Train Speeds Calculation: Opposite and Same Direction Crossings

Let the lengths of the two trains be $L_1$ and $L_2$ meters, and their respective speeds be $S_1$ and $S_2$ meters per second (m/s).

When two trains cross each other, the total distance involved is the sum of their lengths ($L_1 + L_2$). The relationship is given by: Distance = Speed $\times$ Time.

Scenario 1: Crossing in Opposite Directions

When the trains move towards each other, their relative speed is the sum of their speeds: $S_{relative} = S_1 + S_2$. The time taken to cross is 15 seconds.

Using the distance formula:

Scenario 2: Crossing in Same Direction

When the trains move in the same direction, their relative speed is the difference between their speeds: $S_{relative} = |S_1 - S_2|$. Assume $S_1 > S_2$, so the relative speed is $S_1 - S_2$. The time taken to cross is 45 seconds.

Using the distance formula:

Deriving the Speed Relationship

Equate the expressions for the total length ($L_1 + L_2$) from Equation 1 and Equation 2:

Divide both sides by 15:

Simplify the equation:

Rearrange the terms to find the relationship between $S_1$ and $S_2$:

This implies that the speed of one train is twice the speed of the other.

Checking the Options

We need to find the pair of speeds from the options where one speed is double the other:

• Option 1: 20 m/s and 45 m/s ($45 \neq 2 \times 20$)
• Option 2: 15 m/s and 45 m/s ($45 \neq 2 \times 15$)
• Option 3: 30 m/s and 60 m/s ($60 = 2 \times 30$). This satisfies the condition $S_1 = 2S_2$.
• Option 4: 30 m/s and 50 m/s ($50 \neq 2 \times 30$)

Conclusion

The pair of speeds that satisfies the condition derived from the crossing times is 30 m/s and 60 m/s.