Question

A passenger train of length $60\,{\text{m}}$ travels at a speed of $80\,{\text{km/hr}}$ another freight train of length $120\,{\text{m}}$ travels at a speed of $30\,{\text{km/hr}}$. The ratio of times taken by the passenger train to completely cross the freight train when: (1) they are moving in the same direction, and (ii) in the opposite directions is…….?
A. $\dfrac{5}{2}$
B. $\dfrac{{25}}{{11}}$
C. $\dfrac{3}{2}$
D. $\dfrac{{11}}{5}$

Answer 1

Use the formula for speed of an object. This formula gives the relation between the distance travelled by the object and the time required to travel this distance. First determine the relative speed of the passenger train with respect to freight train when they are moving in the same and opposite direction. Substitute these values in the formula for speed and calculate the ratio of times.

Formula used:
The speed $v$ of an object is given by
$v = \dfrac{d}{t}$ …… (1)
Here, $d$ is the distance travelled by the object and $t$ is the time required to travel the same distance.

Complete step by step answer:
We have given the length of the passenger train is $60\,{\text{m}}$ and the speed of the passenger train is $80\,{\text{km/hr}}$.
${L_P} = 60\,{\text{m}}$
${v_P} = 80\,{\text{km/hr}}$
Also, we have given that the length of the freight train is $120\,{\text{m}}$ and the speed of the freight train is$30\,{\text{km/hr}}$.
${L_F} = 120\,{\text{m}}$
${v_F} = 30\,{\text{km/hr}}$

(i) Let us first determine the time required for the train to travel the complete length of the freight train in the same direction as that of the freight train. The relative speed ${v_S}$ of the passenger train with respect to the freight train when they are moving in the same direction is given by
${v_S} = {v_P} - {v_F}$
Substitute $80\,{\text{km/hr}}$ for ${v_P}$ and $30\,{\text{km/hr}}$ for ${v_F}$ in the above equation.
${v_S} = \left( {80\,{\text{km/hr}}} \right) - \left( {30\,{\text{km/hr}}} \right)$
$\Rightarrow {v_S} = 50\,{\text{km/hr}}$
Hence, the relative speed of the passenger train with respect to the freight train when they are both moving in the same direction is $50\,{\text{km/hr}}$.

The distance travelled by the passenger train is equal to the length of the passenger train and length of the freight train.
$d = {L_P} + {L_F}$
Rewrite equation (1) for the ${v_S}$.
${v_S} = \dfrac{{{L_P} + {L_F}}}{{{t_1}}}$
Here, ${t_1}$ is the time to cross the freight train in the same direction.
Rearrange the above equation for ${t_1}$.
${t_1} = \dfrac{{{L_P} + {L_F}}}{{{v_S}}}$

(ii) Let us now determine the time required for the train to travel the complete length of the freight train in the opposite direction as that of the freight train. The relative speed ${v_O}$ of the passenger train with respect to the freight train when they are moving in the opposite direction is given by
${v_O} = {v_P} + {v_F}$
Substitute $80\,{\text{km/hr}}$ for ${v_P}$ and $30\,{\text{km/hr}}$ for ${v_F}$ in the above equation.
${v_O} = \left( {80\,{\text{km/hr}}} \right) + \left( {30\,{\text{km/hr}}} \right)$
$\Rightarrow {v_O} = 110\,{\text{km/hr}}$
Hence, the relative speed of the passenger train with respect to the freight train when they are moving in the opposite direction is $110\,{\text{km/hr}}$.

The distance travelled by the passenger train is equal to the length of the passenger train and length of the freight train.
$d = {L_P} + {L_F}$
Rewrite equation (1) for the ${v_S}$.
${v_O} = \dfrac{{{L_P} + {L_F}}}{{{t_2}}}$
Here, ${t_2}$ is the time to cross the freight train in the opposite direction.
Rearrange the above equation for ${t_2}$.
${t_2} = \dfrac{{{L_P} + {L_F}}}{{{v_O}}}$

Let us determine the ratio $\dfrac{{{t_1}}}{{{t_2}}}$.
$\dfrac{{{t_1}}}{{{t_2}}} = \dfrac{{\dfrac{{{L_P} + {L_F}}}{{{v_S}}}}}{{\dfrac{{{L_P} + {L_F}}}{{{v_O}}}}}$
$\Rightarrow \dfrac{{{t_1}}}{{{t_2}}} = \dfrac{{{v_O}}}{{{v_S}}}$
Substitute $50\,{\text{km/hr}}$ for ${v_S}$ and $110\,{\text{km/hr}}$ for ${v_O}$ in the above equation.
$\Rightarrow \dfrac{{{t_1}}}{{{t_2}}} = \dfrac{{110\,{\text{km/hr}}}}{{50\,{\text{km/hr}}}}$
$\therefore \dfrac{{{t_1}}}{{{t_2}}} = \dfrac{{11}}{5}$
Therefore, the ratio of the times is $\dfrac{{11}}{5}$.

Hence, the correct option is D.

Note: The students should be careful while determining the relative velocities of the passenger train with respect to the freight train when the passenger train and freight train are moving in the same direction and opposite direction. If these velocities are determined correctly then the final ratio of the times will also be incorrect.