The velocity of three molecules, are `2 km s^(-1), 4 km s^(-1), 6 km s^(-1)`. Find
root mean square velocity.

Answer

To find the root mean square (RMS) velocity of the three molecules with velocities \( V_1 = 2 \, \text{km/s} \), \( V_2 = 4 \, \text{km/s} \), and \( V_3 = 6 \, \text{km/s} \), we can follow these steps:

### Step 1: Square the velocities
First, we need to square each of the velocities:
\[ V_1^2 = (2 \, \text{km/s})^2 = 4 \, \text{km}^2/\text{s}^2 \]
\[ V_2^2 = (4 \, \text{km/s})^2 = 16 \, \text{km}^2/\text{s}^2 \]
\[ V_3^2 = (6 \, \text{km/s})^2 = 36 \, \text{km}^2/\text{s}^2 \]

### Step 2: Calculate the mean of the squared velocities
Next, we find the mean of these squared values. Since we have three velocities, we will divide the sum of the squared velocities by 3:
\[ \text{Mean of squared velocities} = \frac{V_1^2 + V_2^2 + V_3^2}{3} = \frac{4 + 16 + 36}{3} = \frac{56}{3} \, \text{km}^2/\text{s}^2 \]

### Step 3: Take the square root of the mean
Finally, we take the square root of the mean of the squared velocities to find the RMS velocity:
\[ V_{\text{rms}} = \sqrt{\frac{56}{3}} \, \text{km/s} \]
Calculating this gives:
\[ V_{\text{rms}} \approx \sqrt{18.67} \approx 4.32 \, \text{km/s} \]

### Final Answer
Thus, the root mean square velocity \( V_{\text{rms}} \) is approximately:
\[ V_{\text{rms}} \approx 4.32 \, \text{km/s} \]

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