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A car goes from $A$ to $B$ (without changing direction), during first part of journey its average speed is $V_1$ and for second part of journey average speed is $V_2$ . The ratio path length of second part to the path length of first part is $\sqrt {\frac{{{V_2}}}{{{V_1}}}} $ . In that case the average speed of total path is ....... of average speeds of both parts. Choose correct option of fill.
Airthmatic mean
Geometric mean
Harmonic mean
Information is insufficient to answer
Solution
$\frac{S_{2}}{S_{1}}=\sqrt{\frac{V_{2}}{V_{1}}}, \quad V_{avg}=\frac{S_{1}+S_{2}}{t_{1}+t_{2}}=\frac{S_{1}+S_{2}}{\frac{S_{1}}{V_{1}}+\frac{S_{2}}{V_{2}}}$
$=\frac{1+\frac{S_{2}}{S_{1}}}{\frac{1}{V_{1}}+\frac{1}{V_{2}} \times \frac{S_{2}}{S_{1}}}=\frac{1+\sqrt{\frac{V_{2}}{V_{1}}}}{\frac{1}{V_{1}}+\frac{1}{V_{2}} \sqrt{\frac{V_{2}}{V_{1}}}}=\sqrt{V_{2} V_{1}}$
