Velocity 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 velocity $\mathbf v$ of $p$ can be expressed as:

$\mathbf v = r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$

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 $p$

Let the radius vector $\mathbf r$ from the origin to $p$ be expressed as:

$(1): \quad \mathbf r = r \mathbf u_r$

$\text {(2)}: \quad$

$\ds \dfrac {\d \mathbf u_r} {\d \theta}$

$=$

$\ds \mathbf u_\theta$

$\text {(3)}: \quad$

$\ds \dfrac {\d \mathbf u_\theta} {\d \theta}$

$=$

$\ds -\mathbf u_r$

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

$\ds \mathbf v$

$=$

$\ds \dfrac {\d \mathbf r} {\d t}$

$\ds $

$=$

$\ds r \dfrac {\d \mathbf u_r} {\d t} + \mathbf u_r \dfrac {\d r} {\d t}$

$\ds $

$=$

$\ds r \dfrac {\d \mathbf u_r} {\d \theta} \dfrac {\d \theta} {\d t} + \mathbf u_r \dfrac {\d r} {\d t}$

$\ds $

$=$

$\ds r \dfrac {\d \theta} {\d t} \mathbf u_\theta + \dfrac {\d r} {\d t} \mathbf u_r$

substituting from $(2)$ and $(3)$

$\blacksquare$

1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.21$: Newton's Law of Gravitation: $(4)$