Problem Statement:
Two identical blocks A and B of mass M each are kept on each other on a smooth horizontal plane. There exists friction between A and B. A bullet of mass m hits the lower block with a horizontal velocity v and goes embedded into it. Find work done by friction between A and B?

Solution:

Given:
Mass of block A = Mass of block B = M
Mass of bullet = m
Velocity of bullet = v

Approach:
To find the work done by friction between blocks A and B, we need to determine the relative displacement between the two blocks. We can analyze the problem in two stages:

1. Collision Stage:
In this stage, the bullet collides with block A and gets embedded into it. Due to the collision, the system experiences an impulse, which causes the blocks to move together with a common velocity, let's say V.

2. Sliding Stage:
In this stage, the blocks A and B move together with a common velocity V. The friction between the blocks opposes their relative motion. We need to calculate the work done by this friction force.

Let's calculate each stage in detail.

1. Collision Stage:
During the collision, the momentum of the system is conserved. Therefore, we can write the equation:

(m * v) + (M * 0) = (M + m) * V

Simplifying the equation, we get:

m * v = (M + m) * V

V = (m * v) / (M + m)

2. Sliding Stage:
During the sliding stage, the friction force between blocks A and B opposes their relative motion. The work done by friction is given by the equation:

Work = Friction force * Displacement

The friction force can be calculated using the equation:

Friction force = μ * Normal force

The normal force on block A is equal to its weight, which is M * g, where g is the acceleration due to gravity.

The displacement during the sliding stage is given by the equation:

Displacement = V * t

where t is the time taken during the sliding stage.

Now, let's calculate the work done by friction.

Work = (μ * M * g) * (V * t)

Substituting the value of V from the collision stage equation, we get:

Work = (μ * M * g) * ((m * v) / (M + m)) * t

Now, we have the expression for the work done by friction between blocks A and B.