In the figure shown, a semi-cylindrical frictionless track of radius R, centred at point O. A particle is released from point A(OA = R/2). Find the coefficient of restitution e, if the velocity of the particle becomes horizontal just after collision with the track.

Just before collision, the components of velocity along the normal and along the tangent arevn=usin⁡θ and vt=ucos⁡θThe component of velocity in common tangent direction will remain unchanged as in this direction there is no impulse.Hence, vt=usin⁡θ           …(i)Velocity of the particle just after collision along common normal reactionvn′=evn=eucos⁡θ           …(ii)After collision the particle moves in horizontal direction, hence the net vertical component of particle velocity should be zero for whichevncos⁡θ=vtsin⁡θ           …(iii)From (i), (ii) and (iii) we gete(ucos⁡θ)⋅cos⁡θ=(usin⁡θ)⋅sin⁡θor e=tan2⁡θ=tan2⁡π6=13