Question

If two trains 120m and 80m in length are running in opposite directions with velocities $42kmh{{r}^{-1}}$ and $30kmh{{r}^{-1}}$ . In what time, will they completely cross each other?

Answer 1

We will calculate the time taken to completely cross each other by using relative speed of one train with respect to the other. We will subtract the speed of one train with another to calculate the relative speed of one train with respect to another. We will do the same for relative distance to be travelled by one train with respect to the other.

Complete answer:
Let us first assign some terms that we are going to use later.
Let the first train be train A and the second train be train B.
The speed of train A is given to be :
$\Rightarrow {{V}_{A}}=42kmh{{r}^{-1}}$
And, the speed of the train B is given to be:
$\Rightarrow {{V}_{B}}=30kmh{{r}^{-1}}$
Now, let us calculate the relative speed of one with respect to the other. We will calculate the speed of train B with respect to train A (say ${{V}_{BA}}$ ). Since, the trains are running opposite their velocities will add up.
$\begin{aligned} & \Rightarrow {{V}_{BA}}=(42+30)kmh{{r}^{-1}} \\\ & \Rightarrow {{V}_{BA}}=72kmh{{r}^{-1}} \\\ \end{aligned}$
Which is also equal to:
$\begin{aligned} & \Rightarrow {{V}_{BA}}=\dfrac{72\times 1000}{3600}m{{s}^{-1}} \\\ & \Rightarrow {{V}_{BA}}=20m{{s}^{-1}} \\\ \end{aligned}$
Now, we will calculate the total relative distance (say d) that has to be travelled by the train:
$\begin{aligned} & \Rightarrow d=(120+80)m \\\ & \Rightarrow d=200m \\\ \end{aligned}$
Therefore, the time taken by the trains to cross each other is given by:
$\Rightarrow t=\dfrac{200}{20}s$
$\therefore t=10s$
Hence, the time taken by the trains to cross each other is calculated to be 10 seconds.

Note:
We could have calculated the time taken by each train to cross each other but it would have taken up a lot more time and calculation. So, we preferred the use of relative velocity and relative distance that one train has to travel with respect to the other. This not only saved us some time but also reduced our calculations and made the solution easier.