The velocities of ten molecules of any gas are given `v,0,2v,4v,3v,2v,v,3v,5v,v`. Calculate their root mean square velocity.

Answer

To calculate the root mean square (RMS) velocity of the given molecules, we will follow these steps:

### Step 1: List the velocities
The velocities of the ten molecules are:
- \( v_1 = v \)
- \( v_2 = 0 \)
- \( v_3 = 2v \)
- \( v_4 = 4v \)
- \( v_5 = 3v \)
- \( v_6 = 2v \)
- \( v_7 = v \)
- \( v_8 = 3v \)
- \( v_9 = 5v \)
- \( v_{10} = v \)

### Step 2: Square each velocity
Next, we will square each of these velocities:
- \( v_1^2 = v^2 \)
- \( v_2^2 = 0^2 = 0 \)
- \( v_3^2 = (2v)^2 = 4v^2 \)
- \( v_4^2 = (4v)^2 = 16v^2 \)
- \( v_5^2 = (3v)^2 = 9v^2 \)
- \( v_6^2 = (2v)^2 = 4v^2 \)
- \( v_7^2 = v^2 \)
- \( v_8^2 = (3v)^2 = 9v^2 \)
- \( v_9^2 = (5v)^2 = 25v^2 \)
- \( v_{10}^2 = v^2 \)

### Step 3: Sum the squared velocities
Now, we will sum all the squared velocities:
\[ \text{Sum} = v^2 + 0 + 4v^2 + 16v^2 + 9v^2 + 4v^2 + v^2 + 9v^2 + 25v^2 + v^2 \]
Calculating this gives:
\[ \text{Sum} = (1 + 0 + 4 + 16 + 9 + 4 + 1 + 9 + 25 + 1)v^2 = 70v^2 \]

### Step 4: Divide by the number of molecules
Since there are 10 molecules, we will divide the sum by 10:
\[ \text{Mean} = \frac{70v^2}{10} = 7v^2 \]

### Step 5: Take the square root
Finally, we take the square root of the mean to find the root mean square velocity:
\[ V_{\text{rms}} = \sqrt{7v^2} = \sqrt{7}v \]

### Final Answer
The root mean square velocity of the gas molecules is:
\[ V_{\text{rms}} = \sqrt{7}v \text{ m/s} \]

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