Average Power

What power is

Power is the rate of doing work — how much work is done per unit time:

$P = \frac{W}{t}$

Two engines may do the same total work, but the one that does it faster has greater power.

Units

$[P] = \frac{\text{J}}{\text{s}} = \text{W (Watt)}$

One Watt is one Joule per second.

Other common units:

• Kilowatt (kW): $1 \text{ kW} = 1000 \text{ W}$
• Horsepower (hp): $1 \text{ hp} = 746 \text{ W}$ (used for engines and motors)
• Kilowatt-hour (kWh): unit of energy (not power) — $1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$

Average vs instantaneous power

$P = \frac{W}{t}$ gives the average power over time interval $t$.

If power varies, the average power:

$P_{avg} = \frac{\text{total work done}}{\text{total time}}$

Instantaneous power is derived in the next entry.

Relation to force and velocity

For constant force and velocity:

$P = \frac{W}{t} = \frac{Fd}{t} = F \cdot \frac{d}{t} = Fv$

This gives the instantaneous power when force and velocity are constant. The full instantaneous power formula $P = \vec{F} \cdot \vec{v}$ handles varying quantities.

Examples

Climbing stairs: A 60 kg person climbs 10 m of stairs in 20 s:

$W = mgh = 60 \times 10 \times 10 = 6000 \text{ J}$

$P = \frac{6000}{20} = 300 \text{ W}$

Car engine: A car engine delivers 50 kW of power. In 1 hour, total work done:

$W = P \times t = 50000 \times 3600 = 1.8 \times 10^8 \text{ J}$

Power and efficiency

Machines are not 100% efficient — some input power is lost to friction and heat. Efficiency:

$\eta = \frac{P_{output}}{P_{input}} \times 100\%$

A motor with 80% efficiency and input power 1 kW delivers only 800 W of useful mechanical power.