A smooth hemisphere of mass m and radius R is at rest. A smooth solid sphere of mass 2m and radius R moving with velocity `V_(0)` between two horizontal smooth surfaces separated by a distance slightly greater than 2R as shown in figure. Solid sphere collides with the hemisphere. if coefficient of restitution is `(1)/(2)`, then

Options

A. The speed of hemisphere after collision is `V_(0)`
B. The speed of solid sphere after collision is `(V_(0))/(2)`
C. The loss in kinetic energy of the system is `(1)/(4)mV_(0)^(2)`
D. The final kinetic energy of hemisphere is `(1)/(4)` th the initial kinetic energy of sphere

Answer

A, B, C

Explanation

`2MV_(0)=2MV_(1) +MV_(2)`
`((1)/(2)=e=(V_(2)cos theta - V_(1)cos theta))/(V_(0)cos theta)`

`V_(1)=(2-e)(V_(2))/(3)=(V_(0))/(2)`
`V_(2)=(1+e)(2V_(0))/(3)=V_(0)`
` K.E = (1)/(2)2m V_(0)^(2)`
`K.E =(1)/(2)2m xx(V_(0)^(2))/(4)+(1)/(2) xxM xxV_(0)^(2)`
Loss in K.E `= (1)/(4) mV_(0)^(2)`.