Editing Rotating reference frame (section)

=== Relation between velocities in the two frames ===
A velocity of an object is the time-derivative of the object's position, so

:<math>\mathbf{v} \ \stackrel{\mathrm{def}}{=}\   \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \ .</math>

The time derivative of a position <math>\boldsymbol{r}(t)</math> in a rotating reference frame has two components, one from the explicit time dependence due to motion of the object itself in the rotating reference frame, and another from the frame's own rotation.  Applying the result of the previous subsection to the displacement <math>\boldsymbol{r}(t),</math> the [[Velocity|velocities]] in the two reference frames are related by the equation

:<math> 
\mathbf{v_i} \ \stackrel{\mathrm{def}}{=}\   
\left({\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}}\right)_{\mathrm{i}} \ \stackrel{\mathrm{def}}{=}\   
\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = 
\left[ \left(\frac{\mathrm{d}}{\mathrm{d}t}\right)_{\mathrm{r}} + \boldsymbol{\Omega} \times \right] \boldsymbol{r} = 
\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)_{\mathrm{r}} + \boldsymbol\Omega \times \mathbf{r} = 
\mathbf{v}_{\mathrm{r}} + \boldsymbol\Omega \times \mathbf{r} \ ,
</math>
where subscript <math>\mathrm{i}</math> means the inertial frame of reference, and <math>\mathrm{r}</math> means the rotating frame of reference.