In 1 km races, runner 1 on track 1 (with time 2 min, 27.95 s ) appears to be faster than runn

step 1 So we've got one kilometer races. One runner has a time of two minutes and 27 .95 seconds. The o

In 1 km races, runner 1 on track 1 (with time 2 min, 27.95...

In $1 \mathrm{~km}$ races, runner 1 on track 1 (with time $2 \mathrm{~min}, 27.95 \mathrm{~s}$ ) appears to be faster than runner 2 on track $2(2 \mathrm{~min}, 28.15 \mathrm{~s})$. However, length $L_{2}$ of track 2 might be slightly greater than length $L_{1}$ of track 1. How large can $L_{2}-L_{1}$ be for us still to conclude that runner 1 is faster?

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Key Concepts

Average Speed

Average speed is the ratio of the distance covered to the time taken. In the context of races, it is used to compare the performance of different runners by calculating how quickly they complete a given distance, regardless of slight variations in track conditions or measurement uncertainties.

Unit Conversion

Converting given time units (such as minutes and seconds) into a consistent format is crucial when comparing performance metrics. This ensures that calculations based on time, such as speed or the impact of additional distance, are accurate and comparable across different data sets.

Track Length Variation

Variations in the actual distance of race tracks can introduce small errors or discrepancies when comparing performance. Recognizing and accounting for these differences is key in real-world measurements, as even a slight increase in track length for one runner can affect the interpretation of who is actually faster.

Inequality Analysis

To determine a threshold such that one runner can still be considered faster, despite possible differences in track lengths, an inequality is set up comparing effective speeds. This analysis is an application of error or sensitivity analysis, where the maximum allowable difference in track length is derived to ensure a valid performance comparison.

Transcript

00:02 So we've got one kilometer races.

00:13 One runner has a time of two minutes and 27 .95 seconds.

00:27 The other runner has a time of two minutes and 28 .15 seconds.

00:57 But we're saying that we want to conclude that runner 1 is faster than runner 2.

01:10 Well, let's convert this to seconds.

01:14 Two minutes is 120 seconds.

01:17 So that would be 147 .95 seconds.

01:24 And this would be 148 .15 seconds.

01:31 Seconds okay um speed is distance divided by time this is t1 and this is t2 okay so so i could write that as speed equals l1 over t1 and speed and speed and speed two and is l2 over t2.

02:30 All right.

02:35 So we want speed 1 to be greater than speed 2.

02:50 Now, we're trying to solve for l2 minus l1.

02:58 So if i subtract l2, actually everything's positive.

03:17 So i'm just going to cross multiply and not reverse, not worry about reversing the inequality.

03:28 L1t2 must be greater than l2t1.

03:38 Now, how are we going to get a change in distance? i don't think that really worked.

03:56 So let's try something else.

04:05 No, that's not going to work either.

04:09 I was thinking about making a common denominator, but i would just end up with about the same thing...