A ball of mass m falls symmetrically on a set of three hemispheres placed in contact with each other on smooth horizontal ground from a height h. Mass of each hemisphere is m/2 and radius of each body is R. Neglecting any rebounding of hemisphere. If sphere comes to rest after the simultaneous collision with three hemispheres then ratio of KE of sphere before collision and KE of any hemisphere after collision is P; find the value of P.
Physics · Mixed Chapter
JEE Mains

Question

A ball of mass m falls symmetrically on a set of three hemispheres placed in contact with each other on smooth horizontal ground from a height h. Mass of each hemisphere is m/2 and radius of each body is R. Neglecting any rebounding of hemisphere. If sphere comes to rest after the simultaneous collision with three hemispheres then ratio of KE of sphere before collision and KE of any hemisphere after collision is P; find the value of P.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: In top view, the three hemispheres touch, making an equilateral triangle of side 2R. The falling ball's centre is above its centroid, so each contact normal has horizontal component 1/√3 and vertical component √(2/3). Step 2: Let initial ball speed be v and impulse magnitude at each contact J. If the ball stops, vertical impulse balance gives 3J√(2/3)=mv. Step 3: Each hemisphere (mass m/2) receives horizontal impulse J/√3. Hence its speed u=(J/√3)/(m/2)=√2v/3. Step 4: Initial ball KE=½m…