Introduction to Calculus

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For a particle moving along a curve $C$ traced by $\boldsymbol{r}(t)$ , the derivative of $\boldsymbol{r}(t)$ carries a physical interpretation used throughout this module.

Velocity

\vec{v}(t) = \boldsymbol{r}'(t)

$\vec{v}(t)$ is tangent to $C$ and points in the direction of motion.

Speed

v(t) = \lVert \vec{v}(t) \rVert = \lVert \boldsymbol{r}'(t) \rVert

Acceleration

\vec{a}(t) = \vec{v}'(t) = \boldsymbol{r}''(t)

$\vec{a}(t)$ measures how the velocity vector changes, both in magnitude (speed) and in direction.

Written by

Sahithyan K.

October 8, 2026 1 min read

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