I know that planets can't have infinite energy, due to the law of conservation of energy.
However, I'm confused because I see a contradiction and it would be great if someone could explain it.
Energy is defined as the capacity to do work. Work is defined as Force x Distance. Force is defined as Mass x Acceleration. Thus, if we accelerate a mass for some distance by using some force, we are doing work, and we must have had energy in order to do that work.
In orbit, planets change direction, which is a change in velocity, which is an acceleration. Planets have mass, and they are moving over a particular distance. Thus, work is being done to move the planets.
In an ideal world, planets continue to orbit forever. Thus, infinite work will be done on the planets as they orbit.
How can infinite work be done (or finite work over an infinite time period, if you'd like to think of it that way) with a finite amount of energy?
Where is the flaw in this argument?