Kinematics: Describing Motion

The Language of Motion

Kinematics describes how things move, without asking why. Motion is a chain of derivatives and integrals linking three core quantities:

Quantity	Symbol & Definition	Rate of Change
Position	$x$ : where an object is	—
Velocity	$v = \frac{dx}{dt}$ : rate position changes	First derivative
Acceleration	$a = \frac{dv}{dt}$ : rate velocity changes	Second derivative

Speed is the magnitude of velocity. Integrating acceleration recovers velocity, and integrating velocity recovers position.

Motion built by running the sum, not read off a slope

Constant Acceleration & Pitfalls

For constant acceleration (like gravity $g \approx 9.8\,\text{m/s}^2$ ), calculus yields three exact equations of motion:

$v = v_0 + a t$
$x = x_0 + v_0 t + \tfrac{1}{2} a t^2$
$v^2 = v_0^2 + 2a(x - x_0)$

Where $x_0$ and $v_0$ are initial position and velocity.

Common pitfall: Velocity and acceleration are independent. A tossed ball at its peak has $v = 0$ yet is still accelerating downward at $-g$ . Zero velocity does not mean zero acceleration. Likewise, constant velocity means zero acceleration.

The Language of Motion

Constant Acceleration & Pitfalls

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