Displacement velocity and acceleration

Displacement, velocity, and acceleration form a hierarchy linked by differentiation and integration.

Displacement

• Measures the net change in position (final position minus initial position)
• Is always a vector quantity

Average velocity and average acceleration

• Describe changes over finite time intervals
• Are calculated as the ratio of a change in a quantity to the elapsed time: average velocity = Δx/Δt and average acceleration = Δv/Δt

An object accelerates whenever its velocity changes in magnitude, direction, or both — not only when it speeds up.

The transition from average to instantaneous values

The transition from average to instantaneous values is the heart of the calculus-based approach. As the time interval shrinks toward zero, the average value becomes the instantaneous value, which is the derivative.

Instantaneous velocity and acceleration

• Instantaneous velocity is the derivative of position with respect to time: v = dx/dt
• Instantaneous acceleration is the derivative of velocity with respect to time: a = dv/dt = d²x/dt²

The reverse process — integration — recovers position from velocity or velocity from acceleration, with initial conditions fixing the constant of integration.

Mastering differentiation and integration of polynomial time functions is essential for every subsequent topic in AP Physics C: Mechanics.