

• two people climb the same staircase, but one runs while the other walks
• two kettles heat the same amount of water, but one heats it faster
• two motors lift the same load, but one completes the lift more quickly
• two lamps transfer energy continuously but at different rates

The faster energy is transferred, the greater the power.

 

 

 

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Energy and Power Are Different

Energy and power are closely related, but they are not the same quantity.

Energy describes how much energy is transferred or stored.

Power describes how quickly that energy is transferred.

For example, suppose two motors each transfer 10,000 J of energy.

Motor A takes:

20 s

Motor B takes:

5 s

They transfer the same amount of energy, but Motor B transfers it much faster.

Therefore, Motor B has the greater power.

The Power Equation

Power can be calculated using:

P = E/t

where:

• P = power in watts (W)
• E = energy transferred in joules (J)
• t = time in seconds (s)

Power can also be expressed as:

P = W/t

where W represents work done, not watts.

This means:

power = energy transferred ÷ time

or:

power = work done ÷ time

The Watt

The SI unit of power is the watt (W).

One watt means that one joule of energy is transferred every second.

Therefore:

1 W = 1 J/s

A device operating at:

10 W

transfers:

10 J every second

A device operating at:

1000 W

transfers:

1000 J every second

Kilowatts and Megawatts

Large power values are often expressed using larger units.

1 kW = 1000 W

1 MW = 1,000,000 W

1 GW = 1,000,000,000 W

For example:

2000 W = 2 kW

5,000,000 W = 5 MW

Small electrical devices may use only a few watts, while power stations may produce hundreds or thousands of megawatts.

Worked Example: Basic Power Calculation

A machine transfers 600 J of energy in 3 s.

Calculate its power.

Use:

P = E/t

Substitute:

P = 600/3

P = 200 W

The machine has a power output of 200 W.

This means it transfers energy at a rate of:

200 J/s

Worked Example: Electric Device

An electrical device transfers 12,000 J of energy in 60 s.

P = E/t

P = 12,000/60

P = 200 W

The device operates at 200 W.

Worked Example: Larger Machine

A machine transfers 2.4 MJ of energy in 40 s.

First convert megajoules to joules:

2.4 MJ = 2,400,000 J

Then:

P = 2,400,000/40

P = 60,000 W

or:

P = 60 kW

Comparing Power

Suppose two machines perform the same task.

Machine A

Energy transferred = 5000 J
Time = 10 s

P = 5000/10

P = 500 W

Machine B

Energy transferred = 5000 J
Time = 5 s

P = 5000/5

P = 1000 W

Machine B has twice the power because it transfers the same energy in half the time.

Power and Time

For a fixed amount of energy:

shorter time → greater power

longer time → smaller power

For example:

The total energy is identical in every case.

Only the rate of energy transfer changes.

Power and Energy

If the time remains constant:

more energy transferred → greater power

For example:

Power is directly proportional to energy transferred when time remains constant.

Rearranging the Power Equation

Starting with:

P = E/t

we can rearrange the equation to calculate energy or time.

Finding Energy

Multiply both sides by t:

E = Pt

Finding Time

Rearrange:

t = E/P

These three forms are useful:

P = E/t

E = Pt

t = E/P

Worked Example: Finding Energy

A 500 W motor operates for 20 s.

Calculate the energy transferred.

Use:

E = Pt

E = 500 × 20

E = 10,000 J

The motor transfers 10 kJ of energy.

Worked Example: Finding Time

A machine must transfer 24,000 J of energy.

Its power is:

800 W

How long does it take?

Use:

t = E/P

t = 24,000/800

t = 30 s

Power and Work Done

Because work done represents energy transferred:

P = W/t

where:

• P = power
• W = work done
• t = time

Recall:

work done = force × distance

For a constant force acting in the direction of motion:

W = Fd

Therefore:

P = Fd/t

Since:

v = d/t

we can also write:

P = Fv

for a constant force acting in the direction of motion.

Worked Example: Lifting a Load

A motor does 15,000 J of work lifting a load in 5 s.

Calculate its power.

P = W/t

P = 15,000/5

P = 3000 W

or:

P = 3.0 kW

Power and Gravitational Potential Energy

When an object is lifted:

ΔEₚ = mgΔh

If the lift takes time t:

P = mgΔh/t

This allows us to calculate the power required to raise an object.

