system
Given the following post, respond as if you're commenting on Reddit:
human
Why does orbital radius increase with increased velocity?

So apparently our teacher said that if a spaceship increases its velocity for a short amount of time its orbital radius increases, and as a result it takes longer for the spaceship to finish a circle.

But according to V^2 = GM/R if the velocity increases, r has to decrease. So mathematically it doesn't out and it's bugging me.
gpt
The easiest way to see why the size of your orbit will increase is to think about energy.  The total energy is the sum of two terms, kinetic energy (which depends only on your velocity), and potential energy, which depends only on your current location, and which grows with distance.  For example, the potential energy outside a mass M is PE = -GM/R, which *increases* as R increases.  If you suddenly increase your velocity, you have increased your kinetic energy, which means you've increased your total orbital energy.  This means you have more energy to climb higher up in the gravitational potential, i.e. your orbit can travel to larger distance.

The expression you wrote, v^2 = GM/R, is only true for circular orbits.  If you are on an initially circular orbit and suddenly change your orbital velocity, your orbit will change, becoming non-circular.  In general, orbits around isolated masses are elliptical; circular orbits are only possible when your velocity has a magnitude equal to the expression above, and points exactly 90 degrees away from the radial direction.  You should not use that expression for orbits that are not circular.