Let's say we have two planets at a stand still within reasonable distance of each other. They will accelerate towards each other and subsequently collide.

If instead we give them a sufficient (but finite) initial velocity in opposite directions orthogonal to the path between them, they will instead enter into a orbit around each other. In this orbit they will experience continuous acceleration. Thus, for a finite initial velocity I get in return a continuous (and thus infinite, given infinite time) acceleration in return.

Acceleration is work and work takes energy. The energy is kindly supplied by gravity. Is it correctly understood that energy is continuously put into the system, in order to maintain the orbit? And that gravity is thus an infinite source of energy? And that a system of planets + gravity, if given the right initial condition, constitutes a perpetual motion machine? Even if so, it could still be principially impossible to extract surplus energy from the system, eg. for practical uses.

## Best Answer

You go wrong when you say "acceleration is work". Accelerating an object only

sometimesrequires work. Work is only done by a force when the object moves in the direction of the force. If you know what the dot product is then the relevant formula is:\begin{equation} W = \int_{t_0}^{t_1} \vec{F}\cdot \vec{v} dt \end{equation}

If you don't know what the dot product is then all you need to know is its zero when the two directions are at right angles to each other. Taking a circular orbit as an example you can see that the force of gravity acting on the planet is always at right angles to the velocity so the work done is zero.