 

 

 

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Worked Example: Crane

A crane lifts a 500 kg load vertically through 8 m in 10 s.

Use:

g = 9.8 N/kg

First calculate the change in gravitational potential energy:

ΔEₚ = mgΔh

ΔEₚ = 500 × 9.8 × 8

ΔEₚ = 39,200 J

Now calculate power:

P = E/t

P = 39,200/10

P = 3920 W

or:

P = 3.92 kW

Worked Example: Climbing Stairs

A 60 kg student climbs a staircase with a vertical height of 5 m in 6 s.

Use:

g = 9.8 N/kg

GPE gained:

ΔEₚ = 60 × 9.8 × 5

ΔEₚ = 2940 J

Power:

P = 2940/6

P = 490 W

The student's average mechanical power while climbing is approximately 490 W.

Comparing Two Students

Suppose two 60 kg students climb the same 5 m staircase.

Student A takes:

10 s

Student B takes:

5 s

Both gain the same GPE:

2940 J

Student A:

P = 2940/10 = 294 W

Student B:

P = 2940/5 = 588 W

Both perform the same amount of mechanical work, but Student B has twice the average power because the work is completed in half the time.

Power and Kinetic Energy

Power can also describe how quickly kinetic energy changes.

Recall:

Eₖ = ½mv²

If a motor increases an object's kinetic energy, the average power associated with that transfer is:

P = ΔEₖ/t

For an object accelerating from rest:

P = ½mv²/t

if we are calculating the average power over that time interval.

Worked Example: Accelerating a Car

A 1000 kg car accelerates from rest to 20 m/s in 10 s.

Ignore energy losses.

Final kinetic energy:

Eₖ = ½mv²

Eₖ = 0.5 × 1000 × 20²

Eₖ = 200,000 J

Average power:

P = E/t

P = 200,000/10

P = 20,000 W

or:

P = 20 kW

This is the average mechanical power transferred into kinetic energy during the acceleration.

A real engine or motor would need to provide more energy because of friction, air resistance, and other losses.

Average Power and Instantaneous Power

Power can change during a process.

The equation:

P = ΔE/Δt

usually gives average power over a time interval.

For example, a car's engine may produce different power at different moments during acceleration.

Instantaneous power describes the rate of energy transfer at one particular moment.

For many introductory problems, we calculate average power.

Electrical Devices

Electrical appliances are commonly labelled with their power.

For example, a device rated at:

1000 W

transfers approximately:

1000 J every second

while operating at that power.

A 2000 W device transfers energy twice as quickly as a 1000 W device.

 

 

 

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Example: Electric Kettle

A kettle operates at:

2000 W

How much energy does it transfer in 3 minutes?

First convert time:

3 min = 180 s

Use:

E = Pt

E = 2000 × 180

E = 360,000 J

or:

E = 360 kJ

Example: Light Bulb

A lamp operates at 10 W for 60 s.

E = Pt

E = 10 × 60

E = 600 J

The lamp transfers 600 J of energy during that minute.

Not all of this necessarily becomes visible light. Some energy is transferred thermally.

Power Does Not Mean Total Energy

A high-power device does not necessarily transfer more total energy than a low-power device.

It depends on how long each device operates.

For example:

A 1000 W device operating for 10 s transfers:

E = 1000 × 10

E = 10,000 J

A 100 W device operating for 200 s transfers:

E = 100 × 200

E = 20,000 J

The lower-power device transferred more total energy because it operated for much longer.

This is an important distinction:

power = rate

energy = total amount transferred

Power in Vehicles

Vehicle engines and motors are often compared by their power output.

Greater power allows energy to be transferred more rapidly.

This can affect a vehicle's ability to:

• accelerate
• climb hills
• pull heavy loads
• maintain speed against resistive forces

However, power alone does not determine vehicle performance. Mass, gearing, traction, drag, efficiency, and other factors also matter.

 

 

 

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Power in Sports

Power is important in many sports.

Activities requiring high power include:

• sprinting
• jumping
• weightlifting
• cycling
• rowing
• climbing quickly

Two athletes may perform the same amount of work, but the athlete who completes it faster produces greater average mechanical power.

