The following has been bugging me recently and I can’t figure it out:
The orbital equation of an object of negligible mass is:
$$v = \sqrt[2]{MG \over r}$$
So in other words, v decreases as r increases.
Yet I know that in order to increase it's orbital radius (let’s assume a circular starting orbit to keep things simple) a spaceship must accelerate at any point for a given amount of time, then again at once it reaches it’s apoapsis (in order to re-circularise the orbit).
So the spacecraft has gone through two burns in order to accelerate and achieve an orbit of larger radius, yet its final orbital velocity is lower!?!
What happened to the missing velocity?
This is more physics than maths, but the answer is that the acceleration is to act in opposition to gravity - while "acceleration" in plain english usually refers to an increase in speed/velocity, in physics it merely refers to a change in velocity, and can affect both the magnitude and direction of the velocity vector. In the case of the first burn, it changes the velocity from that of the circular orbit to the more elliptical one, and then in the second burn it adjusts to the new orbital velocity.