Solving Train Crossing Problem: Finding Second Train Length

This problem involves calculating the length of a train based on its speed, the length and speed of another train it crosses, and the time taken for the crossing. We'll use the concept of relative speed and the relationship between distance, speed, and time.

Given Information

• Length of the first train (\(L_1\)): 250 m
• Speed of the first train (\(S_1\)): 100 km/h
• Speed of the second train (\(S_2\)): 62 km/h
• Direction: Opposite
• Time to cross (\(T\)): 10 seconds
• Unknown: Length of the second train (\(L_2\))

Understanding Relative Speed

When two trains move in opposite directions, their speeds add up to determine how quickly they cover the distance between them. This is called their relative speed (\(S_{rel}\)).

The formula for relative speed in this case is: \(S_{rel} = S_1 + S_2\)

The total distance (\(D\)) covered when one train crosses another is the sum of their lengths:

\(D = L_1 + L_2\)

The fundamental relationship between distance, speed, and time is:

Distance = Speed \(\times\) Time (\(D = S \times T\))

Step-by-Step Calculation

1. Convert Speeds to Meters per Second (m/s)

Since the time is given in seconds and lengths are in meters, we need to convert the speeds from km/h to m/s. We use the conversion factor: 1 km/h = \(\frac{5}{18}\) m/s.

Speed of the first train (\(S_1\)): \(S_1 = 100 \text{ km/h} = 100 \times \frac{5}{18} \text{ m/s} = \frac{500}{18} \text{ m/s}\)

Speed of the second train (\(S_2\)): \(S_2 = 62 \text{ km/h} = 62 \times \frac{5}{18} \text{ m/s} = \frac{310}{18} \text{ m/s}\)

2. Calculate Relative Speed (\(S_{rel}\))

Add the speeds of the two trains:

\(S_{rel} = S_1 + S_2 = \left( \frac{500}{18} + \frac{310}{18} \right) \text{ m/s}\) \(S_{rel} = \frac{500 + 310}{18} \text{ m/s} = \frac{810}{18} \text{ m/s}\)

Simplifying the fraction:

\(S_{rel} = 45 \text{ m/s}\)

3. Calculate the Total Distance Covered (\(D\))

Use the formula \(D = S_{rel} \times T\). We know \(S_{rel} = 45\) m/s and \(T = 10\) seconds.

\(D = 45 \text{ m/s} \times 10 \text{ s} = 450 \text{ m}\)

This total distance is the sum of the lengths of both trains.

4. Calculate the Length of the Second Train (\(L_2\))

We know that \(D = L_1 + L_2\). We have \(D = 450\) m and \(L_1 = 250\) m.

\(450 \text{ m} = 250 \text{ m} + L_2\)

Rearrange the formula to solve for \(L_2\):

\(L_2 = 450 \text{ m} - 250 \text{ m}\) \(L_2 = 200 \text{ m}\)

Conclusion

The length of the second train is 200 meters.