Inside This Unit: The Full Breakdown
Kinematics describes motion using calculus: velocity is the derivative of position and acceleration the derivative of velocity, while integration recovers velocity and position. Projectile and relative motion follow from treating components independently.
Why it matters
Kinematics establishes the calculus toolkit (derivatives and integrals of motion) used in every later unit. AP Physics C expects you to move fluently between x(t), v(t), and a(t).
Key concepts
- v = dx/dt and a = dv/dt = d²x/dt².
- Integrate a(t) for v(t) and v(t) for x(t), using initial conditions for constants.
- Constant-acceleration equations apply only when a is constant.
- Projectile motion separates into constant-velocity horizontal and constant-acceleration vertical components.
Motion as Calculus
Velocity is the instantaneous rate of change of position (the tangent slope on x(t)); acceleration is the rate of change of velocity. When acceleration varies, you cannot use the constant-a kinematic equations — instead integrate: v(t) = v₀ + ∫a dt and x(t) = x₀ + ∫v dt. On graphs, slope gives derivatives and area gives integrals.
Projectile and Relative Motion
A projectile has constant horizontal velocity and constant vertical acceleration g; analyze the components independently and share only the time. Relative velocity problems are handled by adding or subtracting velocity vectors to change reference frames.
AP exam tip
When acceleration is not constant, never plug into v = v₀ + at — set up the integral instead. Examiners design problems specifically to catch this.
Connections to other units
- Unit 2: Kinematics combines with forces through Newton’s second law.
- Unit 7: SHM is described by the same derivative relationships among x, v, and a.
