Understanding Race Dynamics: Calculating Distance Difference

This question asks us to determine the distance by which Amit beats Rahul in a 700 m race, given their finishing times.

In a race, when one person 'beats' another by a certain distance, it means that when the first person (the winner) finishes the race, the second person is still that calculated distance away from the finish line.

Step-by-Step Solution to the Race Problem

First, let's find the speed of each runner. Speed is calculated as Distance divided by Time.

1. Calculate Amit's Speed

Amit completes the 700 m race in 20 seconds.

Using the formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Amit's speed $= \frac{700 \text{ m}}{20 \text{ s}} = 35 \text{ m/s}$

2. Calculate Rahul's Speed

Rahul completes the 700 m race in 25 seconds.

Using the formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Rahul's speed $= \frac{700 \text{ m}}{25 \text{ s}} = 28 \text{ m/s}$

3. Determine Rahul's Position When Amit Finishes

Amit finishes the race in 20 seconds. In these 20 seconds, Rahul, running at his speed, covers a certain distance.

Using the formula: $\text{Distance} = \text{Speed} \times \text{Time}$

Distance covered by Rahul in 20 seconds $= \text{Rahul's speed} \times 20 \text{ s}$

Distance covered by Rahul in 20 seconds $= 28 \text{ m/s} \times 20 \text{ s} = 560 \text{ m}$

4. Calculate the Distance Amit Beats Rahul By

When Amit reaches the 700 m finish line (at 20 seconds), Rahul has only covered 560 m. The distance by which Amit beats Rahul is the remaining distance Rahul needs to cover to reach the finish line.

Distance Amit beats Rahul by $= \text{Total race distance} - \text{Distance Rahul covered in 20 seconds}$

Distance Amit beats Rahul by $= 700 \text{ m} - 560 \text{ m} = 140 \text{ m}$

Alternatively, we can consider the time difference.

Amit finishes 25 s - 20 s = 5 seconds before Rahul.

In these 5 seconds, Rahul would still be running. The distance Rahul covers in these extra 5 seconds (or the distance he still needs to cover when Amit finishes) is the distance by which Amit beats him.

Distance Rahul covers in 5 seconds $= \text{Rahul's speed} \times \text{Time difference}$

Distance Rahul covers in 5 seconds $= 28 \text{ m/s} \times 5 \text{ s} = 140 \text{ m}$

Both methods give the same result: Amit beats Rahul by a distance of 140 m.

Summary of Calculations

• Amit's Speed: $\frac{700}{20} = 35$ m/s
• Rahul's Speed: $\frac{700}{25} = 28$ m/s
• Distance covered by Rahul in 20 seconds: $28 \times 20 = 560$ m
• Distance difference at 20 seconds: $700 - 560 = 140$ m
• Time difference: $25 - 20 = 5$ seconds
• Distance Rahul covers in 5 seconds: $28 \times 5 = 140$ m

Thus, Amit beats Rahul by 140 m in the 700 m race.

Revision Table: Key Concepts in Race Problems

Additional Information on Race Calculations

Race problems often involve calculating speeds, times, or distances based on the principle of uniform motion. It's important to be consistent with units (meters for distance, seconds for time, meters per second for speed).

• When comparing runners, you can either calculate the distance covered by the slower runner in the time the faster runner takes, or calculate the distance the slower runner covers in the time difference between them. Both methods should yield the same result.
• Understanding the relationship between speed, distance, and time is fundamental to solving these problems. The core formula $\text{Distance} = \text{Speed} \times \text{Time}$ can be rearranged to find any of the three variables if the other two are known.
• These concepts are applicable in various real-world scenarios, not just sports races, but also in calculating travel times, distances, and speeds for vehicles, etc.