Athlete's Run on a Circumferential Track: Distance Covered vs. Displacement

While the initial problem posed may seem unrealistic, understanding the principles of circular motion is crucial in various fields, from athletics to engineering. This article delves into the concepts of distance covered and displacement in a practical scenario.

Problem and Context

Consider an athlete running on a circular track with a diameter of 200 meters. In 40 seconds, the athlete completes one round of the track. The challenge is to calculate the distance covered and the displacement after 2 minutes and 20 seconds and 2 minutes and 40 seconds.

Understanding the Track and Motion

The diameter of the track is given as 200 meters. Therefore, the circumference of the track, which is the distance covered in one complete round, is calculated as follows:

[ text{Circumference} pi times text{diameter} 200pi , text{meters} ]

This implies that in 40 seconds, the athlete covers a distance of 200π meters. Using this, we can determine the distance covered in any given time period:

Calculations for Specific Time Intervals

2 Minutes and 20 Seconds (140 Seconds)

To calculate the distance covered in 140 seconds, first, we find the distance covered in one second:

[ text{Distance per second} frac{200pi}{40} 5pi , text{meters} ]

Hence, the distance covered in 140 seconds is:

[ text{Distance in 140 seconds} 140 times 5pi 700pi , text{meters} ]

Since 700π meters is approximately 2200 meters, the net distance covered is 2200 meters.

The number of laps completed in 140 seconds is:

[ text{Number of laps} frac{140}{40} 3.5 , text{laps} ]

Since the athlete completes 3.5 laps, the final position is opposite the initial position, leading to a net displacement of the diameter of the track, which is 200 meters.

2 Minutes and 40 Seconds (160 Seconds)

Similarly, we first calculate the number of laps completed in 160 seconds:

[ text{Number of laps} frac{160}{40} 4 , text{laps} ]

Since the athlete completes 4 full laps in 160 seconds, he returns to the starting point. Therefore, the net displacement is zero.

The distance covered in 160 seconds is:

[ text{Distance covered} 4 times 200pi 800pi , text{meters} ]

Approximately, this is 2514.29 meters.

Real-World Application

In athletics, understanding the concepts of distance covered (the path length) versus displacement (the straight-line distance between the starting and ending points) is crucial for analyzing performance and optimizing training. The circumference and diameter relationship, (pi times text{diameter}), provide a simple yet powerful tool for such analysis.

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