[Mathew]

If the string gets cut the ball flies out in a straight line directly away from the source of the centripetal force that is trying to pull it inwards.

From the website http://www.fearofphysics.com/BallString/ballstring.html

Look at the pictures here

Which way?

The string exerts an inwards force on the ball, called the centripetal force. This is what keeps the ball flying in a circle. When the string is cut, this force vanishes (no more string, no more force). At this point there are no other forces acting on the ball (forget gravity for now). Why are forces so important? Because no object can change its direction unless a force acts on it (this is Newton's First Law).

So what does the ball have after the string is cut? It still has its speed. The speed it had as it was rounding the circle just as the string was cut. This speed is pointing along the edge of the former circle (not outward along where the string used to be.) Without any forces, an object will always go in the direction of its speed (Newton's First Law again).