Does a hill bring you closer?

As I have mentioned before, sometimes things come up in running that I wonder about mathematically, like time intervals around a track or what you can count as a PR. Both of those questions occurred to me within the past few years, but here is something that I wondered about way back in high school:

I was in a cross country race, running about 50 feet behind a girl from the other team. During a straight, flat section of the course, we had been running the same speed, so this 50-foot gap had remained constant. I knew that a hill was coming up, so I wondered: If we both slow down the same amount on the hill, will the distance between us stay the same, or will it get bigger, or smaller, while we are climbing the hill?

It's a good question. At the time, I didn't know how to figure out the answer (and I didn't ask anyone). But after a few years of high school math and physics, where I learned that (for example) you can easily toss a ball up and down while you are on a train as long as the train isn't accelerating or decelerating, and I learned about the "twin problems" in relativity, I figured out how to think about the problem, and I figured out the answer.

The answer is that the time gap between my opponent and me stayed the same throughout the race, but the distance between us decreased while we were on the hill.

You can think of it like this: Imagine that we were identical twins who run exactly the same speeds, but she started 5 seconds before me. Then the gap between us will be 5 seconds for the whole race. If we both run at a speed of (say) 10 feet/sec on a flat course, then we'll be running 50 feet apart for the flat sections. But if we slow down to 8 feet/sec on the uphills, we'll still be 5 seconds apart, but only 40 feet apart, so the distance between us decreases on the uphill.

If we both speed up to 12 feet/sec on the downhills, the distance will increase to 60 feet on the downhill -- even greater than it was on the flat. But when the course flattens out again, we'll still be 5 seconds apart, so the gap will be back to 50 feet whenever we are both running on a flat section. (If she is on a hill and I haven't gotten there yet, the distance between us will be somewhere between 40 and 50 feet, and the same with the other transitions.)

I even have a way of visualizing this happening. Way back in middle school, our coach had us all running around the indoor track, half a lap sprinting and half a lap jogging. There were 100 kids of widely varying abilities, so we were spread out all around the track, with people crossing the starting and finishing lines at all different times. But the key was that when you got to the line on the track, your speed suddenly went from jogging to sprinting, and then vice-versa. I imagined a screen where the left half was blue and the right half was red, and there were red blood cells flowing across the screen, and as soon as they hit that imaginary line, they changed from red to blue, and sped up. So as soon as they turned blue and sped up, they had to get farther apart. This is what happens when my cross country opponent and I reached the top of the hill and started running down the other side: suddenly, we sped up and got farther apart!