# Acceleration Vector in Polar Coordinates

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## Theorem

Consider a particle $p$ moving in the plane.

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

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

$\mathbf a = \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\theta + \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2} \mathbf u_r$

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:

$(1): \quad \mathbf r = r \mathbf u_r$ $\text {(2)}: \quad$ $\ds \dfrac {\d \mathbf u_r} {\d \theta}$ $=$ $\ds \mathbf u_\theta$ $\text {(3)}: \quad$ $\ds \dfrac {\d \mathbf u_\theta} {\d \theta}$ $=$ $\ds -\mathbf u_r$
$\mathbf v = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$

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

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

 $\ds \mathbf a$ $=$ $\ds \dfrac {\d \mathbf v} {\d t}$ $\ds$ $=$ $\ds r \dfrac {\d^2 \theta} {\d t^2} \mathbf u_\theta + \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} \mathbf u_\theta + r \dfrac {\d \theta} {\d t} \dfrac {\d \mathbf u_\theta} {\d t} + \dfrac {\d^2 r} {\d t^2} \mathbf u_r + \dfrac {\d r} {\d t} \dfrac {\d \mathbf u_r} {\d t}$ Product Rule for Derivatives $\ds$ $=$ $\ds r \dfrac {\d^2 \theta} {\d t^2} \mathbf u_\theta + \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} \mathbf u_\theta + r \dfrac {\d \theta} {\d t} \dfrac {\d \mathbf u_\theta} {\d \theta} \dfrac {\d \theta} {\d t} + \dfrac {\d^2 r} {\d t^2} \mathbf u_r + \dfrac {\d r} {\d t} \dfrac {\d \mathbf u_r} {\d \theta} \dfrac {\d \theta} {\d t}$ Chain Rule for Derivatives $\ds$ $=$ $\ds r \dfrac {\d^2 \theta} {\d t^2} \mathbf u_\theta + \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} \mathbf u_\theta - r \dfrac {\d \theta} {\d t} \mathbf u_r \dfrac {\d \theta} {\d t} + \dfrac {\d^2 r} {\d t^2} \mathbf u_r + \dfrac {\d r} {\d t} \mathbf u_\theta \dfrac {\d \theta} {\d t}$ substituting from $(2)$ and $(3)$ $\ds$ $=$ $\ds \paren {r \dfrac {\d^2 \theta} {\d t^2} + 2 \dfrac {\d r} {\d t} \dfrac {\d \theta} {\d t} } \mathbf u_\theta + \paren {\dfrac {\d^2 r} {\d t^2} - r \paren {\dfrac {\d \theta} {\d t} }^2} \mathbf u_r$ gathering terms

$\blacksquare$