Linear speed is how fast the arc of any given angle is growing. This is useful to determine the linear speed with the relationship to time.

Imagine that you are out for a run, as you run around the circle you will create your linear speed as you go around the circle.

You can define linear speed as such.

$$linear\ speed=\frac { arc }{ time }$$

To abstract this one more level you can use $s$ for the arc and $t$ for the given time to obtain the linear speed.

$$linear\ speed=\frac { s }{ t }$$

REMEMBER: The $s$ can be found by simply multiplying the radius by any given angle $\theta$. See How to find theta of any given angle?

$$s=r\theta$$