Circular Track Meeting Time Calculation

This problem involves calculating the time it takes for two runners, moving in opposite directions on a circular track, to meet for the first time. We can solve this using the concept of relative speed.

Key Information

• Track Length = 400 meters
• Speed of Runner A ($v_A$) = 6 m/s
• Speed of Runner B ($v_B$) = 4 m/s
• Direction: Opposite

Relative Speed Concept

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. This is the speed at which the distance between them decreases.

Relative Speed ($v_{rel}$) = $v_A + v_B$

In this case, $v_{rel} = 6 \text{ m/s} + 4 \text{ m/s} = 10 \text{ m/s}$.

Calculating Meeting Time

For the runners to meet for the first time, the total distance covered by both combined must equal the length of the circular track.

Time = Total Distance / Relative Speed

Time = $\frac{\text{Track Length}}{v_{rel}}$

Time = $\frac{400 \text{ meters}}{10 \text{ m/s}}$

Time = 40 seconds

Conclusion

The runners will meet for the first time after 40 seconds.