Circular Track Meet and Exchange Speeds: A Mathematical Exploration

Imagine two runners, P and Q, starting from the same point on a circular track of 800 meters at the start of a race. Runner P moves at a speed of 6 meters per second, while P’s counterpart, Q, moves at a measurably slower pace of 2 meters per second. Under what conditions would they meet for the first time, and how do they meet again after exchanging their speeds?

Step 1: Calculating the First Meeting

First, it is essential to understand how the runners' relative speed contributes to the timing of their first meeting. Since they are running in opposite directions, their relative speed is the sum of their individual speeds:

Relative Speed: 6 m/s 2 m/s 8 m/s

Using the formula for distance: (text{Time} frac{text{Distance}}{text{Relative Speed}}), we can determine the time it takes for them to meet for the very first time:

Time until first meeting: (frac{800 , text{meters}}{8 , text{m/s}} 100 , text{seconds})

During this 100 seconds, each runner covers a certain distance. Using the formula for distance: (text{Distance} text{Speed} times text{Time}), we calculate the distances covered:

Distance covered by P: 6 m/s times 100 s 600 meters

Distance covered by Q: 2 m/s times 100 s 200 meters

Step 2: Exchanging Speeds and Calculating the Next Meeting

After the first meeting, they exchange their speeds. Now, P runs at 2 meters per second, and Q runs at 6 meters per second. Although their speeds have changed, the relative speed remains the same, as it is the sum of their speeds in opposite directions:

New Relative Speed: 2 m/s 6 m/s 8 m/s

Given that they have completed a full 800-meter lap, they will meet again on the track after the exact same amount of time it took them to meet for the first time:

Time until next meeting: (frac{800 , text{meters}}{8 , text{m/s}} 100 , text{seconds})

Conclusion

Therefore, after exchanging speeds, P and Q will meet again after 100 seconds. The key takeaway is that the relative speed remains constant, which ensures that the time taken for them to meet again is identical to the time taken for their first meeting.

Understanding the dynamics of relative speed and speed exchange in circular track scenarios is crucial for solving many real-world and theoretical problems in physics and mathematics. By breaking down the problem into manageable steps and using the appropriate formulas, students and professionals can easily tackle such questions with confidence.