Why is linear velocity not rad*m /s?

Question

Considering that angular velocity is:
$w = \frac{2\pi}{t} \frac{[rad]}{[s]}$;
and that linear velocity is:
$ v = w \frac{[rad]}{[s]} \times r[m]$;

I would expect linear velocity to be expressed as $v = x\frac{[rad]\times[m]}{[s]}$;
but, as far as I know, it's always expressed in meters per second.

My question is: Where do the radians disappear to?

Answer 1

Well, radian is a dimensionless physical quantity. So, mentioning it while mentioning the unit of any physical quantity is not a big deal. Just like, for the angular velocity or angular frequency $\omega$, you have the unit $rad/s$ or rather $s^{-1}$, since the unit in both cases is $\mathrm{Hz}$ (hertz).

Answer 2

You can consider that $r$ is not strictly the radius, but the conversion factor between the angular speed and the linear speed, in $\frac{[m]}{[rad]}$.

Now imagine that the angles are counted in degrees. You would need to use

$w = \frac{360}{t} \frac{[°]}{[s]}$;

$ v = w \frac{[°]}{[s]} \times r'\frac{[m]}{[°]}$

where

$r'\frac{[m]}{[°]}=\frac{2\pi}{360}r\frac{[m]}{[rad]}$.