Astronaut With Gun and Circular Motion

1. The problem statement, all variables and given/known data
An astronaut (mass mA) in space, tied to the end of a tether of length 10m, is traveling in a circle with constant speed making one revolution every 15seconds.
a) Assume that the astronaut fires a spring loaded gun (containing a ball) towards the center of the circle and when he is located at the origin. Sketch the trajectory of the ball as seen by someone fixed in space. (Overheard perspective)
b) If the gun is directed towards the center of the circle, then regardless of the speed of the ball the astronaut cannot catch the ball. Explain why this is so.
c) Is there any circumstance for which the astronaut can fire the gun and catch the ball? Explain.

2. Relevant equations
Fc = mv^2/r

3. The attempt at a solution
I'm working on this in a group. The general consensus for part A is that the trajectory of the ball will go diagonally off to the right (assuming he is traveling counter-clockwise). Of course I've heard arguments that make sense to me for the ball just going straight to the other side of the circle, and for the ball going at a curve to the right. I got to admit I find it very confusing.

For part b I think we have it in our mind, due to the circle the astronaut has to take and the balls singular direction, but not really sure how to formulate it into a coherent sentence. As for part c we think if he shot the ball at a certain critical angle he might be able to catch it.

Thanks before hand for any help you can offer! If you need to see the picture we have to sketch the trajectory on I can scan it in (though I don't find it all that helpful).
Ok so for part a) we have an astronaut going around with some velocity v and some angular velocity omega. Before the bullet is fired you would agree with me that the bullet is moving with the same speed as the astronaut, right?

Now consider if the astronaut was stationary and fired a bullet. The bullet would move in the direction fired with some velocity.

Now if we add these velocities together at the instant the astronaut fires the bullet this will be the total velocity of the bullet. But this still leaves the question of which direction and path the bullet will take.

Well the astronauts velocity is tangent to the circle and the bullet is fired radially inward. So what can be said about the direction?
This says to me, and I'm really just guessing here that we would add the inward velocity and the tangential velocity so that ball will go diagonally to the right?

Then I would say for part B that no matter how fast the bullet was shot the ball would still adopt the tangential velocity of the astronaut, going at a straight line as opposed to a curve the astronaut would have to be able to speed up in order to catch the ball?

Then for part C the only situation in which the ball could be shot and the astronaut could catch it is if he shot it at some critical angle (I'm guessing 180degrees - opposite his motion at the point) and at the same speed so that the two velocities would cancel out and the ball would remain motionless? Or would that be for a way in which he could never fire the ball and not a way in which he could catch it?
any feedback on how I'm interpreting this?
Hey I hate to keep bumping this but it's due tomorrow and I want to make sure I'm on the right track at least, thanks!
*bump* sorry again.

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