Problem Statement:
A small block of mass m is kept at the left end of a larger block of mass M and length 1. The system can slide on a horizontal road. The system is started towards the right with an initial velocity v. The coefficient of friction between the road and the bigger block is μ1 and that between the blocks is μ2. Find the time elapsed before the smaller block separates from the bigger block.

Solution:
1. Free Body Diagram:
To solve this problem, we need to analyze the forces acting on the blocks. Let's draw a free body diagram for the larger block.

- The weight of the larger block acts vertically downwards.
- The normal force exerted by the road acts vertically upwards.
- The frictional force between the road and the larger block acts in the opposite direction of motion.
- The force exerted by the smaller block on the larger block due to friction acts towards the left.

2. Equations of Motion:
Based on the forces acting on the larger block, we can write the equations of motion.

- ΣF = ma, where ΣF is the net force acting on the block, m is the mass of the block, and a is the acceleration.
- The net force acting on the block is the difference between the frictional force and the force exerted by the smaller block due to friction.

3. Frictional Forces:
The frictional force between the road and the larger block can be calculated using the equation:

- frictional force = μ1 * normal force

The normal force is equal to the weight of the larger block, which is Mg.

- normal force = Mg

The frictional force between the blocks can be calculated using the equation:

- frictional force = μ2 * normal force

The normal force is equal to the weight of the smaller block, which is mg.

- normal force = mg

4. Net Force:
The net force acting on the larger block is the difference between the frictional force and the force exerted by the smaller block due to friction.

- net force = frictional force (between the road and the larger block) - force exerted by the smaller block due to friction

5. Acceleration:
Using the equation ΣF = ma, we can calculate the acceleration of the larger block.

- net force = ma

6. Time of Separation:
The time of separation is the time taken by the smaller block to cover the length of the larger block, which is 1m.

- distance = velocity * time
- time = distance / velocity

7. Conclusion:
To find the time elapsed before the smaller block separates from the larger block, follow these steps:
- Calculate the frictional forces between the road and the larger block, and between the blocks.
- Calculate the net force acting on the larger block.
- Calculate the acceleration of the larger block.
- Use the acceleration to calculate the time taken for the smaller block to cover the length of the larger block, which is the time of separation.