[psyang]

Originally Posted by GGG

In response to psyang thoughts on the dog problem. I butchered the quoting so deleted it all.

Spoiler!

I don’t think it’s a divide by zero error. It’s just you have insufficient information about the initial conditions of the problem because your start at an infinitesimal distance.

Remember the fly problem in reverse if two trains are traveling toward each other and a fly travels in between how long does it take for the fly to get squished.

It doesn’t matter where the fly starts it gets squished when the trains collide. So that problem works backwards and it shows that the end point of the dog (starting point of the fly) doesn’t actually matter and can be anywhere.

It’s essentially behaves like an integration constant, it can be any value. You take an equation take the derivative then take the integral and you don’t get the same
Equation back. You lose information.

So the problem with the information given does have all possible locations as a solution. I don’t think it’s a divide by zero problem.

Spoiler!

The difference with the fly/train puzzle is that the fly will get squashed at different times depending on where it starts. Its position is completely deterministic at any point in time given known initial conditions.

In the dog problem, the dog can be at any location at the same time.

What hammered home the issue for me is the first thought experiment I mentioned - if A starts along the path with a small epsilon headstart (ie. an inifinitely small but non-zero headstart) the problem becomes completely deterministic. But once that headstart hits 0, it becomes non-deterministic.

Another thought experiment: if you take a snapshot at time=epsilon, you will see A has travelled a small amount, B has travelled a smaller amount, and you'd expect C to have travelled between A and B enough times to maintain its speed.

But for C to have travelled between A and B, it would have had to:
1) travel to A
2) travel to B
3) possibly repeat 1 and 2 enough times to ensure distance/time = C's speed.

But think of what is involved in step 1. Right at the start, C couldn't travel to where A is, because A travels slower than C. So it would immediately travel to where B is.

But by the same token, it couldn't travel to where B is by the same argument. It travels 0 distance in 0 time, which is where the divide by zero issue occurs.

It's a tricky problem and, because of the asymptotic nature of it, things like limits don't help.