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(B) Let $S_1$ and $S_2$ be the path lengths of the first and second parts,respectively.Given: $\frac{S_2}{S_1} = \sqrt{\frac{V_2}{V_1}}$.The average s

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Geometric mean

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(B) Let $S_1$ and $S_2$ be the path lengths of the first and second parts,respectively.Given: $\frac{S_2}{S_1} = \sqrt{\frac{V_2}{V_1}}$.The average speed $V_{avg}$ is defined as the total distance divided by the total time:$V_{avg} = \frac{S_1 + S_2}{t_1 + t_2} = \frac{S_1 + S_2}{\frac{S_1}{V_1} + \frac{S_2}{V_2}}$.Dividing the numerator and denominator by $S_1$:$V_{avg} = \frac{1 + \frac{S_2}{S_1}}{\frac{1}{V_1} + \frac{1}{V_2} \cdot \frac{S_2}{S_1}}$.Substituting $\frac{S_2}{S_1} = \sqrt{\frac{V_2}{V_1}}$:$V_{avg} = \frac{1 + \sqrt{\frac{V_2}{V_1}}}{\frac{1}{V_1} + \frac{1}{V_2} \sqrt{\frac{V_2}{V_1}}} = \frac{1 + \sqrt{\frac{V_2}{V_1}}}{\frac{1}{V_1} + \frac{1}{\sqrt{V_1 V_2}}} = \frac{1 + \sqrt{\frac{V_2}{V_1}}}{\frac{\sqrt{V_2} + \sqrt{V_1}}{V_1 \sqrt{V_2}}} = \frac{\sqrt{V_1} + \sqrt{V_2}}{\sqrt{V_1}} \cdot \frac{V_1 \sqrt{V_2}}{\sqrt{V_1} + \sqrt{V_2}} = \sqrt{V_1 V_2}$.Thus,the average speed is the geometric mean of $V_1$ and $V_2$.

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