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Average Speed

A bus travell...

Question

A bus travelling from city A to city B covers half the distance at a speed u. It covers the remaining distance at a speed v for the half the time and at a speed vprime for the other half of the time. Calculate the average speed of the bus for whole journey.

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Solution

$A v e r a g e s p e e d i s g i v e n b y : a v e r a g e s p e e d = \frac{t o t a l d i s \tan c e t r a v e l l e d}{t o t a l t i m e t a k e n} l e t t o t a l d i s \tan c e t r a v e l l e d = x a n d t o t a l t i m e i s t f o r c a s e 1 : d i s \tan c e t r a v e l l e d = x / 2 s p e e d = u t i m e t_{1} = \frac{x}{2 u} f o r c a s e 2 . s_{2} = v t_{2} / 2 a n d f o r c a s e 3 s_{3} = v' t_{2} / 2 w h e r e t_{1} + t_{2} = t a n d s_{2} + s_{3} = x / 2 \frac{t_{2}}{2} (v + v') = \frac{x}{2} t_{2} = \frac{x}{v + v'} a v e r a g e s p e e d = \frac{x}{t_{1} + t_{2}} = \frac{x}{\frac{x}{2 u} + \frac{x}{v + v'}} = \frac{2 u (v + v')}{v + v' + 2 u}$

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