AP Physics C Mechanics Unit 3 Notes: Understanding Power in Work–Energy Systems

Power

The rate at which work is done or energy is transferred/transformed; tells “how fast” energy is delivered.

Average Power

Work (or energy change) over a time interval: P_avg = W/Δt = ΔE/Δt.

Instantaneous Power

Power at a moment in time, defined by a derivative: P = dW/dt = dE/dt.

Watt (W)

SI unit of power: 1 W = 1 J/s = 1 N·m/s.

Horsepower (hp)

Common non-SI unit of power: 1 hp ≈ 746 W.

Work–Time Graph Interpretation

On a W vs. t graph, instantaneous power is the slope: P = dW/dt.

Energy–Time Graph Interpretation

On an E vs. t graph, power is the slope: P = dE/dt.

Power–Time Graph Interpretation

On a P vs. t graph, the area under the curve is work/energy transferred: W = ∫P dt.

Differential Work

Infinitesimal work by a force over displacement: dW = F⃗ · d r⃗.

Force–Velocity Power Relation

Instantaneous power delivered by a force: P = F⃗ · v⃗.

Dot Product (in Power)

Only the component of force parallel to velocity contributes: P = Fv cosθ.

Negative Power

Occurs when a force component is opposite the motion (cosθ < 0); the force removes mechanical energy (e.g., braking, friction).

Zero Power with Perpendicular Force

If force is perpendicular to velocity (θ = 90°), then P = 0 even if the force is large.

Centripetal Force Does No Work (Uniform Circular Motion)

In uniform circular motion, centripetal force is perpendicular to velocity, so it does zero work and delivers zero power.

Work–Energy Theorem

Net work equals change in kinetic energy: W_net = ΔK.

Net Power and Kinetic Energy

Net power equals the rate of change of kinetic energy: P_net = dK/dt.

Translational Kinetic Energy

K = (1/2)mv^2 (for constant mass m).

Power in 1D Translational Motion

For motion along the velocity direction with constant m: Pnet = dK/dt = mv a, consistent with P = Fv and Fnet = ma.

Rotational Mechanical Power

Instantaneous power in rotation: P = τω (torque times angular speed).

Rotational Kinetic Energy

K_rot = (1/2)Iω^2.

Net Rotational Power

Net rotational power equals the rate of change of rotational kinetic energy: Pnet = dKrot/dt.

System Boundary (Power Accounting)

Power is energy per time crossing a chosen system boundary via external forces; changing the system changes what counts as “external” power.

Sum of Powers from External Forces

If multiple external forces act, net power adds: Pnet = Σ(F⃗ext · v⃗).

Efficiency (η)

Ratio of useful output power to input power: η = Pout/Pin (dimensionless, often a percent).

Constant Power Acceleration (Frictionless)

With constant power P applied to mass m starting from rest: v(t) = √(2Pt/m), so speed grows like √t and applied force decreases as F = P/v.