Problem 73 - Master Resource Book in JEE Main Physics

Question

A \(210 \mathrm{~m}\) long train is moving due North at a speed of \(25 \mathrm{~m} / \mathrm{s}\). A small bird is flying due South, a little above the train with speed \(5 \mathrm{~m} / \mathrm{s}\). The time taken by the bird to cross the train is \(\begin{array}{llll}\text { (a) } 6 \mathrm{~s} & \text { (b) } 7 \mathrm{~s} & \text { (c) } 9 \mathrm{~s} & \text { (d) } 10 \mathrm{~s}\end{array}\)

Step-by-Step Solution

Answer

The bird takes 7 seconds to cross the train.

Step 1: Identify Relative Speed

The train moves at a speed of \(25 \mathrm{~m/s}\) northwards while the bird flies south at \(5 \mathrm{~m/s}\). When two objects move in opposite directions, their relative speed is the sum of their speeds. Therefore, the relative speed of the bird with respect to the train is \(25 + 5 = 30 \mathrm{~m/s}\).

Step 2: Calculate Time to Cross the Train

To find the time taken for the bird to cross the train, use the formula \(\text{time} = \frac{\text{distance}}{\text{relative speed}}\). The distance to be covered is the length of the train, \(210 \mathrm{~m}\). Using the relative speed from Step 1, the time taken \( t \) is \( \frac{210}{30} \mathrm{~s} = 7 \mathrm{~s} \).

Key Concepts

Train and Bird Problem

The train and bird problem is a classic example that helps illustrate concepts of relative velocity and the relationship between speed, distance, and time. Imagine you have a train traveling in one direction and a bird flying in the opposite direction. The goal is to determine different parameters like how long it takes the bird to cross the train completely. This scenario showcases how two moving objects with different speeds can be analyzed together to find a solution.

• In this problem, the train is moving at a speed of 25 m/s towards the North.
• A bird is flying above the train towards the South with a speed of 5 m/s.
• To solve such problems, understanding their relative motion is crucial, which involves calculating relative speed.

Analyzing problems like this not only helps in understanding motion but can also be applied to real-life scenarios involving moving vehicles or objects. It demonstrates how varying speeds and directions affect how we perceive movement.

Time Calculation

To solve the train and bird problem, calculating the time taken for the bird to cross the train is essential. This involves using the relative speed of the bird with respect to the train.To find the relative speed:- When two objects move towards each other with different velocities, their relative speed is the sum of their speeds if they are moving in opposite directions.- In this problem, the relative speed becomes 25 m/s + 5 m/s = 30 m/s.Once you have the relative speed, you can determine how long it takes for the bird to completely cross the train. To do this, you need the total distance the bird has to travel over the train's length The formula used is:\[\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}}\]Plug in the numbers from the problem:- Distance = 210 m (the length of the train)- Relative Speed = 30 m/sTherefore, time taken,\[\text{Time} = \frac{210 \text{ m}}{30 \text{ m/s}} = 7 \text{ seconds}\]This calculation shows that it takes the bird exactly 7 seconds to fly over the entire length of the train.

Distance and Speed Relationship

Understanding the distance and speed relationship is key to solving any motion problem. In the train and bird problem, this relationship helps determine how fast or slow an object (like the bird) moves relative to another object (the train) and how much time it takes to cover a specific distance.The fundamental relationship between distance, speed, and time is expressed by the formula:\[\text{Distance} = \text{Speed} \times \text{Time}\]However, when two objects are involved, especially when they move in opposite directions, knowing their relative speed is essential. Relative speed effectively alters the speed for the object in question, in this case, the bird.This concept allows:

• Understanding how combining opposite speeds affects the time to cover a distance.
• Applying the relationship in varied contexts, like traffic analysis or aerodynamic studies.

By knowing the length of the train and using the relative speed, we easily find out how long it takes for the bird to fly from one end to the other. Recognizing these principles is not only helpful in solving textbook exercises but also in solving real-world motion problems.