How fast must Michael be running and how far? (This is basically the same as those "train" questions.)

Too many variables to make an algebraic equation. Pin Michael’s starting point down.

Or just have Kelly and Michael find a 3 mile loop and duplicate the running.

He could do it as few as twice running twice as fast as you. If he started by you (pass #1), did one lap and passed you again (pass #2), and ended by you (pass #3). Since you know he doesn’t start next to you or end next to you, give it an extra quarter mile for some start and end time?

Thanks Empress! That makes sense. Just to clarify, by “you” you mean Kelly, right?
Gail, don’t be such a pessimist! Fluther can figure out anything!

Removed by me.

EVERYTHING IS WRONG. I forgot he was running the opposite direction. Which makes him going slower than I was actually thinking at first. I realized it when I tried to take this a step further and actually figure it out and he was running like the world’s fastest mile.

They are totally running the same rate if they start and stop together and pass once mid-stream. Again, speed him up a bit so that they don’t have to actually start and stop together and add a bit more mileage. But in opposite directions, they could be going the same rate and still see each other three times.

They could be going the same rate and still pass each other three times even though she only runs around once?

I don’t think that’s right. I keep drawing little circles with my fingers, and he has to be going slightly faster, because he has to get in a little extra distance to catch her the third time before she’s done. Right?

Draw it out like they start and stop in the same place. Then if they meet halfway through the loop and again the at the end, they are going the exact same rate. Then, like you said, make him a bit faster so he can pass more often.

Yes, so we are back where I started. He must be going faster and farther – do we know how much? That’s the original question.

Amen, sisters. Ah have faith so ah will check in later. Perhaps calculus since we don’t know Michael’s speed or starting point. He may approach a limit, considering how fast the fastest humans can run three miles.

Well, the true minimum is that it must take him 1 second less to run all of that. But a practical minimum is harder without more information,

Empress, have you never taken calculus? The “true minimum is 1 second less”? You’re joking right?

@EmpressPixie got it right the first time. Since they passed three times, the relative distance covered is between 3 and 4 loops. Kelly does one loop so Michael must have done 2 or 3 loops, meaning that he is traveling 2 to 3 times faster than Kelly.

—@Lost, I don’t think you’re thinking about the fact that they’re going opposite directions ?

Sure I’ve taken calc. What I mean is that the slowest he could be running and still pass her three times has to be about one second faster than her. He comes on the track, passes her immediately, passes her halfway, and passes her right before the end. If he is any at all faster than she is, even a microsecond, this works. And as they start at the same time, if he is any at all faster, he passes her a third time before she finishes. I was simply using “second” as the smallest unit of time.

It is because they are going in opposite directions that their speeds are additive. If I am going 20 mph and someone approaches me at 30 mph, our relative speed is 50 mph. It is the same thing with the circuits. To make things real simple, suppose Michael did one circuit in the opposite direction. Then they would meet twice, once at the midway point and once at the end.

Okay class, once you figure this one out, here is a slightly tricky problem that also involves relative rates. You download a one hour video takes 3 minutes to download for every minute of play time, so the total download time is 3 hours. You would like to start watching the video before it downloads completely. In order to view the video uninterrupted, what is the minimum amount of viewing time that must be downloaded before you can start watching?
Hint: There are two things going on, downloading and viewing. What can you conclude about the times required to do these two activities?

I think he would have to be running a 7–8 minute mile. Assuming that he got 2/3 of a mile by the time she got 1/3 and then he got another 2/3 of a mile by the time she got through another 1/3.

This could change though,m if the passings were in different locations.

I think Kelly and Mike should text each other to go for a smoothie together, so then Kelly will ask “Hey Mike, you passed me by in the park a couple of times this morning, how fast were you running?” Mike: “I don’t know, actually I don’t really care, life is not a math equation Kelly… Why don’t we better see who’s the first to finish the smoothie without having a headache, the loser will pay the movie tickets (smile)”
Kelly: “O.k., but you know you will lose, why do we make a competition? Why don’t you just ask me out, you know I’ll say yes…”
Mike (blush): “O.k. I’ll pick you up at seven, and, I wasn’t running that fast :D”

From Michael; Sorry, Kelly, I am texting from the ER where I have just had someone use the defibrillator on me. Perhaps you should hook up with a less competitive runner. I’d write more but I still can’t breath too well.

EmpressPixie is the most accurate. The fact they are going in opposite directions is more of a distraction than a help to get the solution. The passing 3 times helps establish that as Empress Pixie said Michael only has to go a little faster than Kelly so that they pass when they start, when they cross half way and then just before she finishes.

It is not possible to answer how fast Michael is going but it is possible to state the minimum speed. This speed does not need to be a complex calculation of the speed difference because of opposite directions it is simply based on Kelly’s speed.

So during the 40mins that Kelly ran 1 loop (3miles), Micheal ran for the same time (starting from the first crossing) and distance plus the distance between the last crossing and Kelly finishing her loop. This distance is unknown it could be 10metres it could be half a mile. As we are searching for the minimum we would have to state a small distance which makes the difference to the overal speed negliable over a 3 mile distance.

So the answer is Michael is running the same speed as Kelly (or faster). So if it took Kelly 40mins to run 3miles she could run 4.5 miles in an hour if she kept her speed = 4.5mph

To make the question have a more meaningful answer you could add. At the 3rd pass Kelly is 2/3rds away from completing her circuit. Or that Kelly ran on for another 5mins after passing Michael the 3rd time. Then we could calculate his exact speed.