Understanding the Problem: Train Crossing Concepts

This problem involves calculating the time taken for a train to cross a bridge, which requires us to first determine the length of that train. We are given information about two trains, Train X and Train Y, moving in opposite directions and the time it takes for them to cross each other. This initial information helps us find the unknown length of Train Y. Once we have the length of Train Y, we can calculate the time it needs to cross a bridge of a given length using its speed.

Step-by-Step Solution for Train Crossing Time

Let's break down the problem into parts to solve it systematically.

Step 1: Calculate the Relative Speed of the Trains

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. This is because the distance between them decreases at a rate equal to the sum of their speeds.

• Speed of Train X ($S_X$) = 50 km/h
• Speed of Train Y ($S_Y$) = 76 km/h
• Relative Speed ($S_{relative}$) = $S_X + S_Y$

First, we need to convert the speeds from km/h to meters per second (m/s) for consistency with the given lengths (in meters) and time (in seconds). The conversion factor is $\frac{5}{18}$ (since 1 km = 1000 m and 1 hour = 3600 seconds, so 1 km/h = $\frac{1000}{3600}$ m/s = $\frac{5}{18}$ m/s).

• $S_X = 50 \times \frac{5}{18}$ m/s
• $S_Y = 76 \times \frac{5}{18}$ m/s
• $S_{relative} = (50 + 76) \times \frac{5}{18}$ m/s
• $S_{relative} = 126 \times \frac{5}{18}$ m/s
• $S_{relative} = \frac{126 \times 5}{18} = \frac{7 \times 18 \times 5}{18} = 7 \times 5 = 35$ m/s

The relative speed of the two trains is 35 m/s.

Step 2: Determine the Total Distance Covered When Trains Cross

When two trains cross each other, the total distance covered relative to each other is equal to the sum of their lengths. Let $L_X$ be the length of Train X and $L_Y$ be the length of Train Y.

• Length of Train X ($L_X$) = 345 m
• Let Length of Train Y = $L_Y$ m
• Total distance ($D_{cross}$) = $L_X + L_Y = 345 + L_Y$ meters

Step 3: Use the Crossing Time to Find the Length of Train Y

We know the relative speed and the time taken for the trains to cross each other. We can use the formula: Distance = Speed $\times$ Time.

• Time taken to cross ($T_{cross}$) = 22 seconds
• $D_{cross} = S_{relative} \times T_{cross}$
• $345 + L_Y = 35 \times 22$
• $345 + L_Y = 770$
• $L_Y = 770 - 345$
• $L_Y = 425$ meters

The length of Train Y is 425 meters.

Step 4: Calculate the Time for Train Y to Cross the Bridge

Now we need to find the time it takes for Train Y to cross a bridge of length 905 m. When a train crosses a bridge, the total distance it needs to cover is the sum of its own length and the length of the bridge.

• Length of Train Y ($L_Y$) = 425 m
• Length of the bridge ($L_{bridge}$) = 905 m
• Total distance to cross the bridge ($D_{bridge}$) = $L_Y + L_{bridge} = 425 + 905 = 1330$ meters
• Speed of Train Y ($S_Y$) = 76 km/h

We already converted the speed of Train Y to m/s in Step 1:

• $S_Y = 76 \times \frac{5}{18} = \frac{380}{18} = \frac{190}{9}$ m/s

Now, use the formula Time = Distance / Speed to find the time taken ($T_{bridge}$).

• $T_{bridge} = \frac{D_{bridge}}{S_Y}$
• $T_{bridge} = \frac{1330}{\frac{190}{9}}$
• $T_{bridge} = \frac{1330 \times 9}{190}$
• $T_{bridge} = \frac{133 \times 9}{19}$
• Since $133 = 19 \times 7$, we have:
• $T_{bridge} = 7 \times 9 = 63$ seconds

Train Y will cross the bridge in 63 seconds.

Let's summarize the key values:

The final answer is 63 seconds.

Revision Table: Key Concepts for Train Problems

Additional Information: Solving Train Speed Distance Problems

Problems involving trains often require careful consideration of what 'distance' means in the context of crossing points, platforms, or other trains. The key is to identify the total distance that needs to be covered for the crossing to be complete. For instance, a train finishes crossing a bridge only when its last coach leaves the bridge, meaning it has traveled its own length plus the length of the bridge.

Unit consistency is crucial. Always convert all quantities to a consistent system (like meters and seconds or kilometers and hours) before applying formulas. The conversion factor $\frac{5}{18}$ for km/h to m/s (and $\frac{18}{5}$ for m/s to km/h) is very useful.

Relative speed is a concept used when two moving objects interact. If they move towards each other (opposite directions), their speeds add up to find how quickly the distance between them closes. If they move in the same direction, the relative speed is the difference between their speeds, indicating how quickly the faster object gains on the slower one.

Practicing various types of train problems helps solidify these concepts. Pay attention to the direction of motion and what object is being crossed.