Question

Two trains of equal lengths are running on parallel lines in the same direction at the rate of $ 46km/hr $ and $ 36km/hr $ . The faster train passes the slower train in $ 36\sec $ . What is the length of the trains?
A. $ 50m $
B. $ 72m $
C. $ 80m $
D. $ 82m $

Answer 1

Hint: You can start by defining a vector quantity. Then convert the velocity of the trains which is given in $ km/hr $ to $ m/\sec $ . Now calculate the relative speed of both the trains. Then use the equation $ Velocity = \dfrac{{Displacement}}{{Time}} $ to find out the length of the trains which will be half of the displacement.

Complete step-by-step answer:
A vector is a mathematical quantity that has both a magnitude (size) and a direction. To imagine what a vector is like, imagine asking someone for directions in an unknown area and they tell you, “Go $ 5km $ towards the West”. In this sentence, we see an example of a displacement vector, “\[5km\]” is the magnitude of the displacement vector and “towards the North” is the indicator of the direction of the displacement vector.
A vector quantity is different from a scalar quantity in the fact that a scalar quantity has only magnitude, but a vector quantity possesses both direction and magnitude. Unlike scalar quantities, vector quantities cannot undergo any mathematical operation, instead they undergo Dot product and Cross product.
Some examples of vectors are – Displacement, Force, Acceleration, Velocity, Momentum, etc.
Let the length of each train be $ x $ .
For the faster train to pass the slower train it will have to train a distance of $ 2x $ .
Given
 $ {v_f} = 46km/hr = 46 \times \dfrac{5}{{18}}m/s = $ Velocity of the faster train
 $ {v_s} = 36km/hr = 36 \times \dfrac{5}{{18}}m/s = $ Velocity of the slower train
So the relative velocity of the faster train is
 $ {v_r} = {v_f} - {v_s} $
 $ \Rightarrow {v_r} = 46 \times \dfrac{5}{{18}} - 36 \times \dfrac{5}{{18}} $
 $ \Rightarrow {v_r} = 10 \times \dfrac{5}{{18}} $
 $ \Rightarrow {v_r} = \dfrac{{25}}{9}m/s $
We also know that
 $ Velocity = \dfrac{{Displacement}}{{Time}} $
 $ \Rightarrow \dfrac{{25}}{9} = \dfrac{{Displacement}}{{36}} $
 $ \Rightarrow Displacement = 100m $
Here the displacement is equal to the length of the two trains
 $ \Rightarrow 2x = 100 $
 $ \therefore x = 50m $
Hence, option A is the correct choice

Note – In the solution we used the concept of relative velocity. We used it to determine the velocity of the faster rain with respect to the velocity of the slower train. Amazingly no velocity measured in the universe is absolute, each body that is measured, is measured against the velocity of another body. For example - The velocity of the trains are measured against the velocity of the earth.