Quote from: WarpTech on 06/06/2015 12:56 AMWhat this shows is that constant Power is proportional to a constant Limiting Velocity, not a constant force. ...(clipped)... Given a constant input power, eventually the mass will reach a constant velocity < c. Paradox solved!I may be misunderstanding, but doesn't this privilege a reference frame? If you watch a spaceship using 1 watt moving to the right limited to 1 m/s, and I am driving left at 9 m/s, then in my frame of reference the spaceship is travelling at 10 m/s. That's the same 1 watt of input power leading to a different speed limit depending on who's watching.

What this shows is that constant Power is proportional to a constant Limiting Velocity, not a constant force. ...(clipped)... Given a constant input power, eventually the mass will reach a constant velocity < c. Paradox solved!

...Oops - you did it again.Quotem*c^2*dy/dt = Pinisn't right. You ought to have the total differential because, recall, you have allowed a variable m(t). ThusP_{in} = c^{2}(y dm/dt + m dy/dt)and so forth...Yes, there's a reason I call myself deltaMass But quite frankly, you only muddy the waters by introducing Special Relativity. Your equations ought to work at all speeds.

m*c^2*dy/dt = Pin

...we getv = 2 v_{p}as the velocity at which the kinetic energy becomes greater than the input energy Notice that this is twice the amount previously calculated by @frobnicat for the case of constant mass.

Quote from: Rodal on 06/06/2015 12:44 AM...we getv = 2 v_{p}as the velocity at which the kinetic energy becomes greater than the input energy Notice that this is twice the amount previously calculated by @frobnicat for the case of constant mass.Not really, I mean, what you are arriving at after an approximation justified by the extremely low delta_m in subrelativistic case (mission profile with delta_v << c) is exactly the same V_{be}=2*v_{p} (with v_{p}=1/(thrust/power), for thrust and power measured "at rest") as obtained in Higgins. The 2* factor is not dependant on "added mass" relativistic effect, even if properly accounted for, it is dependant on the, somewhat arbitrary (ill defined) frame from which "velocity" is understood, and from the scheme considered to assess excess energy :- Constant acceleration (or almost constant acceleration if mass is slightly increasing) at constant thrust/constant power, when is Ke accounted in the starting reference frame > Pin*t ? => V_{be}=2*V_{p}- Constant tangential velocity (around an arbitrarily large track, or even on a linear infinite track) at constant thrust/constant power, when is recoverable mechanical power in the frame of the track > Pin ? => V_{be}=V_{p}. This last scheme can easily discard away any "flow of added mass" due to P_{in} as IR waste heat, so we have a perfectly stationary situation. Those losses don't prevent the cycle to run over-unity (as has been taken into consideration in my numerical examples). Please note also that in this last scheme, the apparent CoE breaking is "instantaneous", it can be observed on arbitrarily small time intervals (comparing powers, not energies). Please note also that in this last scheme, the em drive is not accelerating (in the axis of its thrust, and no other acceleration at all with a linear track), so it is in the exact same conditions as when constant (or near constant) thrust was observed at constant Pin against a spring on balances. Unless frame invariance is broken and pushing 10mN on balance arm at constant "0 speed" relative to lab is different from pushing 10mN on some generator's arm at constant 200km/s relative to some arbitrary track.

I suspected later that I'd misinterpreted your intent - apologies. Nevertheless, the dynamics "paradox" doesn't depend on relativistic speeds to manifest itself, and so I think this gamma relation is a red herring. It has to make correct predictions at a few metres/second also.

Quote from: SeeShells on 06/06/2015 04:05 AMhttp://arstechnica.com/science/2015/06/nasa-wants-to-cut-travel-time-to-mars-in-half-with-new-propulsion-tech/SeeShells, You rock. How do you manage to find such seriously interesting doc's?

http://arstechnica.com/science/2015/06/nasa-wants-to-cut-travel-time-to-mars-in-half-with-new-propulsion-tech/

For 2 N/W, the limit is .5 m/s, which equates to a gravitational potential energy per unit mass of ~.125 J/kg.

Quote from: deltaMass on 06/06/2015 08:10 AMI suspected later that I'd misinterpreted your intent - apologies. Nevertheless, the dynamics "paradox" doesn't depend on relativistic speeds to manifest itself, and so I think this gamma relation is a red herring. It has to make correct predictions at a few metres/second also.It does. It is not required to come close to the speed of light before it is necessary to "hover" in a gravitational field. Not any more than the notion that I am moving faster than I was 10 minutes ago because I'm in an accelerated reference frame whilst sitting in my chair. At "Any" thrust-to-power ratio, eventually there is a limiting speed. For 2 N/W, the limit is .5 m/s, which equates to a gravitational potential energy per unit mass of ~.125 J/kg. Once this speed is reached, regardless that the engine is running at constant thrust-to-power, it is simply "hovering" at at a constant velocity potential. It does not exceed this speed without first increasing the input power supplied. It does not start providing output power over and above what was supplied. This is a prime example of Einstein's Equivalence Principle, where the gravitational potential energy per unit mass is v^2. When the maximum potential energy is reached, it hovers at constant speed. Todd

