Introduction:
In this scenario, we have two different gases, A and B, filled in a container at the same temperature T. Gas A contains N molecules with mass m, while gas B contains 2N molecules, each with mass 2m. We are given the mean square velocity of gas B, V^2, and the mean square of the x-component of velocity of gas A, w^2. We need to find the ratio of w^2/v^2.

Understanding Mean Square Velocity:
Mean square velocity is a measure of the average kinetic energy of the gas molecules. It is defined as the average of the squares of individual velocities of all the gas molecules. Mathematically, it can be represented as:

Mean square velocity (V^2) = (1/N) * ∑(v^2), where v is the velocity of each molecule.

Analysis:
1. Number of molecules in gas A: N
Number of molecules in gas B: 2N

2. Mass of each molecule in gas A: m
Mass of each molecule in gas B: 2m

3. We are given V^2, the mean square velocity of gas B, and we need to find the ratio of w^2/v^2.

Calculating the Ratio:
1. Mean square velocity of gas A (w^2) can be calculated using the formula mentioned earlier:

w^2 = (1/N) * ∑(v^2)

Since gas A contains N molecules, the sum will involve N velocities.

2. Let's consider the mean square velocity of the individual molecules in gas A. Since it is the same temperature T, the average kinetic energy of all the gas molecules is the same.

3. According to the kinetic theory of gases, the average kinetic energy of a molecule is given by:

KE = (3/2) * k * T

Where k is the Boltzmann constant and T is the temperature.

4. The kinetic energy is directly proportional to the square of the velocity:

KE = (1/2) * m * v^2

Equating the two equations, we get:

(1/2) * m * v^2 = (3/2) * k * T

Solving for v^2, we get:

v^2 = (3 * k * T) / m

5. Similarly, for gas B, each molecule has a mass of 2m. Using the same equation as above, we get:

V^2 = (3 * k * T) / (2m)

6. Now we can calculate the ratio of w^2/v^2:

(w^2 / v^2) = [((1/N) * ∑(v^2)) / ((3 * k * T) / (2m))]

7. Simplifying the equation, we find that the mass and temperature cancel out, and we are left with:

(w^2 / v^2) = (2 / 3)

Conclusion:
The ratio of w^2/v^2 is 2/3. This means that the mean square of the x-component of velocity of molecules in gas A is two-thirds of the mean square velocity of molecules in gas

2/3