Kinematics: Describing Motion

The Language of Motion

Kinematics describes how things move, without yet asking why. Three quantities capture motion, each the rate of change of the one before — the direct payoff of the calculus in Unit 9:

• Position $x$ — where an object is.
• Velocity $v = \frac{dx}{dt}$ — the rate position changes (with direction; speed is its magnitude).
• Acceleration $a = \frac{dv}{dt}$ — the rate velocity changes.

Velocity is the derivative of position, and acceleration is the derivative of velocity — so integrating acceleration recovers velocity, and integrating velocity recovers position. Motion is a chain of derivatives and integrals.

For the important special case of constant acceleration (like gravity near Earth, $g \approx 9.8\,\text{m/s}^2$), the calculus gives clean equations of motion: $$v = v_0 + a t, \qquad x = x_0 + v_0 t + \tfrac{1}{2} a t^2, \qquad v^2 = v_0^2 + 2a(x - x_0).$$ These predict where an object is and how fast it moves at any time, given its start.

A crucial conceptual point: velocity and acceleration are independent. An object can have zero velocity yet nonzero acceleration — a ball at the top of its flight is momentarily still ($v = 0$) but still accelerating downward ($a = -g$). Confusing "not moving" with "not accelerating" is a classic error.

Common pitfall: conflating velocity with acceleration, or assuming zero velocity means zero acceleration. Velocity is how fast you move; acceleration is how fast the velocity changes — a body can be instantaneously at rest ($v = 0$) while strongly accelerating (the top of a toss). Likewise, constant velocity means zero acceleration, not constant nonzero acceleration.