# What is the relationship between linear speed and angular speed?

Jul 17, 2015

$v = \omega R$

#### Explanation:

Linear velocity $v$ is equal to the angular speed $\omega$ times the radius from the center of motion $R$.

We can derive this relationship from the arclength equation
$S = \theta R$ where $\theta$ is measured in radians.

$S = \theta R$

Take a derivative with respect to time on both sides
$d \frac{S}{\text{dt"=d theta/"dt}} R$

$d \frac{S}{\text{dt}}$ is linear velocity and $d \frac{\theta}{\text{dt}}$ is angular velocity

So we're left with:
$v = \omega R$