Question

A train travels with a speed of 60km from station A to station B and then comes back with a speed of $80km$ from station B to station A. Find the average speed of the train.
A) $70kmph$.
B) $68.57kmph$.
C) $60kmph$.
D) $80kmph$.

Answer 1

Speed is defined as the ratio of distance that an object covers to the time taken by the object to cover that distance. The average speed of anybody is defined as the total distance of the journey to the total time taken in completing the various journeys.

Formula used: The formula of the average speed is given by,
$\Rightarrow {v_{avg}} = \dfrac{{{d_t}}}{{{t_t}}}$
Where average speed is equal to ${v_{avg}}$, the sum of total distances of the all the journey is equal to ${d_t}$ the sum of total time taken is equal to${t_t}$.
The formula of the speed is equal to,
$\Rightarrow v = \dfrac{d}{t}$.
Where the velocity is v the distance of the journey is d and the time taken is t.

Complete step by step solution:
It is given the problem that a train travels with a speed of $60km$ from station A to station B and then comes back with a speed of $80km$ from station B to station A and we need to find the average speed of the train.
The formula of the speed is equal to,
$\Rightarrow v = \dfrac{d}{t}$.
Where the velocity is v the distance of the journey is d and the time taken is t.
So let us calculate the time taken by train for first journey,
The time taken for first journey is ${t_1}$ the distance be d and the velocity be ${v_1}$,
$\Rightarrow {v_1} = \dfrac{d}{{{t_1}}}$
$\Rightarrow {t_1} = \dfrac{d}{{{v_1}}}$
$\Rightarrow {t_1} = \dfrac{d}{{60}}$………eq. (1)
Since for the forward journey the speed is $60kmph$.
The time taken by the train for the second journey.
The formula of the speed is equal to,
$\Rightarrow v = \dfrac{d}{t}$.
Where the velocity is v the distance of the journey is d and the time taken is t.
Let the time taken for the second journey be ${t_2}$ the distance will be the same d and the velocity be ${v_2}$.
$\Rightarrow {v_2} = \dfrac{d}{{{t_2}}}$
$\Rightarrow {t_2} = \dfrac{d}{{{v_2}}}$
$\Rightarrow {t_2} = \dfrac{d}{{80}}$………eq. (2)
Since the velocity for the return trip is $80kmph$.
Let’s calculate the average speed of the journey.
The formula of the average speed is given by,
$\Rightarrow {v_{avg}} = \dfrac{{{d_t}}}{{{t_t}}}$
Where average speed is equal to${v_{avg}}$, the sum of total distances of the all the journey is equal to ${d_t}$ the sum of total time taken is equal to${t_t}$.
The total distance of the journey is equal to,
$\Rightarrow {d_t} = 2d$………eq. (3)
The total time taken for the journey is equal to,
$\Rightarrow {t_t} = {t_1} + {t_2}$
Replacing the value of the time taken from both the journey in the above relation, using equation (1) and equation (2).
$\Rightarrow {t_t} = {t_1} + {t_2}$
$\Rightarrow {t_t} = \dfrac{d}{{60}} + \dfrac{d}{{80}}$
$\Rightarrow {t_t} = \dfrac{{7d}}{{240}}$………eq. (4)
The average speed of the train is equal to,
$\Rightarrow {v_{avg}} = \dfrac{{{d_t}}}{{{t_t}}}$
Replacing the value of the total distance and total time taken from the equation (3) and equation (4) in above relation.
$\Rightarrow {v_{avg}} = \dfrac{{{d_t}}}{{{t_t}}}$
$\Rightarrow {v_{avg}} = \dfrac{{2d}}{{\left( {\dfrac{{7d}}{{240}}} \right)}}$
$\Rightarrow {v_{avg}} = \dfrac{{\left( {240} \right) \times \left( {2d} \right)}}{{\left( {7d} \right)}}$
$\Rightarrow {v_{avg}} = \dfrac{{\left( {240} \right) \times \left( 2 \right)}}{{\left( 7 \right)}}$
$\Rightarrow {v_{avg}} = 68 \cdot 57kmph$
The average speed of the train is equal to${v_{avg}} = 68 \cdot 57kmph$.

The correct option for this problem is option B.

Note: It is advisable for students to understand and remember the formula of the average speed. If an object covers various journeys then the average speed will be the ratio of the total distance that the object covers to the total time taken in the various journeys.