Kinematics — Motion in a Straight Line

The kinematics chain

Calculus describes motion elegantly. For an object moving along a straight line, three quantities are linked by differentiation and integration: $$\text{displacement } s(t) \xrightarrow{\ \frac{d}{dt}\ } \text{velocity } v(t) \xrightarrow{\ \frac{d}{dt}\ } \text{acceleration } a(t).$$ - Velocity is the derivative of displacement: $v(t) = s'(t)$. It has direction — positive means moving one way, negative the other. - Acceleration is the derivative of velocity: $a(t) = v'(t) = s''(t)$.

The chain works backwards by integration: integrate acceleration to recover velocity, integrate velocity to recover displacement (using known conditions to find each constant).

Speed versus velocity

Velocity is signed (it includes direction); speed is its magnitude, $|v(t)|$. A particle wi