Bird Crossing Trains Relative Motion Physics Problem

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In this physics problem, we explore the concept of relative motion. Specifically, we'll calculate how long it takes a bird, flying at a constant speed, to cross two trains moving towards each other. This problem combines the ideas of relative velocities and the relationship between distance, speed, and time. Understanding these concepts is crucial for solving a variety of physics problems related to motion.

Problem Statement

Two trains, one 250 meters long and the other 350 meters long, are moving towards each other on parallel tracks. The first train is traveling at 72 kilometers per hour, and the second train is moving at 54 kilometers per hour. A bird is flying at a constant speed of 90 kilometers per hour along the direction of the first train. The question is: how long does it take the bird to cross both trains completely?

Understanding Relative Motion

The key to solving this problem lies in understanding the concept of relative motion. When objects are moving relative to each other, their velocities add or subtract depending on the direction of their motion. In this scenario, since the trains are moving towards each other, their relative velocity is the sum of their individual velocities. Similarly, the bird's velocity relative to each train depends on the bird's direction of flight and the train's direction of motion.

To calculate the time it takes for the bird to cross the trains, we need to consider the bird's speed relative to each train and the total distance the bird needs to cover. The total distance includes the lengths of both trains because the bird needs to completely pass both of them.

Converting Units

Before we proceed with the calculations, it's important to ensure that all units are consistent. The lengths of the trains are given in meters, while the speeds are given in kilometers per hour. To maintain consistency, we'll convert the speeds from kilometers per hour to meters per second. The conversion factor is:

1 kilometer per hour (km/h) = (1000 meters / 1 kilometer) / (3600 seconds / 1 hour) = 5/18 meters per second (m/s)

Using this conversion factor, we can convert the speeds of the trains and the bird:

  • Speed of the first train: 72 km/h * (5/18 m/s per km/h) = 20 m/s
  • Speed of the second train: 54 km/h * (5/18 m/s per km/h) = 15 m/s
  • Speed of the bird: 90 km/h * (5/18 m/s per km/h) = 25 m/s

Calculating Relative Velocities

Now that we have the speeds in consistent units, we can calculate the relative velocities. Let's first consider the bird's velocity relative to the first train. Since the bird is flying in the same direction as the first train, we subtract the train's velocity from the bird's velocity:

  • Relative velocity of the bird with respect to the first train (V_b1) = Speed of bird - Speed of the first train = 25 m/s - 20 m/s = 5 m/s

Next, we need to determine the bird's velocity relative to the second train. Since the trains are moving towards each other, the bird is effectively flying against the second train's motion. Therefore, we add the speeds of the bird and the second train:

  • Relative velocity of the bird with respect to the second train (V_b2) = Speed of bird + Speed of the second train = 25 m/s + 15 m/s = 40 m/s

Calculating the Total Distance

The total distance the bird needs to cover to cross both trains is the sum of the lengths of the two trains:

  • Total distance = Length of the first train + Length of the second train = 250 m + 350 m = 600 m

Determining the Time to Cross Both Trains

Now, this is a crucial part. The bird needs to cross both trains. We can consider two phases. First, crossing the first train, and then, crossing the second train.

Phase 1: Crossing the First Train

  • To completely cross the first train, the relevant relative speed is the bird's speed relative to the first train which we've calculated, V_b1 = 5 m/s. The distance is the length of the first train, 250 m.
  • Time taken to cross the first train (t_1) = Distance / Relative speed = 250 m / 5 m/s = 50 seconds

During this time (50 seconds), the second train is also approaching. So, let's calculate how much distance the second train covers in this time.

  • Distance covered by the second train while the bird crosses the first train = Speed of the second train * Time = 15 m/s * 50 s = 750 meters

Phase 2: Crossing the Second Train

  • Now, to cross the second train, the bird needs to cover the length of the second train (350 m). However, the initial distance between the bird (after crossing the first train) and the end of the second train is now reduced because the second train has moved 750 meters closer during Phase 1. If the trains were initially far apart, this 750 m would contribute to the overall calculation. However, the core idea is that the bird needs to cross the length of the second train relative to the combined speed.
  • The bird's speed relative to the second train is V_b2 = 40 m/s.
  • Time taken to cross the second train (t_2) = Distance / Relative speed = 350 m / 40 m/s = 8.75 seconds

Total Time

Therefore, the total time it takes for the bird to cross both trains is the sum of the times taken in both phases:

  • Total time = t_1 + t_2 = 50 seconds + 8.75 seconds = 58.75 seconds

A More Direct Approach

There's a more direct approach to solve this that avoids breaking it into two phases. While the above method provides a good understanding, consider this:

  • The bird eventually needs to cover the total length of both trains at a relative speed when considering the second train. The critical thing is the bird is flying against the oncoming train in the second part.
  • The total distance is still 600 m.
  • Think of the bird effectively trying to outrun the closing distance caused by the second train.
  • However, this approach becomes complex quickly because the relative speeds change depending on which train the bird is interacting with.

Key Point: The initial phase calculation is necessary for complete accuracy. We cannot simply use one combined relative speed calculation for the entire event.

Alternative Simplified (But Less Accurate) Approach and Why It's Problematic

One simplified, but ultimately incorrect, approach some might attempt is: Calculate a weighted average relative speed, or to try and determine a single "effective" relative speed for the entire 600 m. This would involve combining the relative speeds in some way (e.g., averaging or weighting based on train length). However, this fails to account for the changing interaction and is not physically accurate.

Why this simplified approach doesn't work:

  • The bird's interaction with each train is distinct, governed by its instantaneous relative velocity.
  • Averaging speeds doesn't accurately represent the time spent in each interaction.

Formula for Time Calculation

The fundamental formula used in this problem is the relationship between distance, speed, and time:

Time = Distance / Speed

We applied this formula in different contexts, using relative velocities and the total distance to be covered.

Final Answer

Therefore, it takes the bird 58.75 seconds to completely cross both trains.

Conclusion

This problem highlights the importance of understanding relative motion and how it affects the calculation of time, distance, and speed. By carefully considering the velocities of the objects relative to each other, we can accurately solve complex physics problems. The conversion of units and clear identification of each phase of the problem are crucial steps to arrive at the correct solution. While simplified approaches might seem tempting, a thorough analysis considering each interaction is necessary for accuracy.

This type of problem often appears in introductory physics courses and is a good example of how real-world scenarios can be modeled using physics principles. By mastering these concepts, students can build a strong foundation for more advanced topics in physics and engineering.

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