Question:
The speeds of the seven molecules are 1, 3, 5, 7, 2, 4 and 6 km/s respectively. The ratio of their r.m.s. velocity and average velocity will be
The speeds of the seven molecules are 1, 3, 5, 7, 2, 4 and 6 km/s respectively. The ratio of their r.m.s. velocity and average velocity will be
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Compute rms and mean of the given list.
Updated On: Oct 1, 2026
- \(2\sqrt{5}:1\)
- \(\sqrt{5}:1\)
- \(1:2\sqrt{5}\)
- \(\sqrt{5}:2\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
Average speed is the plain mean. The rms speed is the square root of the mean of squares.
Step 2: Key Formula or Approach
Speeds: 1, 2, 3, 4, 5, 6, 7 km/s.
Step 3: Detailed Explanation
Mean: \(\dfrac{1+2+3+4+5+6+7}{7}=\dfrac{28}7=4\).
Mean of squares: \(\dfrac{1+4+9+16+25+36+49}{7}=\dfrac{140}7=20\), so rms \(=\sqrt{20}=2\sqrt5\).
\[ \frac{v_{rms}}{v_{avg}}=\frac{2\sqrt5}{4}=\frac{\sqrt5}{2} \]
Final Answer:
The ratio is \(\sqrt5:2\), option (D). \[ \boxed{\sqrt5:2\ \text{(D)}} \]
Average speed is the plain mean. The rms speed is the square root of the mean of squares.
Step 2: Key Formula or Approach
Speeds: 1, 2, 3, 4, 5, 6, 7 km/s.
Step 3: Detailed Explanation
Mean: \(\dfrac{1+2+3+4+5+6+7}{7}=\dfrac{28}7=4\).
Mean of squares: \(\dfrac{1+4+9+16+25+36+49}{7}=\dfrac{140}7=20\), so rms \(=\sqrt{20}=2\sqrt5\).
\[ \frac{v_{rms}}{v_{avg}}=\frac{2\sqrt5}{4}=\frac{\sqrt5}{2} \]
Final Answer:
The ratio is \(\sqrt5:2\), option (D). \[ \boxed{\sqrt5:2\ \text{(D)}} \]
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