1.1 Position, Velocity, and Acceleration

Position, Velocity, Acceleration as Vectors

Position x(t) locates an object relative to an origin. Velocity v(t) = dx/dt is the rate of change of position — its sign shows direction. Acceleration a(t) = dv/dt = d²x/dt² is the rate of change of velocity. In AP Physics C, calculus links these quantities directly: differentiate position to get velocity, differentiate velocity to get acceleration.

Integration Goes the Other Way

Since acceleration is the derivative of velocity, velocity is the integral of acceleration over time, plus an initial velocity constant. Similarly, position is the integral of velocity, plus an initial position constant. Graphically, the area under an a(t) vs t curve gives Δv, and the area under a v(t) vs t curve gives Δx.

Graphical Interpretation

On a position-time graph, the slope equals instantaneous velocity. On a velocity-time graph, the slope equals instantaneous acceleration, and the area under the curve equals displacement. Zero slope on x(t) means the object is momentarily at rest; zero slope on v(t) means constant velocity (zero acceleration).

Key terms

Instantaneous velocity

v(t) = dx/dt; the derivative of position with respect to time.

Instantaneous acceleration

a(t) = dv/dt = d²x/dt²; the derivative of velocity or second derivative of position.

Finding velocity from acceleration

v(t) = v₀ + ∫a(t) dt; integrate acceleration over time and add initial velocity.

Slope of a position-time graph

Equals the instantaneous velocity at that point in time.

Area under a velocity-time graph

Equals the displacement of the object over that time interval.

Try an AP-style question

x(t) = 3t² − 2t (m). What is the velocity at t = 2 s?

1. 6 m/s
2. 12 m/s
3. 10 m/s
4. 8 m/s