Help Needed with Physics Problem (2 runners on circular track)?

Question

Two runners start in opposite directions at the northernmost point of a circular track 20.0 m in radius. If the speed of each runner is constant and is 5.31 km/h for Runner A and 3.44 km/h for Runner B, where will they meet? Where will they meet the second time? Assume that they go around the track on essentially the same path, with Runner A initially headed west and Runner B initially headed east.

Can anyone help? Thank you!!!

Answer 1

They will meet around 76 meters from northermost point in direction of runner A (runner B will have travelled around 49 meters)
And again, same thing for the second time, treat that point as starting point, and they will meet another 76 meters in direction of A's travel (by this time, A will have made a full circle but B would not have yet)
To understand the problem, lets simplify it by saying that the speed of A is 2 KM/hr and B is 1 KM/hr. For any given time constant, A will travel twice the distance than B i.e. speed of A divided by speed of B. So when they meet, the total distance travelled by A will be double than that of A. So, if the total distance travelled is 3 KM, then 2 KM was covered by A and 1 KM was covered by B.
In this question, using the same logic, when they meet for the first time, they both have travelled for the same 'time'. We also know that they have together made a full circle, so the total distance travelled is the circumference of the circle i.e. 2*pi*radius = 2*22/7*20 = approx 125 meters.
And now to get the distance travelled by A, simply get the ratio of A's speed with is (5.31/ (5.31+3.44)) = approx 0.6068
Multiply this number by the total distance of 125 meters and you get the distance travelled by A when it meets B, which is about 76 meters.
Would have been easier to expain the answer diagramattically, but important point to note here is the ratio of speed of A and B and that they will be travelling for the same time when they meet....

Answer 2

They will meet around 76 meters from northermost point in direction of runner A (runner B will have travelled around 49 meters)
And again, same thing for the second time, treat that point as starting point, and they will meet another 76 meters in direction of A's travel (by this time, A will have made a full circle but B would not have yet)
To understand the problem, lets simplify it by saying that the speed of A is 2 KM/hr and B is 1 KM/hr. For any given time constant, A will travel twice the distance than B i.e. speed of A divided by speed of B. So when they meet, the total distance travelled by A will be double than that of A. So, if the total distance travelled is 3 KM, then 2 KM was covered by A and 1 KM was covered by B.
In this question, using the same logic, when they meet for the first time, they both have travelled for the same 'time'. We also know that they have together made a full circle, so the total distance travelled is the circumference of the circle i.e. 2*pi*radius = 2*22/7*20 = approx 125 meters.
And now to get the distance travelled by A, simply get the ratio of A's speed which is (5.31/ (5.31+3.44)) = approx 0.6068
Multiply this number by the total distance of 125 meters and you get the distance travelled by A when it meets B, which is about 76 meters.

Though it would have been easier to expain the answer diagramattically, but important point to note here is the ratio of speed of A and B and that they will be travelling for the same time when they meet....
You can try this question with putting simple whole numbers for speed ..
(This question would have been little difficult if both A and B were travelling in the same direction... but even then the same logic could be used with some variation)

Answer 3

You set their angular velocities equal to each other but since there is no acceleration and they are on the same path (in opposite directions) everything cancels out and you get a ratio 3.44/5.31 = .64783.

So they’ll meet at 360 * .64783 = 233.22 degrees from north. Or 81.4 meters along the arc for runner A initially heading West (track circumference = Pi*40m) and 44.25 meters along the arc for runner B starting East.
For their next meeting, simply add another 233.22 degrees from their first meeting point for 106.44 (2*233.22-360) degrees from North.

Answer 4

Let's use 20 km radius, as you have speed in km/h.

They will meet when they have collectively travelled 20km. So, each hour they travel 8.75 km. And 20/8.75 = 80/35 = 16/7.

So, when they first meet, A will have travelled 5.31*16/7 km around the circle.

When they next meet, A will have travelled 5.31*2*16/7 km around the circle.

Is that what you need?

Answer 5

20m radius means 40pi length(`125meters).
since runner 1 is going 1.54 times as fast as runner 2, he will cover 1.54 times the distance. so x = the slower runner,and 1.54x = the faster runner and x+1.54x=the total distance of the track so they pass around the 50 meter mark on the first pass. Just rotate the track 50 meters and start from there for the second pass.

Answer 6

Track circ 2*20.0m*3.14=125.7m
Figure they travel 8.75km/hr, runner A travels 39.3% of that.
Figure the runner will cover 39.3% of the circumference while
runner B covers the other 60.7%. So runner A will cover 49.4m when he meets runner B. For repeated passes, just add another 49.4m (98.8m for two passes).

If you're doing this one for school, don't forget your level of significant figures (3). This would bring the track circumference to 126 and runner A's distance to 49.5m, 99.0m for two laps.

If you draw out all the numbers, you'd get:
2*20.0m*3.141592653589793...=125.664m
125.664m*39.3143%=49.4039m (98.8078m for two).

Answer 7

I answered this in your other thread using equations, and I expressed the answers in degrees because you had a circular track, and the question was they met on two occasions. It seemed to me that to answer the question with a circular track, are better than distances traveled.

In this thread, adashi and jaina both had it, expressing answers in terms of distance. The first three answerers had interesting approaches. But Figurehead was wrong in asserting that angular velocities are equal. They're not, and his answer is wrong. He did, however, use angles (degrees), which I think is best.

Answer 8

Dennis K got it. Pretty clever what he did for the 2nd revolution.
ns220 did some clever stuff too but forgot about calculating circumpherence.