A small particle of mass m is released from a height h on a large smooth sphere kept on a perfectly smooth surface as shown in the figure. Collision between particle and sphere is perfectly inelastic. Determine the velocities of particle and sherre after collision.

A small particle of mass m is released from a height h on a large smoo

1.	A small particle of mass m is released from a height h on a large smooth sphere kept on a perfectly smooth surface as shown in the figure. Collision between particle and sphere is perfectly inelastic. Determine the velocities of particle and sherre after collision.
Answer» Solution :Collision takes lace between smooth bodies, therefore, tangential velocity of particle is unchanged. `v_(r)=usintheta` and `u=sqrt(2gh)` when `sintheta=(R/2)/R=1/2` or `theta=30^(@)` `v_(t)=u/2` …………i From conservation of MOMENTUM along `X`-axis, `P_(i)=P_(f)` `0+0=MV+mv_(n)sintheta+mv_(t)costheta`..............iii from the definitation of coefficient of restitution. `e=(Vsintheta-(-v_(n)))/(ucoshteta-0)=0` [`e=0,` for a perfectly inelastic collision] or, `Vsintheta=-v_(n)` `v_(n)=-V/2` .............iii On substitution expression for `v_(n)` and `v_(t)` in eqn ii we obtain `MV=m(-V/2)1/2+m(u/2)(sqrt(3))/2` `(M+m/4)V=m(usqrt3)/4` where `u=sqrt(2gh)` or `V=(msqrt(6gh))/((4M+m))` `v=sqrt(v_(n)^(2)+v_(t)^(2))=sqrt((6m^(2)gh)/(4(4M+m)^(2))+(gh)/2)`