Two particles `A` and `B`, each of mass `m`, are connected by a light rod of length `L` lying on a smooth horizontal surface. A particle of mass `m` moving with speed `v_(0)` strikes to the end of rod and sticks to `A` as shown . Find
(a) the velocity of the center of mass,
(b) the angular velocity about the center of mass of system (`A+B+` particle) and
(c) the linear speeds of `A` and `B` immediately after collision.

Explanation

When the particle sticks to `A`, the location of center of mass

`x_(c.m.)=(2mxx0+mxxL)/(2m+m)=(L)/(3)`
Conservation of linear momentum
`mv_(0)=(2m+m)v_(c.m.)implies v_(c.m.)=(v_(0))/(3)`
Conservation of angular momentum about `C`
`mv_(0)x_(c.m.)=I_(c)omega`
`mv_(0)(L)/(3)=[2mx_(c.m.)^(2)+m(L-x_(c.m.))^(2)]omega`
`=[2m((L)/(3))^(2)+m((2L)/(3))^(2)]omega`
`=(2)/(3)mL^(2)omegaimplies omega=(v_(0))/(2L)`
linear speed of
`A: v_(A)=v_(c.m.)+x_(c.m.)omega=(v_(0))/(3)+(L)/(3)xx(v_(0))/(2L)=(v_(0))/(2)`
`B: v_(B)=v_(c.m.)-(L-x_(c.m.))omega=(v_(0))/(3)-(2L)/(3)xx(v_(0))/(2L)=0`