If three molecules have velocities `1kms^(-1),1kms^(-1) and 1kms^(-1)`, the ratio of the rms speed and average speed is:-

Options

A. 1
B. `1.14`
C. `1.41`
D. `1.2`

Explanation

To find the ratio of the root mean square (RMS) speed and the average speed for three molecules with the same velocity of \(1 \, \text{km/s}\), we can follow these steps:

### Step 1: Calculate the Root Mean Square (RMS) Speed

The formula for the root mean square speed \(v_{\text{rms}}\) is given by:

\[ v_{\text{rms}} = \sqrt{\frac{V_1^2 + V_2^2 + V_3^2}{N}} \]

where \(V_1\), \(V_2\), and \(V_3\) are the velocities of the molecules, and \(N\) is the number of molecules.

Given:
- \(V_1 = 1 \, \text{km/s}\)
- \(V_2 = 1 \, \text{km/s}\)
- \(V_3 = 1 \, \text{km/s}\)
- \(N = 3\)

Substituting the values, we have:

\[ v_{\text{rms}} = \sqrt{\frac{(1)^2 + (1)^2 + (1)^2}{3}} = \sqrt{\frac{1 + 1 + 1}{3}} = \sqrt{\frac{3}{3}} = \sqrt{1} = 1 \, \text{km/s} \]

### Step 2: Calculate the Average Speed

The formula for the average speed \(v_{\text{avg}}\) is given by:

\[ v_{\text{avg}} = \frac{V_1 + V_2 + V_3}{N} \]

Substituting the values, we have:

\[ v_{\text{avg}} = \frac{1 + 1 + 1}{3} = \frac{3}{3} = 1 \, \text{km/s} \]

### Step 3: Calculate the Ratio of RMS Speed to Average Speed

Now, we can find the ratio of the RMS speed to the average speed:

\[ \text{Ratio} = \frac{v_{\text{rms}}}{v_{\text{avg}}} = \frac{1 \, \text{km/s}}{1 \, \text{km/s}} = 1 \]

### Final Answer

The ratio of the root mean square speed to the average speed is \(1\).

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