Solve Distance Problem A & B: 6 & 10 Min Circular Mile Track

[bergausstein]

A and B can run around a circular mile track in 6 and 10 minutes respectively. If they start at the same instant from the same place, in how many minutes will they pass each other if they run around the track (a) in the same direction, (6) in opposite directions?

can you help solve the first part of the question? thanks!

 

[MarkFL]

I think I would first observe that the size of the track is irrelevant, and we can let the radius of the track be 1 unit. Then, I would describe the position of the runners parametrically, with time = $t$, as the parameter, measured in minutes. Let the runners begin at (1,0) and move in a counter-clockwise direction. Then their positions can be given by:

\(\displaystyle x(t)=\cos\left(\frac{2\pi}{T}t \right)\)

\(\displaystyle y(t)=\sin\left(\frac{2\pi}{T}t \right)\)

where $T$ is the period of their motion, i.e., the time (in minutes) it takes for them to complete one circuit of the track.

Then equate the respective coordinates of both runners, and take the first positive solution for $t$.

 

[bergausstein]

MarkFl, I appreciate what you posted above. But it seems that it isn't in the realm of what I can comprehend at this point in time since I'm just beginning to learn algebra. can you show me the simple approach to this problem? thanks!

 

[MarkFL]

Yes, now that I think about it more, there is a much simpler approach. :D

Let $C$ be the circumference of the track, and using distance = rate times time, and subscripting the faster runner's parameters with a 1 and the slower runner with a 2. We may use the fact that when the faster runner laps the slower runner, his distance ran will be one more circumference than the slower runner, and write:

\(\displaystyle d_1=d_2+C\)

Use $d=vt$:

\(\displaystyle v_1t=v_2t+C\)

What are the velocities of the two runners?

 

[CellCoree]

Sure, no problem! For the first part of the question, we can use the concept of relative speed to determine when A and B will pass each other if they are running in the same direction. Since A is faster than B, we can think of A "catching up" to B on the track. This means that the distance between them will decrease by one mile every minute. So, if A and B start at the same time, they will pass each other when the distance between them is one mile.

Since A takes 6 minutes to run one mile, we can say that after 6 minutes, A will have completed one lap and will be back at the starting point. At the same time, B will have run for 6 minutes and will be 6/10 of a mile ahead of A. This means that the distance between them is now 6/10 of a mile (since A is starting at the same point as B). We can now use the relative speed concept again to determine how long it will take for A to catch up to B.

Since the distance between them decreases by one mile every minute, and the current distance between them is 6/10 of a mile, it will take A 6/10 of a minute to catch up to B. This is equivalent to 36 seconds. So, after 6 minutes and 36 seconds, A will pass B on the track.

For the second part of the question, where A and B are running in opposite directions, we can use the concept of combined speed. This means that their individual speeds will add up to determine their combined speed. Since they are running in opposite directions, they will be moving away from each other, so their combined speed will be the sum of their individual speeds.

Since A takes 6 minutes to run one mile, their combined speed will be 1/6 miles per minute. Similarly, B's individual speed is 1/10 miles per minute, so their combined speed is 1/6 + 1/10 = 4/15 miles per minute. This means that the distance between them will increase by 4/15 of a mile every minute.

Using the same logic as before, we can determine that after 6 minutes, A will have completed one lap and will be back at the starting point. At the same time, B will have run for 6 minutes and will be 6/10 of a mile ahead of