Let's play carrom

Geometry Level 5

Me and my sister are playing carrom one evening. Since I am a maths student, I try to implement it into carrom as well, I play a quite neat shot that pushes the carrom coin neatly into the hole but not after having some collisions with the board.

The mathematics of the situation is as follows :

Imagine the boundaries of the carrom board to be the line segements \(xy(x-1)(y-1)=0 , 0 \le x,y \le1 \) and the holes to be the points \( (0,0) , (1,0) , (0,1) , (1,1) \)

A carrom coin is pushed from the point \( \left (\dfrac{1}{2} , 0 \right ) \) with a non-zero velocity. Such that it collides with the carrom board firstly at the point \(\left ( \dfrac{2}{15} , 1 \right )\) with the carrom board. Find the total number of collisions the coin makes with the carrom board before going into any of the hole.

Details and Assumptions

1)The size of coin and the hole are negligible, that is they are very small(and the size of the hole is just greater than that of coin. That for the coin to go into the hole it must go exactly into the hole. No near cases must be considered)

2) There is no friction on the carrom board. Hence the coin must go on and on, till it reaches the hole. Also while calculating total number of collisions count the first collision too( that is collision at the point \(\left (\dfrac{2}{15},1 \right )\)

3) The collisions the coin makes with the carrom board are elastic that is angle of incidence is equal to angle of reflection.

This is original.
Image credit: Wikipedia Bernard Gagnon

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