Question

A bus traveling along a straight highway covers one-third of the total distance between two places with a velocity $20km{h}^{-1}$. The remaining part of the distance was covered with a velocity of $30km{h}^{-1}$ for the first half of the remaining time and with velocity $50km{h}^{-1}$ for the next half of the time. Find the average velocity of the bus for its whole journey.

Answer 1

Step 1: Given data

A bus covers one-third of the total distance between two places with a velocity $20km{h}^{-1}$.

The remaining part (two-third) of the distance was covered with a velocity of $30km{h}^{-1}$ for the first half of the remaining time.

For the next half of the time, the velocity of the bus is $50km{h}^{-1}$.

Step 2: Find the total time taken by the bus for its whole journey.

Let the total distance be $dkm$.

Let the time taken to cover one-third of the distance with velocity $20km{h}^{-1}$ be $t_1$.

So, $t_1=\frac{displacement}{velocity}=\frac{\left({\displaystyle \frac{d}{3}}\right)}{20}$

$\Rightarrow t_1=\frac{d}{60}h$

Since the remaining part of the distance was covered with a velocity of $30km{h}^{-1}$ for the first half of the remaining time and with velocity $50km{h}^{-1}$ for the next half of the time.

Let the time taken to cover the remaining distance be $2t$.

It means bus covered the remaining distance with a velocity of $30km{h}^{-1}$ for the first $thr$ and it covered the remaining distance with a velocity of $50km{h}^{-1}$ for the next $thr$.

Therefore we can write,

$\Rightarrow 30t+50t=d-\frac{d}{3}$

$\Rightarrow 80t=\frac{2d}{3}$

$\Rightarrow t=\frac{d}{120}$

Hence the total time taken by the bus is,

$\Rightarrow T=t_1+2t$

On putting the required data we get,

$\Rightarrow T=\frac{d}{60}+\frac{2d}{120}$

$\Rightarrow T=\frac{2d}{60}=\frac{d}{30}hr$

Step 3: Find the average velocity of the bus for its whole journey.

Since the formula of average speed is,

$Averagevelocity=\frac{Totaldisplacement}{Totaltimetaken}$

As the total time taken $=T$

On applying the calculated values in the above formula we get,

$\Rightarrow Averagevelocity=\frac{d}{T}$

$\Rightarrow Averagevelocity=\frac{d}{\left({\displaystyle \frac{d}{30}}\right)}$

On further solving the equation we get,

$\Rightarrow Averagevelocity=30km{h}^{-1}$

Hence, the average velocity of the bus for its whole journey is $\mathbf{30}\mathbf{}\mathit{k}\mathit{m}{\mathit{h}}^{\mathbf{-}\mathbf{1}}$.