# What prevents this infinite energy generator from working?

I thought up a device that uses the Oberth effect (wiki) to generate more energy than it consumes. Obviously, I made a mistake/logical error somewhere. Can you help me find it? Here's the thought experiment:

• Suppose we have a mass $$m$$ that can move in a straight tube without any friction. Also, we disregard gravity and any relativistic effects

• The mass is accelerated by a stationary laser at one end of the tube. A laser beam is directed to the mass, which acts as a light sail, resulting in the thrust force: $$F$$

• The laser consumes a constant amount of power: $$P_{laser}$$

• The mass is accelerating constantly at: $$a=\frac{F}{m}$$

• The mass's velocity is increasing by: $$v(t)=at$$. Inserting $$a$$ from above: $$v(t)=\frac{F}{m}t$$

• The kinetic energy of the mass is given by $$W_{kin}=\frac{1}{2}mv^2$$. Inserting $$v$$ from above: $$W_{kin}(t)=\frac{1}{2}m(\frac{F}{m}t)^2 = \frac{1}{2m}F^2t^2$$

• The accumulated energy consumed by the laser over time is $$W_{laser}(t) = P_{laser}*t$$

• The kinetic energy grows quadratically with time, while the laser energy grows linearly with time. This means that the kinetic energy will eventually exceed the laser energy

• Energy "break even" is reached, when $$W_{kin} = W_{laser}$$ That is, when $$\frac{1}{2m}F^2t^2=P_{laser} t$$ at time: $$t=\frac{2m}{F^2}P_{laser}$$

After this point in time, leaving the laser on for longer will increase $$W_{kin}$$ by more than the energy the laser consumes. For any combination of mass, laser power or light sail force, this point will be reached if we wait long enough.

So let's leave the laser on for a bit longer than the "break even" point. Then, the mass with $$W_{kin} > W_{laser}$$ is decelerated, and $$W_{kin}$$ harvested by a perfectly efficient linear generator (or any other suitable generator). It charges a battery which provides enough energy for the laser in the next run and even additional energy. (The generator and the battery can even be arbitrarily inefficient, as long as the device runs long enough until $$W_{kin}$$ is sufficiently larger than $$W_{laser}$$.)

This seems impossible. Where did the additional energy come from? What is going on here?

• I'm pretty sure the Oberth Effect is for rocket engines that actually expel propellant to move, so I don't think it's really the Oberth effect when you use radiation pressure. Also, you may want to look up the effects of velocity on radiation pressure, which might be part of the disconnect.
– JMac
Dec 25 '20 at 16:09

That happens because $$P_{diss}$$ depends on t (fact that you are not considering), infact $$P = Fv = F*\frac{Ft}{m}$$ and knowing that $$W_{laser} = \int P dt = \int \frac{F^2t}{m}dt = \frac{F^2t^2}{2m} = W_{kin}$$. So the conservation of energy is verified.