Consider a reference frame which is rotating (for example on a carousel) and a steady inertial frame, with the same origin.
$\vec{r}=\vec{r'}$
$\vec{v}=\vec{v'}+\vec{\Omega}\times\vec{r'}$
$\vec{a}=\frac{d\vec{v'}}{dt}+\frac{d(\vec{\Omega}\times\vec{r'})}{dt}=\vec{a'}+\vec{\Omega}\times\vec{v'}+\frac{d\vec{\Omega}}{dt}\times\vec{r'}+\vec{\Omega}\times\vec{v'}+\vec{\Omega}\times(\vec{\Omega}\times\vec{r'})$
I would like to understand a little deeper why do the two terms $\vec{\Omega}\times\vec{v'}$ appear, from a physical point of view.
I found on this Wikipedia page http://en.wikipedia.org/wiki/Coriolis_force#Causes, an explanation about it.
- One of them (I think the one coming out the derivation of $\vec{\Omega}\times\vec{r'}$, but I'm not sure) is related to the change of the velocity in space. It is there because the velocities of the points on the carousel are different and if the point moves on the carousel it must change its velocity as a consequence. I think I got this one term.
- The other is related to the change on the velocity in time:
The same velocity (in an inertial frame of reference where the normal laws of physics apply) is seen as different velocities at different times in a rotating frame of reference. The apparent acceleration is proportional to the angular velocity of the reference frame (the rate at which the coordinate axes change direction), and to the component of velocity of the object in a plane perpendicular to the axis of rotation.
I do not understand what does this mean actually and the physical meaning of this other term.