The velocities of three molecules are 3v, 4v and 5v. Calculate their root mean square velocity.

Answer

To calculate the root mean square (RMS) velocity of the three molecules with velocities \(3v\), \(4v\), and \(5v\), we will follow these steps:

### Step 1: Square the velocities
We start by squaring each of the velocities:
- For the first molecule: \((3v)^2 = 9v^2\)
- For the second molecule: \((4v)^2 = 16v^2\)
- For the third molecule: \((5v)^2 = 25v^2\)

### Step 2: Sum the squared velocities
Next, we sum the squared velocities:
\[ 9v^2 + 16v^2 + 25v^2 = 50v^2 \]

### Step 3: Calculate the mean of the squared velocities
Now, we take the mean of the squared velocities. Since there are three molecules, we divide the sum by 3:
\[ \text{Mean} = \frac{50v^2}{3} \]

### Step 4: Take the square root of the mean
Finally, we take the square root of the mean to find the root mean square velocity:
\[ V_{\text{RMS}} = \sqrt{\frac{50v^2}{3}} = \frac{\sqrt{50}v}{\sqrt{3}} = \frac{5\sqrt{2}v}{\sqrt{3}} \]

### Final Result
Thus, the root mean square velocity of the three molecules is:
\[ V_{\text{RMS}} = \frac{5\sqrt{2}v}{\sqrt{3}} \]

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