A vehicle travels half the distance $L$ with speed $v_{1}$ and the other half with speed $v_{2}$ . What is the average speed?

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Solution

Let the vehicle travels from $A$ to $B$ . To calculate average speed we will calculate total distance covered and will divide by time interval in which it covers that total distance.
Time taken to travel first half distance $t_{1} = \frac{\frac{L}{2}}{v_{1}} = \frac{L}{2 v_{1}}$
Time taken to travel first half distance $t_{1} = \frac{\frac{L}{2}}{v_{2}} = \frac{L}{2 v_{2}}$
Total time taken $= t_{1} + t_{2}$
$= \frac{L}{2 v_{1}} + \frac{L}{2 v_{2}} = \frac{L}{2} [\frac{1}{v_{1}} + \frac{1}{v_{2}}]$
We know that $v_{a v} =$ Average speed $= \frac{t o t a l d i s t a n c e}{t o t a l t i m e}$
$v_{a v} = \frac{L}{\frac{L}{2} [\frac{1}{v_{1}} + \frac{1}{v_{2}}]} = \frac{2 v_{1} v_{2}}{v_{1} + v_{2}}$

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