First slide

Average Speed

Question

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

Moderate

A

$\frac{v_{1} + v_{2}}{2}$
B

$\frac{v_{1} v_{2}}{v_{1} + v_{2}}$
C

$\frac{2 v_{1} v_{2}}{v_{1} + v_{2}}$
D

$\frac{v_{1}^{2} v_{2}^{2}}{v_{1}^{2} + v_{2}^{2}}$

Solution

Let S be the total distance travelled by the particle. Let $t_{1}$ be the time taken by the particle to cover first half of the distance. Then $t_{1} = \frac{S / 2}{v_{1}} = \frac{S}{2 v_{1}}$ Let $t_{2}$ be the time taken by the particle to cover remaining half of the distance. Then $t_{2} = \frac{S / 2}{v_{2}} = \frac{S}{2 v_{2}}$
Average speed,
$v_{av} = \frac{Total distance travelled}{Total time taken}$
$= \frac{S}{t_{1} + t_{2}} = \frac{S}{\frac{S}{2 v_{1}} + \frac{S}{2 v_{2}}} = \frac{2 v_{1} v_{2}}{v_{1} + v_{2}}$

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