1. Definition

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

While work measures the amount of energy transferred, power measures how quickly that transfer occurs.

2. Average Power

Average power is defined as:

\(P_{avg}=\frac{W}{\Delta t}\)​

where:

• \(P_{avg}\)​ = average power
• \(W\) = work done
• \(\Delta t\) = time interval

A larger power means the same amount of work is done in less time.

3. Instantaneous Power

In AP Physics C, power is often considered at a specific instant.

Instantaneous power is:

\(P=\frac{dW}{dt}\)​

This represents the rate of energy transfer at a particular moment.

4. Power and Force

Since work is defined as:

\(dW=\vec{F}\cdot d\vec{r}\)

we obtain:

\(P=\vec{F}\cdot\frac{d\vec{r}}{dt}\)​

Since

\(\frac{d\vec{r}}{dt}=\vec{v}\)

the instantaneous power becomes:

\(P=\vec{F}\cdot\vec{v}\)

Using the dot product:

\(P=Fv\cos\theta\)

where:

• F = force magnitude
• v = speed
• θ = angle between force and velocity

5. Special Cases

Force Parallel to Velocity

\(\theta=0^{\circ}\)

\(P=Fv\)

Maximum power is delivered.

Force Perpendicular to Velocity

\(\theta=90^{\circ}\)

\(P=0\)

No work is being done, so no power is transferred.

Example:

• centripetal force in uniform circular motion

Force Opposite Velocity

\(\theta=180^{\circ}\)

\(P=-Fv\)

Power is negative because energy is being removed from the object.

Example:

• friction
• braking systems

6. Units of Power

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

\(1W=1\frac{J}{s}\)​

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

7. Mechanical Example

Suppose a machine performs:

\(500J\)

of work in:

\(10s\)

Average power:

\(P=\frac{500J}{10s}=50W\)

8. Power and Motion

For motion at constant speed:

\(P=Fv\)

This equation is commonly used to determine:

• engine power
• motor power
• elevator power

9. Power and Energy

Power describes the rate of energy transfer:

\(P=\frac{\Delta E}{\Delta t}\)​

where $E$ can represent:

• kinetic energy
• potential energy
• mechanical energy
• electrical energy

10. Physical Meaning

Two machines may perform the same amount of work.

The machine that completes the work in less time has greater power.

Therefore:

• work measures how much energy
• power measures how fast energy is transferred

11. Applications in Mechanics

Power is important in analyzing:

• automobiles
• elevators
• cranes
• engines
• athletes performing mechanical work

In many real systems, increasing speed requires greater power because:

\(P=Fv\)

Summary

Power is the rate of doing work or transferring energy.

Average power:

\(P_{avg}=\frac{W}{\Delta t}\)​

Instantaneous power:

\(P=\frac{dW}{dt}\)​​

Force–velocity form:

\(P=\vec{F}\cdot\vec{v}\)​

or

\(P=Fv\cos\theta\)​

Key ideas:

• power measures how quickly energy is transferred
• SI unit is the watt
• power depends on both force and velocity
• positive power adds energy, negative power removes energy

Power provides the connection between work, energy, and time in AP Physics C Mechanics.