# Bouncing Balls

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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