A body covers one-third of its journey with speed $v_{0}$ . The remaining portion of the distance was covered with velocity $v_{1}$ for half the remaining time and with velocity $v_{2}$ for the other half of the remaining time. Assuming the body always travels in a straight line, find the average velocity of body over the whole journey.

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Solution

Average speed $= \frac{t o t a l d i s t a n c e c o v e r e d}{t o t a l t i m e t a k e n}$
At the total distance $= 3 x$
Time taken to cover first one third $(x)$
$= t_{1} = \frac{x}{v_{1}}$
Time taken to cover second one third $= t_{2} = \frac{x}{v_{2}}$
Time taken to cover $3^{r d}$ one third $= t_{3} = \frac{x}{v_{3}}$
Average speed $= \frac{3 x}{\frac{x}{v_{1}} + \frac{x}{v_{2}} + \frac{x}{v_{3}}} = \frac{3}{\frac{1}{v_{1}} + \frac{1}{v_{2}} + \frac{1}{v_{3}}}$ .

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Average speed=total distance coveredtotal time takenAt the total distance=3xTime taken to cover first one third(x)=t1=xv1Time taken to cover second one third=t2=xv2Time taken to cover 3rd one third=t3=xv3Average speed=3xxv1+xv2+xv3=31v1+1v2+1v3.