# The Curvy Rebound: One of the most interesting (Geometric) Probability problems.

An infinitely small ball is placed at a random point on the red line shown on the right. The line is 10 metres long. The semi-circle that stems from this line is its resulting size too. Also positioned 3 metres above the centre of the line is a small circle with a diameter of 1 metre. There is no gravity of any sort by the way.

The ball is shot up into the semi-circle at a random angle from -90 degrees to +90 degrees. After it comes back past the red line, it is all over.

Question-Part 1: What's the chance the 'ball' will contact the yellow circle?

Question-Part 2: Same question as above, except that the curve (which has now extended to become a full circle) rotates (pivot = point at centre of yellow circle) 45 degrees for every 5 metres the ball has travelled.

Background: This question is from the same site (http://www.skytopia.com/project/imath/imath.html) I asked this question on MSE from: Probability of the Center of a Square Being Contained in A Triangle With Vertices on its Boundary

My effort: This problem has vexed me a lot, and that too the first part itself. The second part looks seriously tough. So I'll just tell you the my effort for part 1.

I initially wanted to solve a very simple case, so I took one endpoint of the diameter. I could see that distance from the endpoint to the centre of the yellow circle would be $\sqrt{34}cm$ cm; therefore length of tangents to the circle from the endpoint would be $\sqrt{33.75}cm$ each. I then get ~ 9.84 degrees as the angle subtended by the chord joining the two points of contact (a.k.a. points where tangents from the endpoint touch the yellow circle) on the yellow circle at the endpoint.

So I've evaluated that, for certain, the favourable event here, at one endpoint of the diameter, is at least 9.84 degrees out of a sample space of 180 degrees. But now, I have to evaluate:-

1. Other possibilities of angle which result in the ball to hit the yellow circle when initially at the end point. I hope I;ve been doing the right calculations so far.
2. Since this was just a grossly simplified case, it still doesn't get me close to the answer required for JUST PART ONE, as it (the ball) can lie anywhere on the red line (A.k.a. diameter of the big circle).
3. Part two. I'm pretty sure that this is far above my capability for now.

Sorry for the post being quite long; I'm sure the question is interesting enough. So essentially, to recap, my questions are:-

Question(s): What is the solution to PART 1 and PART 2 of the question? How exactly can one approach this question and be successful in it? Has my approach been alright so far?

Note:

To clarify:-

1. The balls bounces off, without loss of velocity. (There's no GRAVITY as previously stated in the question).
2. The curve which becomes the circle is the semi-circle, and this new circle pivots around the point which happens to be the centre of the yellow circle.
3. The red line rotates along with the circle, and this rotation is constantly clockwise.