Problem Statement:
Two masses, m1 = 4kg and m2 = 2kg, are released from rest and attached to a massless pulley. We need to find out the distance traveled by the center of mass in 9 seconds.

Explanation:
To solve this problem, we need to understand the concept of center of mass and the motion of the masses.

Center of Mass:
The center of mass of a system is the point where the total mass of the system is considered to be concentrated. In this problem, we have two masses, m1 and m2, attached to a pulley. The center of mass of the system will lie somewhere between these two masses.

Motion of Masses:
When the masses are released from rest, they will accelerate due to the force of gravity. The acceleration of each mass can be calculated using Newton's second law:

F = ma

For m1:
m1g - T = m1a1

For m2:
T - m2g = m2a2

Where:
m1 = mass of m1 (4kg)
m2 = mass of m2 (2kg)
g = acceleration due to gravity (9.8 m/s^2)
T = tension in the string connecting the masses
a1 = acceleration of m1
a2 = acceleration of m2

Since the masses are connected by a string, their accelerations must be equal and opposite in direction:

a1 = -a2

By solving the above equations, we can find the value of acceleration (a) and tension (T).

Distance Traveled by the Center of Mass:
The distance traveled by the center of mass can be calculated using the equation:

d = (1/2)at^2

Where:
d = distance traveled by the center of mass
a = acceleration of the masses
t = time (9 seconds)

By substituting the value of acceleration (a) and time (t) in the above equation, we can find the distance traveled by the center of mass.

Conclusion:
By applying the concepts of center of mass and motion of masses, we can solve the given problem. The distance traveled by the center of mass can be calculated using the equation d = (1/2)at^2, where a is the acceleration of the masses and t is the time.