1 The moment of inertia of the rod about A,
2 The centre of mass of the rod is initially at a depth below A
3 At the moment of impact the centre of mass is at a depth below A.
4 Use the energy principle to find the KE ( ) of the rod just before impact, where is its angular velocity.
5 At the moment of impact the only external impulse on the system (rod and particle) is at A. So the angular momentum of the system about A is conserved. (Angular momentum of rod = before impact and after impact where is the angular velocity of the rod after impact; angular momentum of particle = before impact and after impact, where is the initial speed of the particle.)
6 If the collision is perfectly elastic, there is no loss of energy. So you can equate the KE of the system before and after impact.
7 If the collision is perfectly inelastic, and the particle adheres to the rod then there will be loss of energy (so you can't use #6). But the speed, , of the particle will now be = linear speed of the end of the rod.
Can you attempt it now?