Well that explains why you got the wrong answer! Springs, as you now point out, work opposite the way gravity does, in that the longer a spring is, the more energy it take to continue to deform it. (Assuming we mean an ideal spring, not one that’s going to switch to plastic deformation at some point.) So if we were talking about springs, you would be correct that the most efficient time to teleport the spring longer would be when it’s already as long as possible.
But we are not talking about springs, we are talking about gravity, which works differently. (Not only is the function going in a different direction, but also at a different rate. Gravity decreases as the inverse square of the distance, whereas spring force increases linearly with distance.) So your “simplification” is just wrong. You stated:
A weird consequence. Say our spaceship didn’t have a rocket, but instead it had a machine that teleported the ship a fixed distance (say 100m). (A fixed change in position, instead of a fixed change in momentum). In this diagram that is just rotating the arrows 90 degrees. This implies the most efficient time to use the teleporting machine is when you are at the maximum distance from the planet (minimum kinetic energy, maximum potential). Mathematically this is because the potential energy has the same quadratic scaling as the kinetic. Visually, its because its where you are adding the new vector to your existing vector most efficiently.
This is false. It takes more energy to move an object up by 1 meter on the surface of Earth than it does a million km away, because gravity gets weaker as you go further away. So if you want to maximize the gain in potential energy you get from your teleportation machine, you want to use it as close to the planet as possible.
(An easy way to see why this must be true is that an object’s potential energy at infinity is finite, so each additional interval of distance must decrease in energy in order for the sum of all of them to stay finite.)
Well that explains why you got the wrong answer! Springs, as you now point out, work opposite the way gravity does, in that the longer a spring is, the more energy it take to continue to deform it. (Assuming we mean an ideal spring, not one that’s going to switch to plastic deformation at some point.) So if we were talking about springs, you would be correct that the most efficient time to teleport the spring longer would be when it’s already as long as possible.
But we are not talking about springs, we are talking about gravity, which works differently. (Not only is the function going in a different direction, but also at a different rate. Gravity decreases as the inverse square of the distance, whereas spring force increases linearly with distance.) So your “simplification” is just wrong. You stated:
This is false. It takes more energy to move an object up by 1 meter on the surface of Earth than it does a million km away, because gravity gets weaker as you go further away. So if you want to maximize the gain in potential energy you get from your teleportation machine, you want to use it as close to the planet as possible.
(An easy way to see why this must be true is that an object’s potential energy at infinity is finite, so each additional interval of distance must decrease in energy in order for the sum of all of them to stay finite.)