To solve this problem, we'll use the information given and apply the concepts of relative speed and the formula for speed, distance, and time.

Step 1: Analyze the scenario where the trains travel in opposite directions.

When two trains are traveling in opposite directions, their relative speed is the sum of their individual speeds.

$\text{Relative speed} = 60 \text{ km/h} + 50 \text{ km/h} = 110 \text{ km/h}$

Convert this speed to meters per second (m/s) since the time is given in seconds:

$110 \text{ km/h} = \frac{110 \times 1000}{3600} \text{ m/s} = \frac{110000}{3600} \text{ m/s} \approx 30.56 \text{ m/s}$

Since they take 5 seconds to cross each other, the combined length of the two trains $L_1 + L_2$ is:

$L_1 + L_2 = \text{Relative speed} \times \text{time} = 30.56 \text{ m/s} \times 5 \text{ s} \approx 152.78 \text{ meters}$

Step 2: Analyze the scenario where the trains travel in the same direction.

When the trains are traveling in the same direction, their relative speed is the difference between their speeds:

$\text{Relative speed} = 60 \text{ km/h} - 50 \text{ km/h} = 10 \text{ km/h}$

Convert this speed to meters per second:

$10 \text{ km/h} = \frac{10 \times 1000}{3600} \text{ m/s} \approx 2.78 \text{ m/s}$

In this case, the length of the slower train $L_2$ will be overtaken by the faster train in 18 seconds. Thus:

$L_2 = \text{Relative speed} \times \text{time} = 2.78 \text{ m/s} \times 18 \text{ s} \approx 50 \text{ meters}$

Step 3: Determine the length of the faster train $L_1$.

Since $L_1 + L_2 = 152.78$ meters and $L_2 \approx 50$ meters:

$L_1 = 152.78 - 50 = 102.78 \text{ meters}$

Conclusion:

The lengths of the two trains are approximately:

$L_1 = 102.78 \text{ meters}, \quad L_2 = 50 \text{ meters}$

Thus, the correct answer is:

(c) 102.78, 50

Would you like more details or have any questions? Here are some related questions to deepen your understanding:

1. How do you convert km/h to m/s?
2. What is relative speed and how is it used in train problems?
3. Why does the relative speed change depending on the direction of travel?
4. Can the same methods be applied to problems involving boats or planes?
5. How would the problem change if the time to overtake in the same direction was different?

Tip: Always ensure units are consistent when solving problems, particularly when time is given in seconds and speed in km/h.