Question

A bus covers the first half of a certain distance woth speed v1 and the second half with a speed v2.The average speed during the while journey is

Answer 1

☺️☺️☺️
_________________________________________

Given,
Bus covers the first half of a certain distance woth speed v1 and the second half with a speed v2.

Let the total distance covered by the bus be 'x'.

$\underline{\large{\textbf{Step-1}}}$
Find the Time taken by the bus to cover first half of total distance 'x'

Distance covered by the bus = $\frac{x}{2}$

Speed of the bus = $V_{1}$

We have,
$time\: taken= \dfrac{Distance\: Covered}{Speed}$

So, the time taken to cover first half of distance 'x' is given by

$t_{1} = \dfrac{(\frac{x}{2})}{V_{1}}$

$\implies t_{1} = \dfrac{x}{2V_{1}}$

$\underline{\large{\textbf{Step-1}}}$
Find the Time taken by the bus to cover second half of total distance 'x'

Distance covered by the bus = $\frac{x}{2}$

Speed of the bus = $V_{2}$

We have,
$time \: taken= \dfrac{Distance\: Covered}{Speed}$

So, the time taken to cover second half of distance 'x' is given by

$t_{1} = \dfrac{(\frac{x}{2})}{V_{2}}$

$\implies t_{1} = \dfrac{x}{2V_{2}}$

$\underline{\large{\textbf{Step-3}}}$
Find the Average speed of the bus during the whole Journey

We have,
$Average\: speed = \dfrac{Total\: distance\: covered}{Total\: time\: taken}$

$Average\: speed = \dfrac{x}{t_{1}+t_{2}}$

Substituting $t_{1}$ and $t_{2}$

$\implies Average\: speed = \dfrac{x}{(\frac{x}{2V_{1}})+(\frac{x}{2V_{2}})}$

$\implies Average\: Speed = \dfrac{x}{(\frac{xV_{1}+xV_{2}}{2V_{1}V_{2}})}}$

$\implies Average\: speed = \dfrac{x}{(\frac{x(V_{1}+V{2})}{2V_{1}V_{2}})}$

$\implies Average\: speed = \dfrac{1}{(\frac{V_{1}+V{2}}{2V_{1}V_{2}})}$

$\implies Average\: speed = \dfrac{2V_{1}V_{2}}{V_{1}+V{2}}$

Therefore,

Average speed of the bus during the whole Journey is $\dfrac{2V_{1}V_{2}}{V_{1}+V{2}}$

_________________________________________