Question

Consider the following collision problem from the game of golf. A golf club driver is swung and hits a golf ball. The driver has a shaft of length L and a head. To simplify the problem, we will neglect the mass of the driver shaft entirely. The driver head we will assume is a uniform density sphere of radius R and mass M. The golf ball is a uniform density sphere of radius r and mass  m. Initially the ball is at rest, with zero linear velocity, v → 2 i = 〈 0 , 0 , 0 〉, and zero angular velocity, ω → 2 i = 〈 0 , 0 , 0 〉. The driver head has an initial linear velocity of v → 1 i = v 1 i 〈 1 , 0 , 0 〉. Immediately after the collision the golf ball has linear velocity v → 2 f = v 2 f 〈 cos ⁡ θ , sin ⁡ θ , 0 〉and angular velocity ω → 2 f = ω 2 f 〈 0 , 0 , 1 〉. This angular velocity of the golf ball is commonly referred to as "backspin". Recall that the moment of inertia for a sphere rotating about its center of mass is I = 2 M R 2 5.  

You can assume that the parameters described in the previous paragraph are the known quantities in the problem. In the questions that follow, you can write your results in terms of these known values:

Question 1: Prior to the collision the driver head was being swung along a circular path because the golfer was exerting a net torque at the end of the golf club with their hands to rotate the club. This circular motion implies that there is a relationship between the linear velocity of the driver head and its angular velocity. What is the initial angular velocity of the driver head ω → 1 i in terms of v 1 i, L, and R. Express your result as a three-component vector.  

Answer 1

This problem is very easy once you understand the language of representation of velocity and angular velocity vectors in it.

Also there is a lot of information which is not necessary to solve what is asked in the question still I have diagramatically explained the whole problem before and after the collision to get a picture of what is happening.

The real solution starts from THE THEORY OF CIRCULAR MOTION in the image attached for the detailed solution.