Rolling Projectile!

A uniform sphere is given an angular velocity of \(\omega = 7\text{ rad/sec}\) in the clockwise direction and is released from a height of \( h = 22\text{ m}\) towards the ground, which is sufficiently rough. It then collides with the ground, and rebounds at an angle of \(\theta \).

Find the range (in meters) of the projectile thus traced (before the next collision).

Details and Assumptions:

  • The radius of the sphere is \( R = 2\text{ m},\) and \( g = 10\text{ m/s}^2\).
  • Due to the collision, the velocity in the vertical direction is reduced to \( \frac{1}{5} \) of the original magnitude.
  • The ground has sufficient friction such that when the sphere is on the ground, it is in the state of pure rolling. That is, before rebounding, the sphere has already started pure rolling.
  • The initial height measured is from the ground to the center of mass of the sphere.
  • Neglect air drag and buoyancy effect. Neglect all deviation effects on the rotating sphere (due to rotation) when in the air.

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