Worked Example: Weightlifting

An athlete lifts a 100 kg mass vertically through 2 m in 1.5 s.

Use:

g = 9.8 N/kg

Work done against gravity:

W = mgΔh

W = 100 × 9.8 × 2

W = 1960 J

Power:

P = 1960/1.5

P ≈ 1307 W

or approximately:

1.31 kW

Power Stations

Power stations are designed to transfer enormous amounts of energy every second.

For example, a power station producing:

500 MW

has a power output of:

500,000,000 W

This means it transfers approximately:

500,000,000 J of energy every second

when operating at that output.

This demonstrates why watts are a rate, not an amount of energy.

Power and Efficiency

A device may receive energy at one power but provide useful energy at a smaller output power.

For example:

Input power = 1000 W

Useful output power = 800 W

Efficiency:

efficiency = useful output power / input power

efficiency = 800/1000

efficiency = 0.80

or:

80%

The remaining power is being transferred into other forms, often thermal energy and sound.

Worked Example: Motor Efficiency

An electric motor receives:

2500 W

of electrical power.

Its efficiency is:

80%

Useful mechanical power:

useful power = efficiency × input power

useful power = 0.80 × 2500

useful power = 2000 W

Therefore:

500 W

is being transferred into other forms while the motor operates under those conditions.

Energy Flow and Power

Energy flow diagrams can also include power.

For example:

2000 W electrical input → 1600 W useful mechanical output + 400 W thermal/sound

This means that every second:

2000 J enters

1600 J becomes useful mechanical energy

400 J is transferred elsewhere

The energy still balances.

Power and Energy Conservation

Power does not change the law of conservation of energy.

A powerful machine does not create more energy.

It simply transfers energy more rapidly.

For example, two motors may each transfer:

100 kJ

Motor A might take:

100 s

Motor B might take:

20 s

Motor A:

P = 100,000/100 = 1000 W

Motor B:

P = 100,000/20 = 5000 W

Both transfer the same energy.

Motor B simply does so five times faster.

Real-World Comparison: Kettles

Suppose two idealized kettles must transfer 300 kJ to water.

Kettle A

Power = 1000 W

t = E/P

t = 300,000/1000

t = 300 s

Kettle B

Power = 2000 W

t = 300,000/2000

t = 150 s

The higher-power kettle can transfer the required energy in half the time, assuming equal efficiency and identical conditions.

Real-World Comparison: Elevators

Consider two elevators lifting identical loads through the same height.

Both perform the same amount of work.

If Elevator A completes the lift in 20 s and Elevator B completes it in 10 s:

Elevator B has twice the average mechanical power.

This is because:

P = E/t

Halving the time while transferring the same energy doubles the power.

Power and Time Graphs

If a device operates at constant power, energy transferred increases steadily with time.

Since:

E = Pt

a graph of energy against time for constant power is a straight line.

The gradient is:

gradient = ΔE/Δt

Therefore:

gradient = power

A steeper energy-time graph represents a more powerful device.

Variable Power

Not every device operates at constant power.

Suppose a machine's power changes over time.

The total energy transferred depends on the area under a power-time graph.

For constant power:

E = Pt

which is simply:

area = height × width

For changing power, the area under the graph can still represent the total energy transferred.

This becomes useful when analyzing engines, motors, electrical systems, and changing mechanical processes.

Common Mistakes

Confusing Power with Energy

Energy is measured in:

joules (J)

Power is measured in:

watts (W)

Power describes the rate of energy transfer.

Thinking a More Powerful Device Always Uses More Energy

Not necessarily.

Energy depends on both:

power and operating time

because:

E = Pt

Forgetting to Convert Minutes to Seconds

For standard SI calculations:

1 min = 60 s

Therefore:

5 min = 300 s

Confusing Watts and Joules

Watt = power

Joule = energy

Remember:

1 W = 1 J/s

Forgetting to Convert Kilowatts

1 kW = 1000 W

Therefore:

2.5 kW = 2500 W

Using the Wrong Time

Power calculations require the time during which the specified energy transfer occurs.

Assuming Power Is the Same as Force

Power is not force.

Force is measured in:

newtons

Power is measured in:

watts

Forgetting That Power Can Change

Many calculations give average power over a time interval.

