Bullet from circular motion I'm thinking of a problem that I can't verify at this moment, so I'd need your help.
There's a person (Carl) on a circular platform, at a distance R from the center, the platform is moving with angular velocity w.
Carl fires a bullet toward the center of the platform (from his point of view).
The question is: what path will the bullet follow?
I think that it will follow the path in the image (Vb, the big green arrow) because it has a velocity from the circular motion and a velocity given by the gun. 
Vbθ should be w•R.
If so, with the right w, Carl could be hit by his own bullet?

Do you think it could be right?
Thank you :)
 A: Not really.
Using your drawing as a reference, we can see that the horizontal speed of the bullet will always be equal to Carl's linear speed, $V=\omega r$.
Carl's horizontal speed at the moment of shooting (at the bottom of the circle), $V_{h0}$, is also equal to $V$, but, as his trajectory is turning up, his horizontal speed will decrease, since part of the linear speed $V$ will be used to move him up. 

So, Carl would not be able to catch up with the bullet in the horizontal direction, regardless of the relative speeds of the bullet and the platform. Therefore, Carl would not be hit by his own bullet.  

Here is another, less direct (hopefully, more acceptable) answer.
Let's say that Carl fires a bullet at some point A on the circle. Now let's assume that Carl catches up with the bullet at some point B on the circle. 
Answering the following two questions should tell us if this assumption is valid:


*

*Both Carl and the bullet move at constant linear speeds. Which speed is greater?

*Carl's trajectory between A and B is along the circle line, while the bullet's trajectory is along a straight line. Which trajectory is longer? 
