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A particle covers half of its total distance with speed $v_1$ and the rest half distance with speed $v_2$. Its average speed during the complete journey is
A
$\frac{v_1^2v_2^2}{v_1^2 + v_2^2}$
B
$\frac{v_1 + v_2}{2}$
C
$\frac{v_1v_2}{v_1 + v_2}$
D
$\frac{2v_1v_2}{v_1 + v_2}$
Detailed Solution
Let the total distance be 2S, so each half is S.
Time for the first half: $t_1 = \frac{S}{v_1}$; time for the second half: $t_2 = \frac{S}{v_2}$
Average speed = $\frac{\text{total distance}}{\text{total time}} = \frac{2S}{\frac{S}{v_1} + \frac{S}{v_2}}$
$= \frac{2S}{S\left(\frac{v_1 + v_2}{v_1v_2}\right)}$
$= \frac{2v_1v_2}{v_1 + v_2}$
Time for the first half: $t_1 = \frac{S}{v_1}$; time for the second half: $t_2 = \frac{S}{v_2}$
Average speed = $\frac{\text{total distance}}{\text{total time}} = \frac{2S}{\frac{S}{v_1} + \frac{S}{v_2}}$
$= \frac{2S}{S\left(\frac{v_1 + v_2}{v_1v_2}\right)}$
$= \frac{2v_1v_2}{v_1 + v_2}$