Question

A train $ 210 $ m long takes $ 6 $ seconds to cross a man running at a speed of $ 9 $ Km per hour in the direction opposite to that of a train. What is the speed of the train in Km per hour?
A. $ 127 $
B. $ 121 $
C. $ 117 $
D. $ 108 $

Answer 1

Hint: The velocity term gets added when the two objects move towards each other or move in opposite directions and vice versa.

Complete step-by-step answer:
Given information
Length of the train, $ L = 210 $ m
Speed of the man, $ {v_t} = 9 $ Km per hour
Time taken by the train to cross a man, $ t = 6 $ seconds
The velocity is given in Km per hour and time is in seconds. To convert the time in seconds multiplies the time in seconds by a factor of $ \dfrac{1}{{3600}} $ .
Time in hours is,
 $
  t = \dfrac{6}{{3600}} \\
  t = \dfrac{1}{{600}} \\
  $
To convert the length of train to Kilometer, multiply it by a factor of $ \dfrac{1}{{1000}} $ .
Length of train in kilometer is,
 $
  D = \dfrac{{210}}{{1000}} \\
  D = \dfrac{{21}}{{100}} \\
  $
Let the speed of the train be Km per hour.
Both the train and man are moving in opposite direction. Therefore the relative speed will get increased as the speed gets added up, as shown in equation (1)
 $
  {v_r} = {v_t} - \left( { - {v_m}} \right) \\
  {v_r} = {v_t} + {v_m} \cdots \left( 1 \right) \\
  $
 $ \left( { - {v_m}} \right) $ is written because man is moving opposite to the direction of the train or towards the train.
Substitute the value of $ {v_m} = 9 $ in equation (1),
 $ {v_r} = {v_t} + 9 \cdots \left( 2 \right) $
The train will travel a distance equal to its length to cross a man moving in the opposite direction to it.
Distance travelled by the train is given by,
 $ D = {v_r} \cdot t \cdots \left( 3 \right) $
Substitute the value of $ {v_r} = {v_t} + 9 $ , $ t = \dfrac{1}{{600}} $ and $ D = \dfrac{{21}}{{100}} $ in equation (3),
 $ 210= \left( {{v_t} + 9} \right) \times \dfrac{1}{{600}} \cdots \left( 3 \right) $
Solving equation (3) to calculate the value of ,
 $
  \dfrac{{21}}{{100}} = \left( {{v_t} + 9} \right) \times \dfrac{1}{{600}} \\
  {v_t} = \dfrac{{21 \times 600}}{{100}} - 9 \\
  {v_t} = 117 \\
  $
Therefore, the speed of the train is $ {v_t} = 117 $ Km per hour.
So, the correct answer is “Option C”.

Note: The important which should be kept in mind are
When 2 objects with speed $ {v_1} $ and $ {v_2} $ travel in same direction, the relative speed is given by, $ {v_r} = {v_1} - {v_2} $
When 2 objects with speed $ {v_1} $ and $ {v_2} $ travel in opposite direction, the relative speed is given by,
 $
  {v_r} = {v_1} - \left( { - {v_2}} \right) \\
  {v_r} = {v_1} + {v_2} \\
  $
It is important to convert the units in the same system. If the time is in hour and speed is in meter per seconds, then convert the time into seconds by multiplying it with $ 3600 $ .