Calculating Bridge Length: Train Speed, Time, and Distance

This problem involves a train of a certain length crossing a bridge. When a train crosses a bridge, the total distance it covers is equal to the length of the train plus the length of the bridge. The train starts to cross the bridge when its front end reaches the beginning of the bridge and finishes crossing when its rear end leaves the end of the bridge.

We are given the following information:

• Length of the train: $L_{\text{train}} = 253$ m
• Speed of the train: $S = 60$ km/h
• Time taken to cross the bridge: $T = 42$ seconds

We need to find the length of the bridge, $L_{\text{bridge}}$, in meters.

Unit Conversion for Train Speed

The speed is given in kilometers per hour (km/h), but the time is in seconds and the lengths are in meters. We need to convert the speed from km/h to meters per second (m/s) to maintain consistent units for calculation. The conversion factor from km/h to m/s is $\frac{5}{18}$.

Speed in m/s $= 60 \times \frac{5}{18}$ m/s

Speed in m/s $= \frac{10 \times 5}{3}$ m/s

Speed in m/s $= \frac{50}{3}$ m/s

Calculating Total Distance Covered by the Train

The total distance covered by the train while crossing the bridge is calculated using the formula:

$\text{Distance} = \text{Speed} \times \text{Time}$

Total distance $D = S \times T$

$D = \frac{50}{3} \text{ m/s} \times 42 \text{ s}$

$D = 50 \times \frac{42}{3}$ m

$D = 50 \times 14$ m

$D = 700$ m

This total distance of 700 m is the combined length of the train and the bridge.

Finding the Length of the Bridge

As explained earlier, the total distance covered by the train to cross the bridge is the sum of its own length and the length of the bridge.

Total distance $D = L_{\text{train}} + L_{\text{bridge}}$

We know $D = 700$ m and $L_{\text{train}} = 253$ m. We can rearrange the formula to find $L_{\text{bridge}}$:

$L_{\text{bridge}} = D - L_{\text{train}}$

$L_{\text{bridge}} = 700 \text{ m} - 253 \text{ m}$

$L_{\text{bridge}} = 447 \text{ m}$

Thus, the length of the bridge is 447 meters.

Revision Table: Train Crossing Bridge Problem

Additional Information: Problems Involving Trains

Train problems are common in aptitude tests and physics. They often involve relative speed and the concept of distance covered based on the train's length and the length of the object being crossed.

• Crossing a point object (pole, person): The distance covered is just the length of the train itself ($D = L_{\text{train}}$).
• Crossing an object with length (bridge, tunnel, platform, another train): The distance covered is the sum of the train's length and the object's length ($D = L_{\text{train}} + L_{\text{object}}$).
• Trains moving in the same direction: The relative speed is the difference between their speeds ($S_{\text{relative}} = |S_1 - S_2|$).
• Trains moving in opposite directions: The relative speed is the sum of their speeds ($S_{\text{relative}} = S_1 + S_2$).

Always ensure that all units (speed, time, distance) are consistent before performing calculations.