Analysis of the Problem:

We have a system consisting of two blocks - a smaller block of mass m and a larger block of mass M. The blocks are placed one after the other on a horizontal road and are initially at rest. The system is started towards the right with an initial velocity v. We need to find the time elapsed before the smaller block separates from the larger block.

Steps to Solve the Problem:

1. Identify the forces acting on the system
2. Determine the net force on the system
3. Apply Newton's second law to find the acceleration of the system
4. Use the acceleration to find the time taken for separation

Forces Acting on the System:

1. Weight of the smaller block (mg) acting vertically downwards
2. Normal force on the smaller block (N) acting vertically upwards
3. Frictional force between the smaller block and the larger block (f1) acting towards the left
4. Weight of the larger block (Mg) acting vertically downwards
5. Normal force on the larger block (N') acting vertically upwards
6. Frictional force between the larger block and the road (f2) acting towards the left

Net Force on the System:

The net force on the system is the sum of all the forces acting on it. In this case, the net force is given by:

Net force = f1 + f2

The frictional force between the smaller block and the larger block is given by:

f1 = u/2 * N

where u is the coefficient of friction between the two blocks.

The frictional force between the larger block and the road is given by:

f2 = u * N'

where u is the coefficient of friction between the larger block and the road.

Acceleration of the System:

Using Newton's second law, the acceleration of the system can be calculated using the formula:

Net force = (m + M) * a

where a is the acceleration of the system and (m + M) is the total mass of the system.

Substituting the values for the net force and simplifying, we get:

u/2 * N + u * N' = (m + M) * a

Time Taken for Separation:

The smaller block will separate from the larger block when the frictional force between them becomes zero. This happens when the frictional force f1 becomes equal to the maximum frictional force f2. Thus, we can equate the two frictional forces:

u/2 * N = u * N'

Using this equation, we can solve for N' in terms of N:

N' = (u/2) * N / u

Substituting this value of N' in the equation for acceleration, we get:

u/2 * N + u * (u/2) * N / u = (m + M) * a

Simplifying the equation, we find:

N = (2mu / u + 1) * (m + M) * a

Now, we can use the equation for N to find the time taken for separation. The time taken for separation is the time taken for the smaller block to cover the length of the larger block, which is 1.

Using the equation for displacement, s = ut + (1/2)at