Train Bridge Calculation: Step-by-Step Solution

This problem involves a train crossing a bridge, requiring us to calculate the length of the bridge. We'll use the fundamental relationship between speed, distance, and time, along with necessary unit conversions to solve this train problem.

Understanding the Problem Elements

Let's list the given information from the question:

• Train Length: \(L_{\text{train}} = 120 \, \text{m}\)
• Train Speed: \(V = 60 \, \text{km/hr}\)
• Time to Cross Bridge: \(T = 24 \, \text{seconds}\)
• Goal: We need to find the length of the bridge, \(L_{\text{bridge}}\), in meters.

Speed Conversion from km/hr to m/s

Before we can use the speed in our calculations, we must ensure all units are consistent. The train's speed is given in kilometers per hour (km/hr), but the time is in seconds, and lengths are in meters. Therefore, we need to convert the speed from km/hr to meters per second (m/s).

The conversion factors are:

• \(1 \, \text{kilometer} = 1000 \, \text{meters}\)
• \(1 \, \text{hour} = 3600 \, \text{seconds}\)

To convert speed from km/hr to m/s, we multiply the speed by \(\frac{1000}{3600}\), which simplifies to \(\frac{5}{18}\).

Let's calculate the train's speed in m/s:

\(V = 60 \, \text{km/hr}\)

\(V = 60 \times \frac{5}{18} \, \text{m/s}\)

\(V = \frac{300}{18} \, \text{m/s}\)

\(V = \frac{50}{3} \, \text{m/s}\)

Distance Covered by the Train

When a train crosses a bridge, the total distance that the train travels is not just the length of the bridge, but also its own length. This is because the entire train must pass over the bridge. So, the total distance covered by the train from the moment its front end touches the bridge until its rear end leaves the bridge is the sum of the train's length and the bridge's length.

Let the length of the bridge be \(L_{\text{bridge}}\).

Total Distance (\(D\)) = Length of Train (\(L_{\text{train}}\)) + Length of Bridge (\(L_{\text{bridge}}\))

Substituting the given train length:

\(D = 120 \, \text{m} + L_{\text{bridge}}\)

Applying the Speed, Distance, Time Formula

Now, we can use the fundamental formula that relates speed, distance, and time:

Distance = Speed \(\times\) Time

\(D = V \times T\)

Substitute the expressions for total distance, the calculated speed, and the given time into this formula:

\(120 + L_{\text{bridge}} = \left(\frac{50}{3} \, \text{m/s}\right) \times (24 \, \text{s})\)

Now, perform the multiplication:

\(120 + L_{\text{bridge}} = 50 \times \frac{24}{3}\)

\(120 + L_{\text{bridge}} = 50 \times 8\)

\(120 + L_{\text{bridge}} = 400\)

Calculating Bridge Length

Finally, to find the length of the bridge, we need to isolate \(L_{\text{bridge}}\) in the equation:

\(L_{\text{bridge}} = 400 - 120\)

\(L_{\text{bridge}} = 280 \, \text{m}\)

Thus, the length of the bridge is 280 meters.