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Dear ajit ,
you can apply the concept of relative velocity in case of angular motion also. so here the angular velocityof b w.rt a is w in clockwise direction.
therefore
a) for first collision - time taken is the time when the body b moves a full circle w.r.t to a if we are applying relative concepts ....... so wrelative = vrelative/R ; wrelative = R/Rt1 ; t1 =1 / wrelative
therefore t1 = 1/w
for third collision : consider the scenario when the first hit takes , the velocities are exchanged i.e the angular velocities are also exchanged . so now A has 2w and B has w as angular velocity. so now in the same time A will hit B from behind. So , taking symmetry third collision shud take 3t1 as time. therefore t3 = 3t1. i.e 3/w
b)before first collision the angular velocity of A was w . so distance travelled = velocity * t1
= wR * 1/w = R
between first and second collision -- distance travelled = 2WR*1/w = 2R
so total distance travelled = 3R
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