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

Options

A. `sqrt((50)/(3))v`
B. `sqrt((5)/(2))v`
C. `7/2v`
D. `5/2v`

Explanation

To calculate the root mean square (RMS) velocity of three molecules with given velocities, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the velocities of the molecules**:
- The velocities are given as \( v_1 = 3v \), \( v_2 = 4v \), and \( v_3 = 5v \).

2. **Use the formula for root mean square velocity**:
- The formula for the root mean square velocity \( V_{rms} \) for \( n \) molecules is:
\[ V_{rms} = \sqrt{\frac{v_1^2 + v_2^2 + v_3^2}{n}} \]
- Here, \( n = 3 \) since there are three molecules.

3. **Calculate the squares of the velocities**:
- Calculate \( v_1^2 \), \( v_2^2 \), and \( v_3^2 \):
\[ v_1^2 = (3v)^2 = 9v^2 \]
\[ v_2^2 = (4v)^2 = 16v^2 \]
\[ v_3^2 = (5v)^2 = 25v^2 \]

4. **Sum the squares of the velocities**:
- Now, add these squares together:
\[ v_1^2 + v_2^2 + v_3^2 = 9v^2 + 16v^2 + 25v^2 = 50v^2 \]

5. **Substitute into the RMS formula**:
- Substitute the sum of squares into the RMS formula:
\[ V_{rms} = \sqrt{\frac{50v^2}{3}} \]

6. **Simplify the expression**:
- Factor out \( v^2 \) from the square root:
\[ V_{rms} = \sqrt{\frac{50}{3}} \cdot v \]

7. **Final result**:
- Thus, the root mean square velocity is:
\[ V_{rms} = \frac{\sqrt{50}}{\sqrt{3}} \cdot v = \frac{5\sqrt{2}}{\sqrt{3}} \cdot v \]

### Final Answer:
\[ V_{rms} = \frac{5\sqrt{2}}{\sqrt{3}} \cdot v \]

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