Question

A train 100m long travelling at 40m/s starts overtaking another train 200m long travelling at \[30m{{s}^{-1}}\] . The time taken by the first train to pass the second train completely is:
(A). \[30s\]
(B). \[40s\]
(C). \[50s\]
(D). \[60s\]

Answer 1

Hint: As the train has to overtake the other train in speed as well as in length, the relative velocity of the train with respect to the other velocity will be calculated. Using the formula for speed, we calculate the time taken to travel the relative distance. Convert the units as required.

Formula used:
 \[\text{s =}\dfrac{x}{t}\]

Complete step by step solution:
Relative velocity is defined as the velocity of a body with respect to another body at rest or moving with some velocity. Similarly, the relative distance is the distance covered by a body with respect to another body. Relative velocity is given by-
 \[{{v}_{r}}={{v}_{1}}-{{v}_{2}}\]
 \[{{v}_{r}}\] is the relative velocity
 \[{{v}_{1}}\] is the velocity of the object for which relative velocity is to be calculated
 \[{{v}_{2}}\] is the velocity of the object with respect to which the relative velocity of the other object is to be calculated.
A \[100m\] train is trying to overtake a \[200m\] long train. Therefore, the relative distance it has to cover is-
 \[100+200=300m\]
The relative velocity between the two trains will be-
 \[{{v}_{r}}=40-30=10m{{s}^{-1}}\]
Using the formula of speed, we get,
 \[\text{s =}\dfrac{x}{t}\]
Here, \[s\] is the speed
 \[x\] is the distance covered
 \[t\] is time taken
 Substituting the values of relative distance and speed in the above equation, we, get,
 \[\begin{align}
  & {{v}_{r}}=10 \\
 & 10=\dfrac{300}{t} \\
 & \Rightarrow t=30s \\
\end{align}\]
Therefore, the train covers the total distance and overtakes the other train in \[30s\] .

So, the correct answer is “Option A”.

Note: When the bodies are moving in the same direction, the relative velocity of one body with respect to the other decreases, hence the velocities are subtracted. Similarly, when the bodies are moving in opposite directions the relative velocity increases so the velocities get added. The relative velocity in 2 dimensions, represents the velocity of one vector as seen from the frame of the other vector.