Understanding the Train Crossing Problem

This problem involves calculating the uniform speed of a train based on the time it takes to cross two different bridges. When a train crosses a bridge, the total distance the train covers is the sum of the length of the bridge and the length of the train itself.

Let:

• $L_T$ be the length of the train in meters (m).
• $S$ be the uniform speed of the train in meters per second (m/s).

We are given two scenarios:

1. The train crosses a 1000 m long bridge in 52 seconds.
2. The train crosses a 700 m long bridge in 40 seconds.

We can use the formula: Distance = Speed × Time

Formulating Equations for Speed and Length

For the first scenario (crossing the 1000 m bridge):

• Total distance covered = Length of train + Length of bridge 1 = $L_T + 1000$ m
• Time taken = 52 seconds
• Using the formula, we get the equation:

$$L_T + 1000 = S \times 52 \quad \cdots (1)$$

For the second scenario (crossing the 700 m bridge):

• Total distance covered = Length of train + Length of bridge 2 = $L_T + 700$ m
• Time taken = 40 seconds
• Using the formula, we get the equation:

$$L_T + 700 = S \times 40 \quad \cdots (2)$$

Calculating the Train's Speed

Now we have a system of two linear equations with two variables ($L_T$ and $S$). We can solve this system to find the speed $S$. A simple way to do this is by subtracting Equation (2) from Equation (1):

$(L_T + 1000) - (L_T + 700) = (S \times 52) - (S \times 40)$

Simplify the equation:

$$1000 - 700 = S \times (52 - 40)$$

$$300 = S \times 12$$

Now, solve for $S$:

$$S = \frac{300}{12}$$

$$S = 25 \text{ m/s}$$

Converting Speed to km/h

The question asks for the speed in kilometers per hour (km/h). To convert speed from meters per second (m/s) to kilometers per hour (km/h), we multiply by the conversion factor $\frac{18}{5}$.

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

Speed = $25 \times \frac{18}{5}$ km/h

Speed = $5 \times 18$ km/h

Speed = $90$ km/h

Therefore, the uniform speed of the train is 90 km/h.