Alright so…I'll take a stab at this. Hopefully someone who understands this better will help.

Basically, your biggest determining factor in V speeds is weight. That's almost it, actually. As u/pinkdispatcher mentions, V1 can never be higher than Vr. At some point, at any (legal) weight, the plane is coming off the ground. At some point on your way up to rotation, it's considered more dangerous to stop the take-off than continue, even if something's on fire. That point can be very low if you're very heavy (harder to stop a heavy plane) or faster if you're light (easier to stop) and that has a lot to do with structure, brake energy, and climb considerations with respect to terrain (u/E2TheCustodian touched on that).

Now, hopefully through my mediocre explanation, you can see that the actual length of the runway doesn't really come into that. Because it's about how close you are to flying speed based on weight. The way runway does factor into this is what your maximum permissible takeoff weight will be (which will then effect the V speeds for the reasons I already mentioned). If you have a longer runway you have more space to build and subsequently stop the plane. That means you can allow for a heavier weight. If the runway is shorter, V1 won't fundamentally change for a given weight but you might remove weight so that your speeds, and thereby distances required, shorten accordingly.

TL;DR - It's a blend of structural limits, terrain clearance considerations on climb out and enroute with consideration to a possible engine failure, and weight that determines V1. The actual length of a piece of pavement doesn't apply. You can either make it work on that runway or you need a longer one.

I hope that made sense...