A machine's instantaneous power may vary during that interval.

Check Your Understanding

1. Define power.
2. Explain the difference between energy and power.
3. State the equation for power.
4. What is the SI unit of power?
5. What does 1 W mean?
6. A machine transfers 1200 J in 4 s. Calculate its power.
7. A device transfers 15,000 J in 30 s. Calculate its power.
8. A 2 kW machine operates for 10 s. Calculate the energy transferred.
9. A 500 W device transfers 20,000 J. How long does it operate?
10. Convert 3.5 kW to watts.
11. Convert 2 MW to watts.
12. Two machines each transfer 5000 J. Machine A takes 20 s and Machine B takes 5 s. Compare their power outputs.
13. Explain why completing the same work in less time requires greater average power.
14. A 50 kg student climbs vertically 4 m in 5 s. Calculate the student's average mechanical power using g = 9.8 N/kg.
15. A crane lifts a 1000 kg load through 6 m in 20 s. Calculate its useful mechanical power.
16. A 1500 W kettle operates for 3 minutes. Calculate the energy transferred.
17. A 1000 kg car accelerates from rest to 30 m/s in 15 s. Calculate the increase in kinetic energy.
18. Calculate the average power transferred into kinetic energy in Question 17.
19. Why would the actual engine or motor need to transfer more energy than calculated in Question 17?
20. A machine has an input power of 5000 W and a useful output power of 4000 W. Calculate its efficiency.
21. A motor is 75% efficient and receives 8 kW. Calculate its useful output power.
22. Explain how a high-power motor differs from a low-power motor performing the same task.
23. A power station produces 300 MW. How many joules does it transfer each second?
24. Explain why a 100 W device can transfer more total energy than a 1000 W device.
25. Describe how power could be measured experimentally by having students climb a staircase.

Key Terms

Power — Rate at which work is done or energy is transferred.

Watt (W) — SI unit of power; one joule transferred per second.

Joule (J) — SI unit of energy.

Energy — Quantity that can be stored and transferred within physical systems.

Work done — Energy transferred when a force causes displacement.

Rate — Change in a quantity per unit time.

Kilowatt (kW) — Unit of power equal to 1000 W.

Megawatt (MW) — Unit of power equal to 1,000,000 W.

Gigawatt (GW) — Unit of power equal to 1,000,000,000 W.

Average power — Total energy transferred divided by the time interval over which the transfer occurs.

Instantaneous power — Rate of energy transfer at a particular moment.

Input power — Rate at which energy enters a device or system.

Useful output power — Rate at which energy is transferred into the desired output.

Efficiency — Fraction of input energy or power transferred into a useful output.

Energy-time graph — Graph showing how transferred energy changes with time; its gradient represents power.

Power-time graph — Graph showing how power varies with time; its area represents energy transferred.

Key Takeaways

• Power describes how quickly energy is transferred.
• Power can also be described as the rate at which work is done.
• The main equation is P = E/t.
• Power is measured in watts (W).
• One watt equals one joule per second.
• Energy and power are different quantities.
• Energy measures the amount transferred; power measures the rate of transfer.
• Two devices can transfer the same amount of energy while having different power outputs.
• For the same energy transfer, completing the process in less time requires greater average power.
• The equation can be rearranged to E = Pt and t = E/P.
• Kilowatts, megawatts, and gigawatts are commonly used for larger power values.
• Power can be combined with kinetic, gravitational potential, and elastic potential energy calculations.
• A motor lifting an object can be analyzed using P = mgΔh/t.
• Average power during acceleration can be found from P = ΔEₖ/t.
• Higher power does not automatically mean greater total energy consumption; operating time also matters.
• Efficiency can be calculated by comparing useful output power with input power.
• Real machines often have a lower useful output power than input power because some energy is dissipated.
• The gradient of an energy-time graph represents power.
• The area under a power-time graph represents energy transferred.
• Power is one of the most useful quantities for comparing how quickly different machines, devices, and people can transfer energy.

The central relationship is:

P = E/t

Remember:

same energy + less time = greater power

and:

1 W = 1 J/s

A powerful system does not necessarily transfer more total energy. It transfers energy at a faster rate.