Two trains of length 200 m and 400 m run on parallel lines. When they run in the same direction, it takes 30 seconds for the train with the higher speed to overtake the other train, and when they travel in the opposite directions, it takes them 6 seconds to cross each other. What are the speeds (in km/h) of the two trains?

Correct option is C

Given:

Length of first train, $$L_1$$​= 200 m

Length of second train, $$L_2$$​= 400 m

Time taken to overtake when moving in the same direction, $$t_1 =$$​30 s

Time taken to cross each other when moving in opposite directions, $$t_2 = 6$$​s

Concept Used:

Same direction: Relative speed = $$v_1 - v_2$$​

Opposite directions: Relative speed =$$v_1 + v_2$$​

Solution:

Let the speeds of the two trains be $$v_1$$​ m/s and $$v_2$$ m/s, where $$v_1 > v_2$$​

​$$\text{Relative speed} = v_1 - v_2$$​​

Sum of lengths of both trains =$$L_1 + L_2 = 200$$​ + 400 = 600 m

​$$\text{Time} = \frac{\text{Total distance}}{\text{Relative speed}} \implies 30 = \frac{600}{v_1 - v_2}$$​​

​$$v_1 - v_2 = \frac{600}{30} = 20 \quad \text{(Equation 1)}$$​​

Relative speed =$$v_1 + v_2$$​​

Time = $$\frac{\text{Total distance}}{\text{Relative speed}} \implies 6 = \frac{600}{v_1 + v_2}$$​​

$$v_1 + v_2 = \frac{600}{6} = 100 \quad \text{(Equation 2)}$$​

Adding both equations:

​$$(v_1 - v_2) + (v_1 + v_2) = 20 + 100 \implies 2v_1 = 120 \implies v_1 = 60 \text{ m/s}$$​​

Subtracting Equation 1 from Equation 2: 

​$$(v_1 + v_2) - (v_1 - v_2) = 100 - 20 \implies 2v_2 = 80 \implies v_2 = 40 \text{ m/s}$$​​

​$$v_1 = 60 \text{ m/s} \times \frac{18}{5} = 216 \text{ km/h}$$​​

​$$v_2 = 40 \text{ m/s} \times \frac{18}{5} = 144 \text{ km/h}$$​​