Kinematics and the Language of Motion

Motion Through Calculus

Physics is where the mathematics of the preceding units meets the physical world, and kinematics — the description of motion — is calculus made concrete. Three quantities describe how an object moves, each the derivative of the one before (Unit 5):

• Position $x(t)$ — where the object is.
• Velocity $v = \frac{dx}{dt}$ — the rate of change of position (a vector, with direction; speed is its magnitude).
• Acceleration $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ — the rate of change of velocity.

The relationship runs both ways via integration (Unit 6): integrating acceleration recovers velocity, and integrating velocity recovers position. Motion is a chain of derivatives going one way and integrals going the other — the Fundamental Theorem of Calculus embodied in physics.

For the important case of constant acceleration (a falling body near Earth, $g \approx 9.8\,\text{m/s}^2$), integrating gives the clean equations of motion: $$v = v_0 + at, \qquad x = x_0 + v_0 t + \tfrac{1}{2}at^2, \qquad v^2 = v_0^2 + 2a(x - x_0).$$ These predict position and velocity at any time from the initial state.

A conceptual essential: velocity and acceleration are independent quantities. An object can have zero velocity yet nonzero acceleration — a ball at the peak of its flight is momentarily at rest ($v = 0$) while still accelerating downward ($a = -g$). "Not moving" is not "not accelerating." Vectors matter too: velocity and acceleration have direction, so an object moving in a circle at constant speed is still accelerating, because its velocity's direction is changing.

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). And constant speed around a curve still involves acceleration, because velocity is a vector whose direction is changing.