Given a point P at (0, R/2), with constant velocity V, after T time, it could move to P’. Now there’s a circle wall centered at (0,0), with radius R. Collision with the wall would cause the object to lose its velocity component perpendicular to the tangent line at the contact point. The point starts with the identical state from P, hits point C, and slides to P2 after T time, because of the collision with the circle wall.
Assuming ∠COP = theta, R, length of CP’ is known, what’s ∠COP2 = phi?
After contacting with C, the point would move in uniform circular motion, with velocity
V*cos(theta). Then, we could calculate the arc length using
V * cos(theta) * CP' / V == CP' * cos(theta). On the other hand,
phi * R is the arc length as well. Therefore,
phi * R = CP' * cos(theta).
Thank to helpful discussions with Ji-Yao Chen.