A thin uniform rod of mass `m` and length `l` is kept on a smooth horizontal surface such that it can move freely. At what distance from centre of rod should a particle of mass `m` strike on the rod such that the point `P` at a distance `l/3` from the end of the rod is instantaneously at rest just after the elastic collision?

Options

A. `1//2`
B. `1//3`
C. `1//6`
D. `1//4`

Answer

A

Explanation

Let the impulse due to the particle be `J,J=mv_(CM)`
`implies v_(CM)=J/m`
Angular impulse about centre of mass is
`Jx=(ml^(2))/12 omegaimplies omega=(12Jx)/(ml^(2))`
For point `A` to be at rest instantaneously,
`vecv_("trans")+vecv_("rot")=0`
or `v_(cm)=(omegal)/6`

Thus `J/m=l/6xx(12Jx)/(ml^(2))`
which gives `x=l/2`