Problem:
A block of mass m is connected to another body of mass M by a massless spring of spring constant k. The blocks are kept on a smooth horizontal plane. Initially, the blocks are at rest and the spring is unstretched when a constant force F starts acting on the block of mass M to pull it. Find the maximum extension of the spring?

Solution:
The given problem can be solved using the concept of conservation of energy. Initially, the total energy of the system is zero as there is no motion. When the force is applied, the block of mass M starts moving and the spring starts stretching.

Step 1: Find the work done by the external force F
The external force F does work on the block of mass M and the work done is given by:
W = F * s
where s is the distance moved by the block of mass M.

Step 2: Find the increase in potential energy of the spring
The spring starts stretching due to the force applied on the block of mass M. The increase in potential energy of the spring is given by:
U = 0.5 * k * x^2
where x is the extension of the spring.

Step 3: Find the increase in kinetic energy of the block of mass M
The block of mass M starts moving due to the force applied on it. The increase in kinetic energy of the block of mass M is given by:
K = 0.5 * M * v^2
where v is the velocity of the block of mass M.

Step 4: Equate the increase in potential energy and kinetic energy
The increase in potential energy of the spring is equal to the increase in kinetic energy of the block of mass M. Therefore, we have:
0.5 * k * x^2 = 0.5 * M * v^2

Step 5: Find the maximum extension of the spring
The maximum extension of the spring can be found by using the equation of motion:
v^2 = u^2 + 2as
where u is the initial velocity of the block of mass M, s is the distance moved by the block of mass M and a is the acceleration of the block of mass M.

As the initial velocity of the block of mass M is zero, we have:
v^2 = 2as

Substituting this in the equation obtained in step 4, we get:
0.5 * k * x^2 = 0.5 * M * (2as)

Simplifying, we get:
x = sqrt((2MF)/k)

Therefore, the maximum extension of the spring is sqrt((2MF)/k).

Conclusion:
The maximum extension of the spring is sqrt((2MF)/k).