Question

A good train and a passenger train are running on parallel tracks in the same direction. The driver of the goods train observes that the passenger train coming from behind overtakes and crosses his train completely in 60 sec. Whereas a passenger on the passenger train marks that he crosses the goods train in 40 sec. If the speeds of the trains be in the ratio 1:2. Find the ratio of their lengths.

Answer 1

AnswEr :

$\textsf{Let the Speed of Good Train and Passenger} \\ \textsf{Train be \bf{x} \textsf{and, }2x \textsf{Length of Good Train be}} \\ \bf{L_1}\textsf{and, Length of Passenger Train be} \:L_2$

• As Trains are moving in same Direction, therefore relative speed will be (Passenger Train – Good Train) = (2x – x)
• Time by Good Train = 60 sec
• Time by Passenger Train = 40 sec

$\rule{130}{2}$

$\bigstar \: \large\boxed{\sf Time\:Taken=\dfrac{Distance}{Speed}}$

$\textbf{\textdagger}\:\underline{\large{\textit{Case 1 :}}}$

$\textsf{The driver of the goods train observes that the} \\ \textsf{passenger train coming from behind overtakes} \\ \textsf{and crosses his train completely in 60 seconds.}$

$\longrightarrow \sf Time\:Taken = \dfrac{Distance}{Speed}\\\\\\\longrightarrow \sf Time\:Taken = \dfrac{Train_1+Train_2}{Relative\:Speed}\\\\\\\longrightarrow \sf 60 \:sec. = \dfrac{L_1+L_2}{2x - x} \qquad \quad\dfrac{\qquad}{}eq.(1)$

$\rule{200}{1}$

$\textbf{\textdagger}\:\underline{\large{\textit{Case 2 :}}}$

$\textsf{Passenger on the passenger train marks} \\ \textsf{that he crosses the goods train in 40 seconds.}$

$\longrightarrow \sf Time\:Taken = \dfrac{Distance}{Speed}\\\\\\\longrightarrow \sf Time\:Taken = \dfrac{Train_1}{Relative\:Speed}\\\\\\\longrightarrow \sf 40 \:sec. = \dfrac{L_1}{2x - x} \qquad \quad\dfrac{\qquad}{}eq.(2)$

$\rule{200}{2}$

$\bigstar \:\underline{\textsf{On dividing eq. [1] by eq. [2], we have :}}$

$:\implies \sf \cancel\dfrac{60\:sec.}{40\:sec.}=\dfrac{\dfrac{L_1+L_2}{ \cancel{2x-x}}}{\dfrac{L_1}{\cancel{2x-x}}}\\\\\\:\implies \sf \dfrac{3}{2}=\dfrac{L_1+L_2}{L_1} \\\\\\:\implies \sf 3 \times L_1=2 \times (L_1+L_2)\\\\\\:\implies \sf 3L_1=2L_1+2L_2 \\\\\\:\implies \sf 3L_1-2L_1 = 2L_2 \\\\\\:\implies \sf L_1=2L_2 \\\\\\:\implies \sf \dfrac{L_1}{L_2} = \dfrac{2}{1} \\\\\\:\implies\boxed{\pink{\sf L_1 :L_2 =2 :1}}$

⠀

∴ Hence, Ratio of their lengths will be 2 : 1