Let's play cricket

There is a hypothetical cricket match going on in kolkata between India vs Australia. Presently there is a spinner of height 2 metre bowling first ball of the over to a right hander.He is bowling a leg stump line, his ball deceives the batsmen and he gets bowled.

In the xyz,co-ordinate space consider the pitch to be the region 0x20,2y2,z=00 \le x \le 20, -2 \le y \le 2,z=0

Consider stumps to be the lines

0z0.5,x=20,y=0.1250 \le z \le 0.5,x=20,y=0.125

0z0.5,x=20,y=00 \le z \le 0.5,x=20,y=0

0z0.5,x=20,y=0.1250 \le z \le 0.5,x=20,y=-0.125

The ball is released from the point (0,0.125,2)(0,-0.125,2) with a velocity 30i^30 \hat{i} m/s, also given an angular velocity of ωi^,ω0-\omega \hat{i}, \omega \ge 0.

The question is simple, if the maximum value of ω\omega, for which the batsmen can get bowled is =a(b+cb)d=\frac { a(b+c\sqrt { b } ) }{ d } rad/s where a,b,c,da,b,c,d are integers, a,da,d are co-prime,bb is not divisible by any square prime number then find a+b+c+da+b+c+d.

Details and Assumptions

1) Assume ball a solid sphere of radius 0.01m,g=10ms20.01 m, g= 10 m{s}^{-2}.Consider the pitch to be sufficiently rough and the collision to be elastic.

2) All the co-ordinates are given in metres.

3) If in the trajectory of the ball it is hitting the stumps, consider the batsmen bowled.

4) Consider it as a point mass while calculating it's trajectory

Image credit: Wikipedia John Sutton
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