On a horizontal ground, there is a fixed platform of height ...

Question

On a horizontal ground, there is a fixed platform of height . On the platform, there is a smooth fixed ramp. The top of the ramp is also at a height above the platform's surface, and the bottom of the ramp is tangential to the platform. A small ball A, starting from rest, slides down the ramp and reaches the smooth horizontal surface of the platform. It then collides inelastically with a stationary ball B on the platform, and the two balls stick together. The combined mass slides off the platform horizontally, and the landing point is exactly half the platform height ( ) away from the platform's edge horizontally. Neglect air resistance and take g as the acceleration due to gravity. What is the mass ratio of the two balls?

To solve this problem, we analyze the system step by step:

Velocity of ball A after descending the ramp

The ball A slides down the ramp from a height . Its potential energy is fully converted to kinetic energy as the ramp is smooth. Using conservation of energy:

Ball A has a velocity at the bottom of the ramp.

Inelastic collision between balls A and B

Let and be the masses of balls A and B, respectively. Before the collision, ball B is stationary. After the collision, the two balls stick together and move with a common velocity .

Using conservation of momentum:

Substitute into the equation:

Horizontal motion of the combined mass

After the collision, the combined mass slides off the platform and lands a horizontal distance from the edge of the platform. We are told .

The time for the combined mass to fall from the platform (height ) is determined by vertical motion under gravity:

During this time, the combined mass travels horizontally a distance , so:

Substitute and :

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