In a circular race of 840 m, A and B start running in the same direction at the same time from the same point at the speeds of 6 m/s and 12 m/s, respectively. After how much time will they meet next?

Question

Prepp Team · Accepted Answer

Circular Race Speed and Time Calculation

This problem involves calculating the time it takes for two runners, A and B, to meet again on a circular track after starting simultaneously from the same point and running in the same direction. We are given the length of the track and the speeds of both runners.

Understanding the Concepts

Track Length: The total distance of one lap is 840 meters.
Speeds: Runner A's speed ($v_A$) is 6 m/s, and Runner B's speed ($v_B$) is 12 m/s.
Direction: Both runners are moving in the same direction.
Meeting Point: They start at the same point and time. They will meet again when the faster runner completes exactly one full lap more than the slower runner.

Calculating Relative Speed

When two objects move in the same direction, their relative speed is the difference between their individual speeds. This helps us understand how quickly the faster object gains on the slower one.

Relative Speed ($v_{rel}$) = Speed of Faster Runner ($v_B$) - Speed of Slower Runner ($v_A$)

Using LaTeX for the formula:

$$v_{rel} = v_B - v_A$$

Substituting the given values:

$$v_{rel} = 12 \text{ m/s} - 6 \text{ m/s}$$ $$v_{rel} = 6 \text{ m/s}$$

This means Runner B gains 6 meters on Runner A every second.

Determining Time to Meet Next

For Runner B to meet Runner A again, Runner B must cover the entire length of the circular track (840 m) relative to Runner A. In other words, Runner B needs to lap Runner A.

The time taken ($t$) can be calculated using the formula:

$$t = \frac{\text{Distance}}{\text{Relative Speed}}$$

The distance Runner B needs to gain is the track length, which is 840 m.

Step-by-Step Calculation

Identify the speeds: $v_A = 6 \text{ m/s}$ and $v_B = 12 \text{ m/s}$.
Identify the track length (distance for one relative lap): $L = 840 \text{ m}$.
Calculate the relative speed ($v_{rel}$) since they run in the same direction: $v_{rel} = v_B - v_A = 12 - 6 = 6 \text{ m/s}$.
Calculate the time ($t$) for the faster runner to cover one lap relative to the slower runner: $t = \frac{L}{v_{rel}}$.
Substitute the values: $t = \frac{840 \text{ m}}{6 \text{ m/s}}$.
Perform the division: $t = 140 \text{ s}$.