# Infinite acceleration

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.

• Have you thought about why the earth is not infinitely accelerating by this logic? – ACuriousMind Aug 1 '15 at 23:53
• Well, it is? It wants to go straight, but is constantly forced into another path. That takes acceleration. – loldrup Aug 1 '15 at 23:57

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

$$W = \int_{t_0}^{t_1} \vec{F}\cdot \vec{v} dt$$

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.

• I see. No work is done in the system. But still, acceleration happens. Can acceleration happen without a constant influx of energy? – loldrup Aug 2 '15 at 0:02
• @loldrup: Yep. The only accelerations which don't require energy are those at right angles to the velocity (where your direction changes but not your speed). If you change the speed at all then work is done. – or1426 Aug 2 '15 at 0:06
• How come some accelerations take energy and some don't? If you could isolate a given instance of acceleration and study it in the lab, could you then dissect it and tell if it was an energy free acceleration or an energy requiring acceleration? – loldrup Aug 2 '15 at 0:14
• @loldrup. As I said previously accelerations which change your speed require energy. Accelerations which only change your direction do not. – or1426 Aug 2 '15 at 0:23
• There is no fundamental difference between "energy-free acceleration" and "energy-requiring acceleration. Which "type" the acceleration is depends on your reference frame, just like energy depends on your reference frame. But in all reference frames, energy is conserved. – knzhou Aug 2 '15 at 0:53

I think you need to read up a bit on vectors and scalars, how we define things like acceleration.

Vectors are physical quantities that have a magnitude and a direction. Velocity is a vector (speed is its magnitude). Acceleration is also a vector. Acceleration is defined as the rate of change of velocity, basically it measures how velocity changes with time.

When something moves in a circle, its speed does not change, but its direction does therefore velocity also changes.

It can accelerate like this forever, but this does not mean it has infinite acceleration.

Energy only changes when speed changes, because energy and speed are both scalars. So energy is not constantly being put into the system to keep a body in constant motion, work is only done when a force does not act perpendicular to the objects velocity.

• I know all this. What I meant with 'infinite acceleration' was that the integral of all that finite acceleration, will grow to infinity over infinite time. I don't understand why you consider your answer for an answer. The basic message is still: 'some kinds of acceleration are considered energy free and I'm not telling you why'. – loldrup Aug 10 '15 at 5:44
• The KE energy of a particle changes with speed . If an acceleration changes the direction of speed and not the amount of speed then energy won;t change and energy will not be needed to cause the acceleration. Sorry if that is just repetition, but I don't understand how else to explain that some accelerations and energy free. Sure, if you push something travelling towards you in the opposite way, it will first stop then move in the opposite way. Then you could say energy changed, but there is no difference between acceleration that causes energy change and one that does not. – Shaurya Bhave Aug 11 '15 at 13:37
• But energy has to remain conserved. I hope that is clear, when you say the integral of finite acceleration with respect to what? A finite acceleration will increase change velocity indefinitely if it was to go on forever. – Shaurya Bhave Aug 11 '15 at 13:45
• But were accelerating orthogonally to the velocity vector, so, I can't see how we should achieve infinite velocity. The integral of the acceleration will go to infinity, but velocity will not. Conservation of energy is what is at question here. It would be inappropriate to assume the question were trying to answer. – loldrup Aug 11 '15 at 20:01
• Sorry my mistake, I accidentally used the word increase. I meant change velocity indefinitely. This can mean it increases the speed, or change the direction or a combination of both. For objects moving in circular motion, it will just change the direction. So speed will not go to infinity. I still do not understand what you mean by integral of acceleration will go to infinity. The integral of acceleration is velocity. – Shaurya Bhave Aug 13 '15 at 15:27

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?

Consider the general case of one particle orbiting another in an ellipse (this is general because we can a always reduce the two body problem to an effective one body problem, and the general closed orbit is elliptical). The total energy is constant, with energy being exchanged between the kinetic energy of the orbiting particle and the gravitational potential energy of the system. Without external forces there is no energy input into the system.

• Given a perfectly circular orbit, there's no exchange between kinetic and potential energy, yet acceleration still happens. I don't understand why you consider your answer for an answer. The basic message is still: 'some kinds of acceleration are considered energy free and I'm not telling you why'. – loldrup Aug 10 '15 at 5:47
• I've edited my answer to reference the specific questions that I was attempting to address. If you feel my answer is insufficient, please let me know how you think I can improve it. I think it is uncharitable to think that anyone is withholding information from you. A system can accelerate when no work is done. or1426 has presented the expression for work from which you can find whether or not work is done in a given dynamical situation. – d_b Aug 10 '15 at 17:29
• I accept that no work is done in an an orbit. If we accept the premise that acceleration orthogonal to the velocity vector is 'energy free', then we can also agree that no energy is continuously put into the system. Thus the central question becomes: Why is 'velocity-orthogonal' acceleration considered energy free? – loldrup Aug 10 '15 at 20:39

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