Question

A train X starts from Meerut at 4 PM and reaches Delhi at 5 PM. While another train Y starts from Delhi at 4 PM and reaches Meerut at 5:30 PM. The two trains will cross each other at?
A. 4 : 36 PM
B. 4 : 42 PM
C. 4 : 48 PM
D. 4 : 50 PM

Answer 1

Hint: In this question we have to find the time at which the two trains cross each other and for that we must first assume that the distance between Delhi and Meerut is \[x\] and we have to calculate the speed of both the trains as time of journey for both is given, after that we will find the relative velocity, \[{{v}_{net}}={{v}_{A}}-{{v}_{B}}\], of both the trains and calculate the time of completing the journey by one train possessing this speed while other is at rest, the time corresponds to the time of crossing of trains by each other.

Complete step-by-step answer:
In this question we have to find the time at which the two trains cross each other and we are given the time for completion of journey by both the trains and the distance between the two stations is the same for each train by default.
First we need to find the speed of each train and for that let us assume that the distance between both the stations is \[x\].
When acceleration is zero, we can find the speed of any object by using the formula, \[speed=\dfrac{distance}{time}\]
Now the speed of train X is given by, as the journey is completed in 1 hour so we will write the time as 1 hour and the distance is assumed by us,
\[spee{{d}_{X}}=\dfrac{x}{1}\]
Now the speed of train Y is given by, as the journey is completed in \[1.5\] hours so we will write the time as \[1.5\] hours and the distance is assumed by us,
\[spee{{d}_{Y}}=\dfrac{x}{1.5}=\dfrac{2x}{3}\]
Now we need to calculate the relative speed of the trains with respect to each other, we are doing this because this concept enables us to think about the problem as only one train is moving with the net speed of both the trains and the other one is at rest.
By calculating the relative speed we can think the problem as to calculate the time taken by any one train to complete the journey with the net speed while other train is kept standing at it's original place throughout the whole journey of the other train and this time will be same as the time of both the trains crossing each other. This is the concept that we will apply here.
So the formula of calculating the relative speed or the net speed is,
\[{{v}_{net}}={{v}_{A}}-{{v}_{B}}\]
In the formula \[{{v}_{net}}\] is the net speed of one train experienced by another train when both the trains are moving with their respective speeds.
Now applying the formula to our situation, we get,
\[{{v}_{net}}=spee{{d}_{X}}-spee{{d}_{Y}}\]
As both the trains are moving in opposite directions to each other or towards each other then the formula changes to,
\[{{v}_{net}}=spee{{d}_{X}}+spee{{d}_{Y}}\]
It is so because when one object is moving towards second object and after some time the second object starts moving towards object one while it is still moving then the distance covered per unit time or speed by object one will become more as compared to before situation and same phenomena will be experienced by object second, as two objects move towards each other then, their speeds are summed up.
Now putting the known values in this equation, we get,
\[{{v}_{net}}=\dfrac{x}{1}+\dfrac{2x}{3}\]
\[{{v}_{net}}=\dfrac{3x+2x}{3}\]
\[{{v}_{net}}=\dfrac{5x}{3}\]
Well it is the speed experienced by each train of itself or the other train moving towards it.
Suppose \[{{v}_{net}}\] is equal to \[spee{{d}_{X}}\], we can also assume it to be \[spee{{d}_{Y}}\] but the answer will be same so no need to solve two cases.
Now using the relation of speed, distance and time, we get,
\[speed=\dfrac{distance}{time}\]
Applying this formula for train X, we get,
\[spee{{d}_{X}}=\dfrac{distance}{tim{{e}_{X}}}\]
Now we are using net speed instead of speed of train X so the time of journey will also change, which is,
\[{{v}_{net,X}}=\dfrac{distance}{tim{{e}_{X}}}\]
Putting the known values, we get,
\[\dfrac{5x}{3}=\dfrac{x}{tim{{e}_{X}}}\]
\[tim{{e}_{X}}=\dfrac{3x}{5x}\]
\[tim{{e}_{X}}=\dfrac{3}{5}\]
\[tim{{e}_{X}}=0.6\] hours
Therefore, after 0.6 hours or after 36 minutes both the trains will cross each other.
So the time at which both the trains will cross each other is 4 : 36 PM.
So, option A is the correct answer.

Note: In the relative speed concept we are making the speed of object one zero by subtracting its speed magnitude from its speed and adding this speed magnitude in the speed of the second object, so that we have to deal with only one object. When two objects are moving in opposite directions then their speeds are summed up but when they are moving in the same direction then their speeds are subtracted from one another to find the net speed experienced by one object. There is an alternate method in which we can think that the whole distance between these two platforms is 1 and from Delhi till the crossing point of both trains the distance is \[x\] so from Meerut the distance of the crossing point will be \[\left( 1-x \right)\], and we know that the time of travelling of both the trains is same as they started at 4 PM from their respective stations so we will equate the time and find the value of \[x\] and \[\left( 1-x \right)\] distance. After finding the distance we will calculate the time consumed by any one train to reach the crossing point. Also put the values of speed in this method same as shown in above method but this time \[x\] will be replaced by 1.