Understanding Speed and Distance in a Circular Race: A and B's Running Dilemma

In this article, we will explore the concept of speed and distance in the context of a circular race. Specifically, we will delve into a problem involving two runners, A and B, who start from the same point and run in opposite directions on a 550-meter circular path. Let's break down the problem step by step and understand who will reach the starting point first and how far apart they are when A completes the lap.

Step 1: Determine the Distances Covered by A and B

The circular path has a length of 550 meters. When A and B pass each other, A has run 250 meters, but B has a 100-meter head start. Therefore, when they meet, B has run 150 meters (250 meters - 100 meters).

Step 2: Calculate the Speeds of A and B

Let 'v_A' and 'v_B' be the speeds of A and B, respectively. The time taken for both to meet can be expressed as:

time distance / speed

For A running 250 meters and B running 150 meters, the time taken is the same. Therefore, we have:

Cross-multiplying gives us a relationship between the speeds of A and B:

250v_B 150v_A

From this equation, we can express the ratio of the speeds of A and B as:

This implies that A is faster than B with a speed ratio of 5:3.

Step 3: Determine the Time Taken for A and B to Complete the Circular Path

The time taken by A to complete one full lap of 550 meters is:

The time taken by B to complete one full lap of 550 meters is:

Using the speed ratio, we can express v_B in terms of v_A:

Let v_A 5k and v_B 3k for some constant k.

Substituting these into the time equations, we get:

and

Step 4: Determine Who Reaches the Starting Point First

To compare the times:

A's time to complete the lap: B's time to complete the lap: Since

Step 5: Calculate the Distance Apart When A Finishes

Now, let's calculate how far B has run when A completes the lap. The time it takes for A to finish the lap is In that same time, B will have run:

Distance B runs v_B times t_A 3k times At this point, A has completed 550 meters, and B has run 330 meters. The distance between A and B when A finishes the lap is:

Distance apart 550 meters (one lap) - 330 meters (B's distance) 220 meters

Conclusion

In conclusion, A will come first to the starting point, and they will be 220 meters apart when A finishes the lap. This problem demonstrates the importance of understanding speed and distance in the context of circular motion, reinforcing the relationships between time, distance, and speed.