Introduction:
In this problem, we have two blocks of masses 5kg and 2kg placed on a horizontal frictionless surface. These blocks are connected by a spring. An external kick gives a velocity of 14 m/s to the heavier block in the direction of the lighter one. We need to calculate the velocity gained by the center of mass.

Understanding the problem:
To solve this problem, we need to apply the principles of conservation of momentum and conservation of energy. The center of mass is the point at which the total mass of a system is concentrated and moves as if all the mass were concentrated at that point.

Solution:
To find the velocity gained by the center of mass, we can follow these steps:

Step 1: Calculate the initial momentum of the system.
- The initial momentum of the system is the sum of the individual momenta of the two blocks.
- Momentum is defined as the product of mass and velocity.
- The initial momentum of the system can be calculated as:
Initial momentum = (mass of block 1 * velocity of block 1) + (mass of block 2 * velocity of block 2)

Step 2: Calculate the final momentum of the system.
- After the external kick, the blocks will move together as a system.
- The final momentum of the system can be calculated as the sum of the individual momenta of the two blocks.
- Since the blocks move together, their final velocity will be the same.
- The final momentum of the system can be calculated as:
Final momentum = (mass of block 1 + mass of block 2) * velocity of center of mass

Step 3: Apply the principle of conservation of momentum.
- According to the principle of conservation of momentum, the total momentum of an isolated system remains constant if no external forces act on it.
- This means that the initial momentum of the system should be equal to the final momentum of the system.
- Equating the initial and final momenta, we can solve for the velocity of the center of mass.

Step 4: Calculate the velocity of the center of mass.
- By equating the initial and final momenta, we can solve for the velocity of the center of mass.
(mass of block 1 * velocity of block 1) + (mass of block 2 * velocity of block 2) = (mass of block 1 + mass of block 2) * velocity of center of mass
velocity of center of mass = [(mass of block 1 * velocity of block 1) + (mass of block 2 * velocity of block 2)] / (mass of block 1 + mass of block 2)

Step 5: Substitute the given values and calculate the velocity of the center of mass.
- In this problem, the mass of block 1 is 5kg, the mass of block 2 is 2kg, and the velocity of block 1 is 14 m/s.
- Substituting these values into the equation, we can calculate the velocity of the center of mass.

Conclusion:
The velocity gained by the center of mass can be calculated by applying the principles of conservation of momentum and solving for the velocity of the center of mass. In this problem, the velocity

10m/s