Question

Two trains $\text{A}$ and $\text{B}$ have lengths $800\text{m}$ and $1000\text{m}$ respectively and are moving with the velocity of $25\text{m/s}$ and $15\text{m/s}$ respectively. At time $t=0$, engine of train $\text{A}$ is just behind tail of train $\text{B}$. Find the time when train $\text{A}$ completely overtake train $\text{B}?$

• A. $10\text{sec}$
• B. $100\text{sec}$
• C. $18\text{sec}$
• D. $180\text{sec}$

Answer 1

The correct option is D $180\text{sec}$

At $t=0$

Head of train $\text{A}$ is just behind the tail of train $\text{B}$ as shown below.


And at $t=t_o$ $[\text{A}$ overtakes $\text{B}]$

Head of train $\text{B}$ is just behind the tail of train $\text{A}.$


From the two situation we see that in time to relative displacement of train $\text{A}$ w.r.t train $\text{B}$ is

$s_{AB}=800+1000=+1800\text{m}$

Relative velocity of train $\text{A}$ w.r.t train $\text{B}$ is given by

$v_{AB}=v_A-v_B=+25-(+15)$

$\Rightarrow v_{AB}=+10\text{m/s}$

As $v_A$ and $v_B$ are constant, so relative acceleration will be zero.

$\therefore a_{AB}=a_A-a_B=0$

Therefore, time taken to overtake the train $\text{B}$ will be

$t_o=\frac{s_{AB}}{v_{AB}}=\frac{1800}{10}$

$\therefore t_o=180\text{seconds}$

Hence, option $\left(d\right)$ is correct.

$$\begin{array}{c}\text{Why this question?}\\ \text{When relative acceleration is zero then,}\\ \text{Relative velocity}=\frac{Relativedisplacement}{timetaken}\end{array}$$