Question

A train \(270 \mathrm{~m}\) long is running over a bridge of length of \(130 \mathrm{~m}\) with a speed of \(40 \mathrm{~m} / \mathrm{s}\). What is the time taken by this train to cross the bridge? (a) \(6 \mathrm{~s}\) (b) \(16 \mathrm{~s}\) (c) \(10 \mathrm{~s}\) (d) none of these

Short answer

(a) 40 s (b) 15 s (c) 10 s (d) 25 s Answer: (c) 10 s

Step-by-step solution

Identify the given data

We are given the following information: - Train length: 270 m - Bridge length: 130 m - Train speed: 40 m/s

Calculate the total distance the train has to travel to cross the bridge

The train has to travel the combined length of itself and the bridge to completely cross the bridge. Therefore, the total distance it has to travel is the sum of the train length and the bridge length: Total distance = Train length + Bridge length Total distance = 270 m + 130 m = 400 m

Calculate the time taken by the train to cross the bridge

To find the time taken, we will divide the total distance by the speed of the train: Time taken = Total distance / Train speed Time taken = 400 m / 40 m/s Time taken = 10 s So, the train takes 10 seconds to completely cross the bridge. Therefore, the correct answer is (c) \(10 \mathrm{~s}\).

Key concepts

Train Problems

Train problems are classic questions in mathematics that involve understanding the movement of trains with respect to tunnels, bridges, or stations. These problems typically require us to think about how long it takes a train to traverse a specific distance. In the given exercise, the central task is to determine the time taken by a train to fully cross a bridge.

• Components Involved: Two key components must be considered: the length of the moving train and the length of the structure being crossed, such as a bridge.
• Total Distance: To solve such a problem, calculate the total distance covered as the sum of the train length and the additional structure (bridge or tunnel).

In our example, the total distance is obtained by adding the length of the train (270 m) to the length of the bridge (130 m), resulting in a combined distance of 400 m. The objective is to apply this knowledge by calculating how long it takes for the entire train to clear the bridge, which will be discussed in detail in the next sections.

Speed and Distance

Speed and distance calculations help us determine how quickly an object, like our train, can cover a specific distance. Key to solving these problems is understanding the relationship between speed, distance, and time, an essential aspect of arithmetic motion.

• Speed Definition: The speed of an object is how fast it moves per unit of time, such as meters per second (m/s).
• Distance Definition: Distance is the total length that the train travels, which, as in our problem, is 400 meters when considering both the train and the bridge lengths.

Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we find the time needed for the train to cross the bridge. With a speed of 40 m/s, the time taken is \( \frac{400 \text{ m}}{40 \text{ m/s}} = 10 \text{ seconds} \). Understanding this relationship helps us solve similar problems efficiently.

Motion in One Dimension

Motion in one dimension simplifies a problem by allowing us to consider movement along a single straight path, without worrying about changes in direction. In this straightforward view, objects like trains move with a constant speed and in a single line, making calculations more manageable.

• Simplified Mathematics: The one-dimensional model reduces motion to a linear distance, making it easier to apply formulas like \( \text{Distance} = \text{Speed} \times \text{Time} \) or its rearrangements.
• Importance of Uniform Speed: When an object travels at a uniform speed, as our train does, predicting its movement becomes a matter of simple arithmetic rather than complex calculus.

By viewing the train's journey across the bridge in this way, we employ a straightforward calculation, highlighting the beauty of motion in one dimension. Knowing that a constant speed covers a fixed distance in a predictable time helps us tackle textbook problems and real-world situations alike.