Question

Two trains cross each other in \(15\) seconds when they are approaching each other from opposite directions whereas they cross each other in \(45\) seconds when they are moving in same direction. Calculate their possible speeds.

• A. \(20,\ 45\ \text{m/sec}\)
• B. \(15,\ 45\ \text{m/sec}\)
• C. \(30,\ 60\ \text{m/sec}\)
• D. \(30,\ 50\ \text{m/sec}\)

Hint:

For trains, use relative speed. Opposite direction gives \(u+v\), and same direction gives \(u-v\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to identify the individual speeds of two trains given the time taken to cross each other in opposite and same-direction motions.

Step 2: Key Formula or Approach:
Let the speeds of the trains be $v_1$ and $v_2$ (where $v_1 > v_2$), and their combined length be $D$.
• When moving in opposite directions, the relative speed is $(v_1 + v_2)$: \[ D = (v_1 + v_2) \times 15 \]
• When moving in the same direction, the relative speed is $(v_1 - v_2)$: \[ D = (v_1 - v_2) \times 45 \]

Step 3: Detailed Explanation:

• Since the distance covered in both cases is the sum of the lengths of the two trains ($D$), we can equate both equations: \[ 15(v_1 + v_2) = 45(v_1 - v_2) \]
• Simplify the expression: \[ v_1 + v_2 = 3(v_1 - v_2) \] \[ v_1 + v_2 = 3v_1 - 3v_2 \] \[ 4v_2 = 2v_1 \implies v_1 = 2v_2 \]
• The speed of one train must be exactly twice the speed of the other train. Let us test the given options for this 2:1 ratio:
• Option (A): Speeds 20 and 45 (ratio = 2.25)
• Option (B): Speeds 15 and 45 (ratio = 3.0)
• Option (C): Speeds 30 and 60 (ratio = 2.0)
• Option (D): Speeds 30 and 50 (ratio = 1.67)
• Only the speeds in Option (C) satisfy the derived mathematical ratio.

Step 4: Final Answer:
The possible speeds of the trains are 30 m/s and 60 m/s, matching Option (C).