Problem 9 - Fundamentals of Physics

Question

In 1 km races, runner 1 on track 1 (with time \(2 \min .27 .95 \mathrm{~s}\) ) appears to be faster than runner 2 on track \(2(2 \min , 28.15 \mathrm{~s})\). Howcver, length \(L_{2}\) of track 2 might be slightly greater than length \(L_{1}\) of track 1. How large \(\operatorname{can} L_{2}-L_{1}\) be for us still to conclude that runner 1 is faster?

Step-by-Step Solution

Answer

The maximum difference \(L_2 - L_1\) can be less than 200 meters.

Step 1: Convert Times to Seconds

Convert the time of both runners from minutes and seconds into seconds for easier calculations.Runner 1: \[ 2 \text{ min } 27.95 \text{ s} = 2 \times 60 + 27.95 = 147.95 \text{ seconds} \]Runner 2: \[ 2 \text{ min } 28.15 \text{ s} = 2 \times 60 + 28.15 = 148.15 \text{ seconds} \]

Step 2: Understanding the Speed Formula

The speed of each runner can be determined using the formula:\[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]Since runner 1 appears faster, the speed must satisfy:\[ \frac{L_1}{147.95} > \frac{L_2}{148.15} \]

Step 3: Cross-Multiplying

To find the maximum allowable difference \(L_2 - L_1\), we can rearrange and cross-multiply the inequality:\[L_1 \times 148.15 > L_2 \times 147.95\]

Step 4: Rearrangement to Find Maximum Difference

Rearrange the inequality to solve for the difference:\[ 148.15L_1 > 147.95L_2 \]Subtract \(147.95L_1\) from both sides:\[ 0.20L_1 > (L_2 - L_1) \]

Step 5: Final Calculation

Since \(L_1\) is 1 km or 1000 m, calculate the maximum value of \(L_2 - L_1\):\[ L_2 - L_1 < 0.20 \times 1000 = 200 \text{ meters}\]

Step 6: Conclusion

For runner 1 to definitely be faster, the maximum \(L_2 - L_1\) must be less than 200 meters.

Key Concepts

Speed Calculation

Calculating speed is an essential concept in understanding physics problems, especially those involving motion. Speed is a measure of how quickly an object covers a distance. To find the speed of an object, we use the formula:

• Speed = \( \frac{\text{Distance}}{\text{Time}} \)

In our exercise, we are comparing the speeds of two runners in a 1 km race. To determine who is faster, we calculate each runner's speed by dividing the length of the track they run (distance) by the time it takes them to complete it.

For instance, if runner 1 finishes their 1 km on track 1 in 147.95 seconds and runner 2 completes it in 148.15 seconds on track 2, we calculate their speeds as follows:

• Runner 1's Speed = \( \frac{1000 \text{ m}}{147.95 \text{ s}} \)
• Runner 2's Speed = \( \frac{1000 \text{ m}}{148.15 \text{ s}} \)

These calculations help us ascertain which runner is moving faster by evaluating the quotient of distance over time.

Unit Conversion

Unit conversion is a fundamental skill in physics needed for solving problems accurately. It involves changing one unit of measure into another, ensuring consistency across calculations.

In the given problem, the runners' times are provided in minutes and seconds. However, for simpler and more straightforward speed calculations, we convert these times into seconds.

To convert the time from minutes and seconds to just seconds we:

• Multiply the minutes by 60 (since there are 60 seconds in a minute).
• Add the remaining seconds to this product.

This conversion is essential for maintaining uniformity in the time unit when using the speed formula, as it uses time in seconds. For example:

• Runner 1: 2 min 27.95 s = \( 2 \times 60 + 27.95 = 147.95 \text{ seconds} \)
• Runner 2: 2 min 28.15 s = \( 2 \times 60 + 28.15 = 148.15 \text{ seconds} \)

Converting to a single unit minimizes errors and simplifies mathematical operations, increasing the accuracy of your results.

Inequalities in Speed Comparison

Inequalities play a crucial role in comparing speeds when conditions or contexts vary between subjects. In this scenario, the lengths of the tracks might be different, and inequalities are used to account for these discrepancies.

When comparing runners' speeds with different track lengths, a direct inequality helps us understand who is truly faster. We set up the inequality:

• \( \frac{L_1}{147.95} > \frac{L_2}{148.15} \)

This equation ensures that we take the length of the track and the time taken into consideration effectively. Through cross-multiplication, the inequality transforms as follows:

• \( 148.15L_1 > 147.95L_2 \)

Solving this inequality allows us to determine the maximum difference \(L_2 - L_1\) that maintains runner 1's superiority in speed:

• From \(148.15L_1 > 147.95L_2\), simplifying yields \(0.20L_1 > (L_2 - L_1)\).

Understanding and applying inequalities in speed comparison helps us adjust for varying conditions and draw more accurate conclusions about relative speeds.