## Saturday, January 22, 2011

### Mixing speeds

Sometimes, when I'm running, I come up with a mathematical question, such as the question about PRs and average speeds. Here is another one.

I was doing 200m intervals on the indoor track, one lap fast and one lap slow, and I wondered: is it the same time interval between when my coach sees me, and when the people on the exercise bikes see me?

More concretely: Suppose that a runner is alternating fast laps (which take 1 minute) with slow laps (which take 2 minutes). She passes her coach, who is standing at the start/finish line, after 1 minute, then 2 minutes later, then 1 minute later, then 2 minutes later, etc. How about the people on the exercise bikes 1/4 of the way around the track? Do they also see her every 1, 2, 1, 2, minutes? Is there a place you could stand so that you could see the runner with an equal time interval between meetings?

The answer is that depending on where you are standing, the interval between meetings with the runner is different. If you stood halfway around the track (diametrically opposite the start/finish line), then it would always be 1:30 between meetings with the runner: If she passed you on a slow lap, she would then run half of a slow lap (1:00) followed by half of a fast lap (0:30) before seeing you again. Similarly, if she passed you on a fast lap, she would then run half of a fast lap (0:30) followed by half of a slow lap (1:00) before seeing you again.

If she passed the people on the exercise bike on a slow lap, then it would be 3/4 of a slow lap (1:30) and 1/4 of a fast lap (0:15) before she passed them again, so that's 1:45. When she passed them on a fast lap, it would be 3/4 of a fast lap (0:45) and 1/4 of a slow lap (0:30) before she passed them again, so that's 1:15. So the people on the exercise bikes see her every 1:45, then 1:15, then 1:45, then 1:15, etc.

You can get any time intervals you want between (1:00/2:00) and (1:30/1:30) by standing in various locations around the track.

This is basically a mixing problem: Mixing various proportions of fast and slow laps. It is similar to the question: Suppose you have one liter of apple juice, which costs \$1, and one liter of grape juice, which costs \$2. You have to make two fruit punches, each 1 liter. What are the possible costs of the two drinks?

In this case if you keep the juices separate, the punches are \$1 and \$2 (like standing at the start/finish line). If you mix the juices 50%/50%, both punches are \$1.50 (like standing diametrically opposite the start/finish line). If you put 25%/75% in each punch, the costs will be \$1.25 and \$1.75. You can get anything between those by varying the concentrations (like you can get any time intervals between the given ones by standing in different places around the track.)

N.B. My fast and slow intervals were not 1 minute and 2 minutes. That was just for purposes of illustration, to make the numbers easier.