Power and Efficiency
Calculate power as work per time or energy transfer rate. Interpret efficiency and energy loss in machines and real-world systems.
A lesson of Physics, in Energy, Work, Power, and Momentum, about 10 minutes.
By the end of this lesson, you will be able to calculate power from work and time or force and velocity, compute mechanical efficiency, and convert between units of power.
Power
Power: the rate at which work is done or energy is transferred.
$$P = \frac{W}{t} = \frac{\Delta E}{t}$$
Unit: Watt (W) = J/s = kg·m²/s³
Also: 1 horsepower (hp) = 746 W
Alternative formula (when force is constant and $v$ is the velocity):
$$P = Fv$$
(Derived from $P = W/t = Fd/t = Fv$)
Example 1 — Power from Work
A motor lifts a 500 kg elevator 30 m in 20 s. What power does it deliver?
$W = mgh = 500 \times 9.8 \times 30 = 147,000$ J
$P = 147,000/20 = 7350$ W = 7.35 kW
Example 2 — Power from Force and Velocity
A car engine exerts 3000 N at 25 m/s. What is the power output?
$P = Fv = 3000 \times 25 = 75,000$ W = 75 kW ≈ 100 hp
Efficiency
Efficiency: ratio of useful output energy to total input energy.
$$\eta = \frac{E_{\text{output}}}{E_{\text{input}}} \times 100%$$
Real machines always have $\eta < 100%$ because friction, heat, and other non-conservative processes waste energy.
Example 3 — Machine Efficiency
An electric motor inputs 500 W and lifts a 30 kg mass 2 m in 3 s. Find the efficiency.
Useful output power: $P_{\text{out}} = mgh/t = 30(9.8)(2)/3 = 196$ W
$\eta = 196/500 \times 100% = 39.2%$
Summary
Power: $P = W/t$ or $P = Fv$; unit is Watt.
1 hp = 746 W; useful for engine comparisons.
Efficiency: $\eta = P_{\text{out}}/P_{\text{in}} \times 100%$; always < 100% in real systems.