Power

What is power?

In plain words: power tells you how fast work is done. Two people of the same mass who climb the same four floors do the same work against gravity, but the one who climbs faster develops more power. A more powerful engine can lift the same load in less time, or a heavier load in the same time.

Here W is the work done (or the energy transferred) in a time t.

Average and instantaneous power

The average power uses the total work over the whole time. The instantaneous power is its limit as the time interval shrinks to zero, the power at one instant:

If the work is done at a steady rate, the two are equal. If not (a car speeding up, a cyclist sprinting, then easing off), the instantaneous power changes from moment to moment, and the average power is the total work divided by the total time.

Power as force times velocity

The small work done by a force $\vec F$ over a small displacement $d\vec r$ is $\vec F \cdot d\vec r$. Dividing by the small time dt gives the instantaneous power:

and since $d\vec r/dt$ is the velocity $\vec v$:

Here v is the velocity of the force’s point of application in the chosen frame; for a particle it is the particle’s velocity. The angle θ is between the force and that velocity. Only the component of the force along the motion delivers power. If θ is more than 90°, the power is negative: the force takes energy away from the body, as friction or braking does, or as gravity does on a rising ball.

On a level road, a vehicle moving at a steady speed needs a driving force equal to the resistance f, so the power is fv. At steady speed uphill on a slope of angle α, the driving force must also balance the component of the weight along the slope, so the useful driving power is (f + mg sin α) v. These expressions neglect transmission losses; the engine may need to deliver more power.

Power, like work and energy, is a scalar.

Units: the watt, the horsepower and the kilowatt hour

The SI unit of power is the watt (W), named after James Watt, one of the innovators of the steam engine in the eighteenth century:

Its dimensions are [M L^² T^⁻³]. A bigger unit, the horsepower, is still used for the output of cars and motorbikes:

Here hp means mechanical horsepower, approximately 745.7 W in NIST Special Publication 811, Appendix B.9; 746 W is the rounded value used in these examples.

The kilowatt hour (kWh) is the energy transferred in one hour at a power of one kilowatt. Electricity bills count energy in kWh, often called “units”:

Note that kWh is a unit of energy and not of power.

NCERT Physics Part I (Class 11), Section 5.10, page 83

For example, a 100 W bulb that is on for 10 hours uses 100 × 10 = 1000 watt hours, that is 1 kWh.

Power in daily life

• Climbing stairs. Running up a staircase needs more power than walking up, though the work done against gravity is the same.
• Engines and motors. A car’s engine power limits its top speed: on a level road at steady top speed, its useful driving power balances air drag and rolling resistance; transmission losses reduce the power available from the engine.
• Electrical appliances. A heater, a geyser or an air conditioner is rated in watts or kilowatts; at constant power its energy use is power multiplied by operating time; if the power cycles, use the average power over that time.
• Pumps and lifts. The power of a water pump or a lift motor decides how much it can raise, and how fast. Real machines also waste some of their input power; the ratio of useful output power to input power is called the efficiency (a term used in many problems, though not in this NCERT section).

Solved examples

Common mistakes to avoid

Quick revision summary

• Power is the rate of doing work or transferring energy: average power is W/t, instantaneous power is dW/dt.
• Power delivered by a force is the dot product of the force and the velocity, Fv cos θ.
• Power is a scalar, with dimensions [M L^² T^⁻³]. Its SI unit is the watt, 1 W = 1 J s^⁻¹; 1 mechanical hp ≈ 746 W.
• The kilowatt hour is a unit of energy, 3.6 × 10^⁶ J, used on electricity bills.

Questions students ask

What is the dimensional formula of power?

The dimensional formula of power is [M L^² T^⁻³]: power is work divided by time, [M L^² T^⁻²]/[T]. Its SI unit is the watt: 1 W = 1 J s^⁻¹ = 1 kg m^² s^⁻³.

What is the difference between energy and power?

Energy is an amount, measured in joules, while power is the rate at which energy is transferred or work is done, measured in watts (1 W = 1 J s^⁻¹). Two pumps that lift the same tank of water to the same roof transfer the same energy, but the one that takes half the time has twice the power. Energy is constant power (or the average power over the interval) multiplied by time, which is why the kilowatt hour is a unit of energy, not of power.

How do you find the power needed to lift a load at a steady speed?

At a steady speed and with no friction, the lifting force equals the weight mg, so the power needed is P = Fv = mgv, or the work mgh divided by the time taken. For example, lifting 250 kg at 1.2 m s^⁻¹ needs 250 × 9.8 × 1.2 = 2940 W of useful power (with g = 9.8 m s^⁻²). Any friction adds fv, and a real motor must take in more than its useful output, because its efficiency is less than 100%.

Is power a scalar or a vector quantity?

Power is a scalar quantity. It is the dot product of force and velocity, P = F · v = Fv cos θ, so it has a size and a sign but no direction; it is negative when the force opposes the motion, as with brakes.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A 100 W bulb is on for 5 hours. How much energy does it use, in kWh and in joules?

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   100 W × 5 h is 500 watt hours, that is 0.5 kWh. In joules, 0.5 × 3.6 × 10⁶, that is 1.8 × 10⁶ J.

2. Q2. A force of 50 N acts on a body moving at 4 m s^⁻¹, at 60° to its velocity. Find the power delivered by the force.

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   $$\begin{aligned} P &= Fv\cos\theta \\ &= 50 \times 4 \times 0.5 \\ &= 100\ \text{W} \end{aligned}$$

3. Q3. A motor raises a 200 kg load through 10 m at a steady speed in 20 s. Find its power output. Take g = 9.8 m s⁻².

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   $$\begin{aligned} P &= \frac{mgh}{t} \\ &= \frac{200 \times 9.8 \times 10}{20} \\ &= 980\ \text{W} \end{aligned}$$