Physics C: Mechanics · Unit 3: Work, Energy, and Power · 14 min read · Updated 2026-05-11
Work and the Work-Energy Theorem — AP Physics C: Mechanics
AP Physics C: Mechanics · Unit 3: Work, Energy, and Power · 14 min read
1. Work Done by a Force (Constant and Variable)★★☆☆☆⏱ 4 min
For a constant force $\vec{F}$ acting on an object with displacement $\Delta\vec{r}$, work is defined as the dot product:
W = \vec{F} \cdot \Delta\vec{r} = F\Delta r \cos\theta
where $\theta$ is the angle between the force and displacement vectors. If $\theta < 90^\circ$, work is positive (energy added); if $\theta > 90^\circ$, work is negative (energy removed); if $\theta = 90^\circ$, work is zero (no energy transfer).
For a variable force that changes with position, we extend the definition by integrating over infinitesimal displacements. For 1-dimensional motion along the x-axis:
W = \int_{x_1}^{x_2} F_x(x) dx
Geometrically, this integral equals the net area between the force vs. position curve and the x-axis, a common AP exam interpretation.
Exam tip: Always confirm the angle between the force and displacement, not irrelevant angles from problem geometry. Normal force never does work on an object moving along a fixed surface, so you can immediately set its work to zero.
2. The Work-Energy Theorem: Derivation and Statement★★★☆☆⏱ 4 min
The work-energy theorem is derived directly from Newton's second law, giving a fundamental relation between net work done on an object and its change in kinetic energy. It is particularly powerful because it can be used for variable forces without integrating acceleration over time.
W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2
A critical point: only net work (the sum of work from all forces acting on the object) equals the change in kinetic energy. Work done by a single force will not give $\Delta KE$ unless it is the only force acting.
Exam tip: If a problem asks for change in speed after work is done by one force, do not forget to add the work from all other forces (like gravity or friction) before setting net work equal to $\Delta KE$.
3. Net Work from Multiple Forces★★☆☆☆⏱ 2 min
When multiple forces act on an object, there are two equivalent ways to calculate net work: you can calculate work done by each force individually and sum them, or calculate the net force first then find the work done by the net force. Mathematically, this equivalence is written as:
Common special cases tested on the AP exam: work done by kinetic friction on a sliding object is always negative (friction opposes displacement), work done by any force perpendicular to displacement is always zero, and if an object moves at constant speed, net force is zero so net work is zero.
Exam tip: If an object moves at constant speed, you can immediately conclude net work is zero per the work-energy theorem, which is a huge shortcut for problems asking for work done by an unknown force.
4. Exam-Style Worked Applications★★★★☆⏱ 4 min
Common Pitfalls
Why: Students memorize the simplified parallel form and forget the general dot product definition.
Why: Problems often highlight one force (like tension or pushing), leading students to forget other forces contribute to net work.
Why: Students mix up work (area under F-position) and impulse (area under F-time).
Why: Students forget friction points opposite displacement.
Why: Students used to vector components for velocity incorrectly extend this to kinetic energy.
Why: Students default to integrating over time from kinematics problems.