A solid spherical ball of radius 5m is projected to the right from a rough ground with a velocity 10 m/s at an angle \(\theta\) with the horizontal. Also at the same time it is given an angular velocity of 3 rad/s clockwise such that axis of rotation is perpendicular to plane of projectile .

Let horizontal distance travelled by the ball from the point of projection to the point where it made it's second bounce be \(D\). Given that maximum value of \(D\) occurs at \(\theta = \arccos { \frac { (\sqrt { a } -b) }{ c } }\) where \(a,b,c\) are integers, \(a\) is square free, \(b,c\) are co-prime then find \(a+b+c\).

**Assumptions and details**

- Co-efficient of restitution of ground = 0.5
Co-efficient of friction of ground with the ball is \(\frac { 1 }{ 3 }\)

Mass of the ball = 5 kg

- Take \(g=10 \text{ m/s}^2\)

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