Question

A train has a speed of $60\,km/h$ for the first one hour and $40\,km/h$ for the next half hour. Its average speed in $km/h$ is:
A. $50$
B. $53.33$
C. $48$
D. $70$

Answer 1

Hint:Average speed defines the entire distance covered by that object with respect to the time which is taken to finish the movement. To determine the average speed, we need to use the formula for average speed which is found by calculating the ratio of the total distance covered by a train to the time taken to cover that distance.

Formula used:
$\text{Distance} = \text{Speed} \times \text{Time}$
$\text{Average speed}\,({v_{average}})\, = \,\dfrac{\text{Total distance covered by train (d)}}{\text{Total time taken (t)}}$

Complete step by step solution:
In the question, the speed of the train for the first one hours and for the next half hour are $60\,km/h$ and $40\,km/h$ respectively. Assume the total time taken by the train be $t$ and the speed of the train be $v$, then: For train has a speed of $60\,km/h$ for the first one hour:
${v_1} = 60\,km/h\, \\$
$\Rightarrow {t_1} = 1\,hr \\$
Use the formula of distance to find the distance covered by the train,
$\text{Distance} = \text{Speed} \times \text{Time}$
Put the given information in the distance formula, we obtain
${d_1} = 60 \times 1 \\ $
$\Rightarrow {d_1} = 60\,km \\$
For train has speed of $40\,km/h$ for the next half hour:
${v_2} = 40\,km/h\, \\$
$\Rightarrow {t_2} = \dfrac{1}{2}\,hr \\$
Now, put the values of ${v_2} = 40$ and ${t_2} = \dfrac{1}{2}$ in the distance formula, we obtain,
${d_2} = 40 \times \dfrac{1}{2} \\$
$\Rightarrow {d_2} = 20\,km \\$
Add ${d_1}$ with ${d_2}$ to determine the total distance covered by a train,
$d = {d_1} + {d_2} \\
\Rightarrow d= 60 + 20 \\
\Rightarrow d= 80\,km \\$
Similarly, add ${t_1}$ with ${t_2}$ to determine the total time taken by the train,
$t = {t_1} + {t_2} \\
\Rightarrow t= 1 + \dfrac{1}{2} \\
\Rightarrow t= \dfrac{3}{2}\,hr \\$
Now, apply the formula of average speed to determine the average speed of the train:
$\text{Average speed}\,({v_{average}})\, = \,\dfrac{\text{Total distance covered by train (d)}}{\text{Total time taken (t)}}$
Put the obtained values of $d$ and $t$ in the formula of average speed, then:
$\text{Average speed}\,({v_{average}})\, = \,\dfrac{{80}}{{\dfrac{3}{2}}} \\$
$\Rightarrow {v_{average}} = \dfrac{{160}}{3} \\$
$\therefore {v_{average}} = 53.33\,km/h $
Thus, the average speed of a train in $km/h$ is $53.33\,km/h$.

Hence, the correct option is B.

Note: It should be noted that the average speed is calculated by dividing the total distance something travels over the total amount of time it spends travelling. Direction is not considered in the total distance so all distances travelled should be positive numbers even if they are in different directions.