A flask has 10 molecules out of which four molecules are moving at `7 ms^(-1)` and the remaining ones are moving at same speed of `X ms^(-1)`. If rms of the gas is `5 ms^(-1)`, what is `X` ?

Explanation

To solve the problem, we need to find the value of \( X \) given the root mean square (RMS) speed of a gas with 10 molecules, where 4 molecules are moving at \( 7 \, \text{m/s} \) and the remaining 6 molecules are moving at \( X \, \text{m/s} \).

### Step-by-Step Solution:

1. **Understand the RMS Speed Formula**:
The RMS speed (\( v_{\text{rms}} \)) is given by the formula:
\[ v_{\text{rms}} = \sqrt{\frac{1}{N} \sum_{i=1}^{N} v_i^2} \]
where \( N \) is the total number of molecules, and \( v_i \) is the speed of each molecule.

2. **Identify the Given Values**:
- Total molecules, \( N = 10 \)
- Speed of 4 molecules, \( v_1 = 7 \, \text{m/s} \)
- Speed of remaining 6 molecules, \( v_2 = X \, \text{m/s} \)
- Given RMS speed, \( v_{\text{rms}} = 5 \, \text{m/s} \)

3. **Set Up the Equation**:
Using the RMS speed formula, we can set up the equation:
\[ 5 = \sqrt{\frac{1}{10} \left( 4 \times (7^2) + 6 \times (X^2) \right)} \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ 25 = \frac{1}{10} \left( 4 \times 49 + 6 \times X^2 \right) \]

5. **Multiply by 10**:
Multiply both sides by 10 to simplify:
\[ 250 = 4 \times 49 + 6 \times X^2 \]

6. **Calculate \( 4 \times 49 \)**:
Calculate \( 4 \times 49 \):
\[ 4 \times 49 = 196 \]
So the equation becomes:
\[ 250 = 196 + 6X^2 \]

7. **Rearrange the Equation**:
Rearranging gives:
\[ 250 - 196 = 6X^2 \]
\[ 54 = 6X^2 \]

8. **Solve for \( X^2 \)**:
Divide both sides by 6:
\[ X^2 = \frac{54}{6} = 9 \]

9. **Take the Square Root**:
Taking the square root gives:
\[ X = \sqrt{9} = 3 \, \text{m/s} \]

### Final Answer:
The value of \( X \) is \( 3 \, \text{m/s} \).