You can calculate the angular velocity of the rod as follows:
- find the linear velocity of one point on the rod relative to another point on the rod
- then find the component of this velocity at right angles to the rod
- then divide this component by the distance between these two points.
One of these two points may be the centre of the rod and the other one end of the rod; or it may be easier to take the points at opposite ends of the rod, and ignore the centre altogether.
So, what principles can you use to solve your problem? I shall call the moving particle A, the particle on the rod that is struck B, and the other end of the rod C. Let's deal with (a) first: elastic collision. You can say:
- At the moment of impact, there is no external impulse. So the (linear) momentum of the system along and perpendicular to the line AB is unchanged.
- The impulse on A is along the line BA, so A bounces back along BA, the line along which it came. Call this rebound velocity .
- The impulse on C is along the line of the rod, so C moves along the line BC.
- Split the velocity of B into two components: one along the rod ( ) and one perpendicular to the rod ( ). Then the velocity of C = , because it's a rigid rod.
- Finally, there is no loss of kinetic energy if the collision is perfectly elastic.
These will give you three equations, from which you can find, in terms of , the velocity of A, and the two components of the velocity of B (and hence the velocity of C). I make these, respectively:
Bearing in mind what I said initially, the velocity of B relative to C is therefore at right angles to the rod, and the angular velocity of the rod is therefore
In part (b), after impact, A and B coalesce to form a particle with mass . So let the components of velocity of this new particle again be and , with C's velocity, as before, being . We can no longer use the KE equation, but the two components of momentum of the system along and at right angles to AB are still unchanged at impact.
So this time you get two equations, which give (according to my working):
and the angular velocity this time is