Acceleration Vector in Polar Coordinates

Theorem
Consider a particle $p$ moving in the plane.

Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$.

Then the acceleration $\mathbf a$ of $p$ can be expressed as:


 * $\mathbf a = \left({r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + 2 \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} }\right) \mathbf u_\theta + \left({\dfrac {\mathrm d^2 r} {\mathrm d t^2} - r \left({\dfrac {\mathrm d \theta} {\mathrm d t} }\right)^2}\right) \mathbf u_r$

where:
 * $\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$
 * $\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$

Proof
Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:
 * $(1): \quad \mathbf r = r \mathbf u_r$


 * MotionInPolarPlane.png

From Derivatives of Unit Vectors in Polar Coordinates:

From Velocity Vector in Polar Coordinates:
 * $\mathbf v = r \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_\theta + \dfrac {\mathrm d r} {\mathrm d t} \mathbf u_r$

where $\mathbf v$ is the velocity of $p$.

The acceleration of $p$ is by definition the rate of change in its velocity: