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Speed vs Velocity

A man travell...

Question

A man travelled half the distance with a velocity $v_{0}$ . The remaining part of the distance was covered with velocity $v_{1}$ for half the time and with velocity $v_{2}$ for the other half of the time as shown in the Fig. Find the average speed of the man over the whole time of motion.

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Solution

Total distance travelled $= s$
to travel half distance, time taken $= t_{1}$
second half $= t_{2}$
distance travelled in first half $= s / 2$
$t_{1} = \frac{(S / 2)}{V_{0}} = \frac{S}{2 V_{0}}$
time taken to cover second half is $t_{2}$
$∴$ $\frac{S}{2} = \frac{t_{2} V_{1}}{2} + \frac{t_{2} V_{2}}{2}$
$S = t_{2} (V_{1} + V_{2})$
$t_{2} = (\frac{S}{V_{1} + V_{2}})$
total time taken $= t_{1} + t_{2}$
$= \frac{S}{{2 V}_{0}} + \frac{S}{(V_{1} + V_{2})}$
average speed $= \frac{t o t a l d i s t a n c e}{t o t a l t i m e}$
$= \frac{S}{\frac{S}{2 V_{0}} + (\frac{S}{V_{1} + V_{2}})}$
average speed $= \frac{{2 V}_{0} (V_{1} + V_{2})}{2 V_{0} + V_{1} + V_{2}}$

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