Work, Energy, and Power

What this unit covers

The topics below follow the published Physics C: Mech course framework for Unit 3. This unit is worth 15–25% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Work & the Work-Energy Theorem14 min · 3 objectivesDefine work as the line integral W = ∫F·dx and compute it for a variable force · State and apply the work-energy theorem W_net = ΔKE · Compute the work done by a position-dependent force such as a spring
• Potential Energy & the Force-Energy Relationship14 min · 3 objectivesDefine potential energy for a conservative force as U = −∫F dx · Recover the force from a potential energy function using F = −dU/dx · Read a potential energy curve to locate equilibria and judge their stability
• Conservation of Energy14 min · 3 objectivesState conservation of mechanical energy for systems acted on only by conservative forces · Solve motion problems using KE_i + U_i = KE_f + U_f · Account for energy removed by friction using the work of nonconservative forces
• Power12 min · 3 objectivesDefine power as the rate of doing work, P = dW/dt · Compute instantaneous power as the dot product P = F·v · Distinguish average power from instantaneous power

Formulas in Unit 3

Every term in Unit 3

All 19 terms we publish for Work, Energy, and Power, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Instantaneous power

P = dW/dt = F·v. Average power is total work divided by total time, and the two differ whenever force or speed varies.

Work as an integral

W = ∫F·dr. For a varying force, this is the area under the force-position curve, and F·d is only the constant-force case.

Work-energy theorem

W_net = ΔKE. Derived by integrating ΣF = ma with respect to position.

Force from potential energy

F = −dU/dx. Force points down the potential energy gradient, which is why objects move toward potential minima.

Potential energy curves

Minima are stable equilibria and maxima unstable. The slope gives the force, and the difference between total energy and U gives the kinetic energy.

Turning points

Where total energy equals potential energy, so kinetic energy is zero and the object reverses direction.

Gravitational potential energy in general

U = −GMm/r, taking zero at infinity. The near-Earth mgh is the small-Δr approximation of this.

Escape velocity

v_esc = √(2GM/R), found by setting total mechanical energy to zero. Independent of the escaping object's mass.

Spring potential energy

U = ½kx², the integral of the restoring force kx over displacement.

Conservative forces and path independence

A force is conservative when its work around any closed loop is zero, which is exactly when a potential energy function can be defined.

When to use energy rather than forces

Energy avoids solving for time and handles varying forces and curved paths; forces are needed when time or an individual force is asked for.

Work done by friction

Negative and equal to −f·d where d is the PATH LENGTH, not the displacement. Friction is not conservative, so the path matters.

Work done by a variable force from a graph

The signed area under the force-position curve. Area below the axis represents negative work.

Stability from a potential curve

A minimum of U is stable equilibrium, a maximum unstable, and a flat region neutral. Small displacements from a minimum produce a restoring force.

Reading kinetic energy from a potential curve

KE = E − U at each position. Where the horizontal total-energy line meets the curve, KE is zero and the object turns around.

Deriving force from U(x)

F = −dU/dx, so the force is the negative slope of the potential energy curve. A steeper curve means a stronger force.

Escape energy

The minimum energy to reach infinity with zero speed, so ½mv² = GMm/R. Independent of launch direction for a non-rotating body.

Power delivered by a constant force

P = Fv, so a vehicle at constant speed against constant drag draws constant power while doing zero net work.

Energy in a spring-mass system

Total energy ½kA² is constant. At displacement x, KE = ½k(A² − x²).

What examiners penalize here

• Move fluently in both directions: integrate a conservative force to get U (U = −∫F dx), and differentiate U to get the force (F = −dU/dx). On a U-versus-x graph, remember the force points downhill and equilibria sit where the slope is zero.
• Use P = W/Δt when you know a total amount of work over an interval, and P = Fv (or F·v) when you want the power at a specific instant or speed. A force perpendicular to the velocity — like the centripetal force on a satellite — does zero work and delivers zero power.

Practice Physics C: Mech

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 7 units of AP Physics C: Mechanics

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations