The Problem:
Two blocks of masses 10 kg and 4 kg are connected by a spring of negligible mass and placed on a frictionless horizontal surface. An impulse gives a velocity of 14 m/s to the heavier block in the direction of the lighter block. We need to find the velocity of the center of mass (COM).

Solution:
To find the velocity of the center of mass, we need to use the principle of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant if no external forces are acting on it.

Step 1: Analyzing the System:
We have a system consisting of two blocks connected by a spring. The blocks are on a frictionless surface, and there are no external forces acting on the system. Thus, the system is isolated, and momentum will be conserved.

Step 2: Understanding Impulse and Momentum:
Impulse is defined as the change in momentum of an object. It is given by the product of the force acting on the object and the time for which the force acts. In this case, the impulse given to the heavier block causes it to have a velocity of 14 m/s in the direction of the lighter block.

Step 3: Applying Conservation of Momentum:
Let the initial velocity of the lighter block be V1 and the initial velocity of the heavier block be V2. The final velocity of the lighter block is V1' and the final velocity of the heavier block is V2'. Since the two blocks are connected by a spring, they move together as one system.

According to the principle of conservation of momentum:
(initial momentum) = (final momentum)
(mass of lighter block * initial velocity of lighter block) + (mass of heavier block * initial velocity of heavier block) = (mass of lighter block * final velocity of lighter block) + (mass of heavier block * final velocity of heavier block)

In this case, the initial velocity of the lighter block is 0 since it is at rest initially. The final velocity of the lighter block is the same as the final velocity of the heavier block since they move together as one system. Let's denote this final velocity as Vf.

(0) + (10 kg * V2) = (4 kg * Vf) + (10 kg * Vf)

Simplifying this equation, we get:
10 kg * V2 = 14 kg * Vf

Step 4: Finding the Velocity of the Center of Mass:
The velocity of the center of mass can be found by taking the weighted average of the velocities of the two blocks, where the weights are given by their respective masses.

Let V_COM be the velocity of the center of mass. Then,
V_COM = (10 kg * V2 + 4 kg * Vf) / (10 kg + 4 kg)
V_COM = (10 kg * V2 + 4 kg * Vf) / 14 kg

Substituting the value of Vf from the previous equation, we get:
V_COM = (10 kg * V2 + 4 kg * (10 kg * V2) / 14 kg
V_COM = (10 kg * V2 + 40 kg * V2) / 14 kg
V_COM = (50 kg * V2) / 14 kg