Question

Two men are running in the same direction with a speed of $6$ km/hr and $7\dfrac{1}{2}$ km/hr. A train running in the same direction crosses them in $5$ sec and $5\dfrac{1}{2}$ sec respectively. The length and the speed of the train are:
A) $22.5$ m and 25 km/hr
B) $23$ m and $25$ km/hr
C) $22.92$ m and $22.5$ km/hr
D) $24$ m and $22.5$ km/hr

Answer 1

Hint: We will first start with determining the relative speed of the train and man. We will assume the general expressions for each term and derive the two different simultaneous equations and solve the equations to calculate each variable. Also, note that the units are different so we need to make the proper conversion of units.

Complete step-by-step answer:
Let us start with simple assumptions.
Let the length of the train be $x$ m.
Also let us assume that length of the train is $y$ km/hr.
Now we know that to convert km/hr into m/s we multiply by $\dfrac{5}{{18}}$ .
As the train is running in the same direction the relative speed of the first man with respect to the train is the difference of the speeds.
Therefore, we will first calculate the relative speed with respect to first man is calculated as below:
$ \Rightarrow $$\left( {x - 6} \right){\text{ km/hr}} = \left( {x - 6} \right) \times \dfrac{5}{{18}}{\text{ m/s}}$
Note that $7\dfrac{1}{2} = 7.5$ .
Similarly, the relative speed with respect to second man is calculated as below:
$ \Rightarrow $$\left( {x - 7.5} \right){\text{ km/hr}} = \left( {x - 7.5} \right) \times \dfrac{5}{{18}}{\text{ m/s}}$
Now we observe that length of the train $ = $ Distance travelled in both cases.
Distance travelled is the product of relative speed and time taken by the train.
Note that we have converted all the quantities in the similar units.
Thus, using the obtained values we can write,
Length of the train $ = \left( {x - 6} \right) \times \dfrac{5}{{18}} \times 5$ .
Distance travelled $ = \left( {x - 7.5} \right) \times \dfrac{5}{{18}} \times 5.5$
But we know that both the quantities are equal.
$ \Rightarrow $$\left( {x - 6} \right) \times \dfrac{5}{{18}} \times 5 = \left( {x - 7.5} \right) \times \dfrac{5}{{18}} \times 5.5$
We can simplify this equation as:
$ \Rightarrow $$5x - 30 = 5.5x - 41.25$
Solving for $x$ we get,
$ \Rightarrow $$x = 22.5{\text{ km/hr}}$
Now calculate the length of the train as follows:
$ \Rightarrow $$\left( {\left( {22.5 - 6} \right) \times \dfrac{5}{{18}} \times 5} \right) = 22.92{\text{ m}}$
Thus, length of the $22.92$ m and speed of the train $22.5$ km/hr.

Thus, option C is the correct option.

Note: Observe that the given quantities were not in similar units, so we first converted them into the same units. If you don’t convert them, it will lead to the wrong conclusion. Also, while calculating the distance we are using the relative speed and not the individual speed of the train or any of the people.