A body covers the first one-third distance with speed v1- the second one-third distance with speed v2- and the last one-third distance with speed v3- The average speed of the body is

Step 1:Given data: Speed for the first one third distance= v1Speed for the second one third distance= v2Speed for the last one third distance= v3Let the total d ...

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Standard XII

Physics

Finding Average Velocity

A body covers...

Question

A body covers the first one-third distance with speed v₁, the second one-third distance with speed v₂, and the last one-third distance with speed v₃. The average speed of the body is

$\frac{v_{1} + v_{2} + v_{3}}{3}$

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$\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$

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$\frac{1}{v_{1}} + \frac{1}{v_{2}} + \frac{1}{v_{3}}$

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$\frac{3}{v_{1}} + \frac{2}{v_{2}} + \frac{1}{v_{3}}$

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Solution

The correct option is B
$\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$

Step 1:Given data:
Speed for the first one-third distance= v₁
Speed for the second one-third distance= v₂
Speed for the last one-third distance= v₃
Let the total distance be = 3d
Step 2:Finding time:
We know $S p e e d = \frac{d i s \tan c e}{t i m e}$
Time taken for the first one-third distance = $\frac{d}{v_{1}}$
Time taken for the second one-third distance = $\frac{d}{v_{2}}$
Time taken for the last one-third distance = $\frac{d}{v_{3}}$
Total time taken for the whole journey = $\frac{d}{v_{1}} + \frac{d}{v_{2}} + \frac{d}{v_{3}} = \frac{d (v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1})}{v_{1} v_{2} v_{3}}$
Step 3:Finding the average speed:
Average speed = $\frac{T o t a l d i s \tan c e}{T o t a l t i m e}$
$= \frac{3 d}{\frac{d (v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1})}{v_{1} v_{2} v_{3}}} = \frac{3 v_{1} v_{2} v_{3}}{(v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1})}$
Thus, the average speed of the body is $\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$ .
Hence, option B is correct.

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