Power

Overview

Power is the rate at which work is done or energy is transferred. It measures how quickly energy is converted from one form to another.

Definition

Average Power

$P_{\text{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$

Instantaneous Power

$P = \frac{dW}{dt} = \frac{dE}{dt}$

Units

Unit	Symbol	Equivalent
Watt	W	J/s = kg·m²/s³
Kilowatt	kW	1000 W
Horsepower	hp	746 W
Megawatt	MW	$10^6$ W

Power in Terms of Force and Velocity

For a constant force:

$P = \vec{F} \cdot \vec{v} = Fv\cos\theta$

When force is parallel to velocity:

$P = Fv$

Energy-Power Relationship

Energy transferred over time:

$E = P \times t$

Common unit: kilowatt-hour (kWh)

$1 \text{ kWh} = 1000 \text{ W} \times 3600 \text{ s} = 3.6 \times 10^6 \text{ J} = 3.6 \text{ MJ}$

Mechanical Power

Rotational Power

$P = \tau\omega$

Where $\tau$ is torque and $\omega$ is angular velocity

Power and Constant Velocity

When an object moves at constant velocity against friction:

$P = f_k \times v = \mu_k mgv$

Power to Climb

Power needed to climb at velocity $v$:

$P = mgv$

Efficiency

The ratio of useful output power to input power:

$\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\%$

$\eta = \frac{E_{\text{out}}}{E_{\text{in}}} \times 100\%$

Energy Losses

$P_{\text{loss}} = P_{\text{in}} - P_{\text{out}} = P_{\text{in}}(1 - \eta)$

Human Power Output

Activity	Typical Power
At rest	~80 W
Walking	~200 W
Cycling	~400 W
Sprinting	~500-2000 W
Maximum sustained	~300-400 W

Examples

Example 1: Motor Power

A motor lifts a 500 kg load 20 m in 10 seconds. Find power.

$W = mgh = 500 \times 9.8 \times 20 = 98{,}000 \text{ J}$

$P = \frac{W}{t} = \frac{98{,}000}{10} = 9{,}800 \text{ W} = 9.8 \text{ kW}$

Example 2: Car Engine

A car moves at 30 m/s against 1000 N of friction. Find power needed.

$P = Fv = 1000 \times 30 = 30{,}000 \text{ W} = 30 \text{ kW} = 40.2 \text{ hp}$

Example 3: Climbing Stairs

A 70 kg person climbs 15 m of stairs in 20 s. Find average power.

$P = \frac{mgh}{t} = \frac{70 \times 9.8 \times 15}{20} = 515 \text{ W}$

Example 4: Electric Bill

A 100 W light bulb runs for 5 hours. Find energy used in kWh.

$E = Pt = 100 \text{ W} \times 5 \text{ h} = 500 \text{ Wh} = 0.5 \text{ kWh}$

Example 5: Efficiency

An engine produces 60 kW from fuel supplying 200 kW. Find efficiency.

$\eta = \frac{60}{200} \times 100\% = 30\%$

Power in Transportation

Terminal Velocity

At terminal velocity, driving force equals drag:

$P = F_{\text{drive}} \times v = F_{\text{drag}} \times v$

Drag Power (at high speed)

Air resistance increases with $v^2$, so power to overcome it:

$P_{\text{drag}} \propto v^3$

This is why fuel consumption increases rapidly with speed.

Acceleration Power

Power needed to accelerate:

$P = Fv = mav$

Note: Power required increases with velocity even at constant acceleration.

Key Relationships Summary

Quantity	Formula
Average power	$P = W/t$
Instantaneous power	$P = Fv$
Rotational power	$P = \tau\omega$
Energy from power	$E = Pt$
Efficiency	$\eta = P_{\text{out}}/P_{\text{in}}$