@Rodal : regarding the straw man aspect of the argument I strongly object. It's not because the apparent CoE breaking is put forward again and again by sceptics (including some sceptics that haven't really gave the experiments and papers the minimum attention span they deserve) that it is less relevant. We don't care about words, paradoxical or not.1/ If the em drive effect is a black box that gives an (averaged) constant thrust for constant power (regardless of constant speed or not constant speed) at better than 1/c ratio, then it can yield more energy than it uses. This is only paradoxical if one can't explain/measure where this excess energy is coming from (what is tapped).2/ If the em drive effect is a black box that don't give a constant thrust for constant power, such that gained Ke<Pin*t, then it becomes not much more useful than, say, a Vasimr scheme where ejection speed is optimised to a given end deltaV goal. If you disagree please set-up a quick comparison with facts and order of magnitude figures. Paradox resolved, so to speak, but all papers that I know of that advocate both em drive for space flight and insisting on apparent respect of CoE (not requiring a mysterious excess energy being tapped) fail to apply the inequality Ke<Pin*t in proposed mission profiles. So, in those papers, it is not a problem of paradox but either unforgivable sloppiness at best, or a deliberate omission of an inconvenient inconsistency in the claims.So, either case, CoE aspects around EM drive claims is not a red herring, it is at the core of why it won't work as claimed, as far as deep space flight is concerned. Proponents say explicitly 2/ and implicitly use 1/ ...[added] BTW I don't see what second law as to do with that...

Quote from: WarpTech on 06/06/2015 01:35 PMQuote from: deltaMass on 06/06/2015 08:10 AMI suspected later that I'd misinterpreted your intent - apologies. Nevertheless, the dynamics "paradox" doesn't depend on relativistic speeds to manifest itself, and so I think this gamma relation is a red herring. It has to make correct predictions at a few metres/second also.It does. It is not required to come close to the speed of light before it is necessary to "hover" in a gravitational field. Not any more than the notion that I am moving faster than I was 10 minutes ago because I'm in an accelerated reference frame whilst sitting in my chair. At "Any" thrust-to-power ratio, eventually there is a limiting speed. For 2 N/W, the limit is .5 m/s, which equates to a gravitational potential energy per unit mass of ~.125 J/kg. Once this speed is reached, regardless that the engine is running at constant thrust-to-power, it is simply "hovering" at at a constant velocity potential. It does not exceed this speed without first increasing the input power supplied. It does not start providing output power over and above what was supplied. This is a prime example of Einstein's Equivalence Principle, where the gravitational potential energy per unit mass is v^2. When the maximum potential energy is reached, it hovers at constant speed. ToddTodd, what should we add to the section on energy conservation in the EM Drive wiki:http://emdrive.wiki/Energy_Conservationas per your latest thoughts (regarding the comparison with a photon rocket and regarding the gamma relationship) ?Is this what you would like to have added?Quote from: WarpTechParadox solved!Constant Power is proportional to a constant Limiting Velocity, not a constant force. You can't have a system where you have constant N/kW and it just keeps accelerating. If N/kW is held constant, then it will reach a limiting velocity. Therefore, the photon rocket equation F = P/c is wrong! No such equation can exist when you are accelerating a mass. Nature won't allow it. Given a constant input power, eventually the mass will reach a constant velocity < c. Note: it seems to me that this "paradox" was never a paradox (a statement that apparently contradicts itself and yet might be true), but like most paradoxes in Physics it just addresses a possible misunderstanding. The misunderstanding of making the unwarranted assumptions that 1) one can have constant acceleration at constant power ad infinitum and 2) that one can ignore the 2nd law of thermodynamics.The paradox can be viewed as a strawman. The so-called typical "attacking a straw man" argument creates the illusion of having completely refuted or defeated an opponent's proposition by replacing it with a different proposition (i.e., "stand up a straw man") and then to refute or defeat that false argument ("knock down a straw man") instead of the original proposition.The different proposition, advanced by this strawman, is that EM Drive necessarily means constant acceleration at constant power (WRONG) and that EM Drive means one can ignore the 2nd Law of Thermodynamics (WRONG).As @Notsosureofit wrote, the arguments behind this "paradox" are circular ("pun intended"). They are based on unwarranted assumptions.All that the "paradox" proves, again, is that one cannot have free energy. It certainly does not prove that there cannot be an EM Drive that respects the 2nd Law of Thermodynamics. The main argument against the EM Drive has always rested on conservation of momentum. That is what needs to be addressed: conservation of momentum. There is no author I know of that has proposed abandoning the law of conservation of momentum. All authors have come up with one way or another to satisfy conservation of momentum, but there is no convincing proof to this date.

Paradox solved!Constant Power is proportional to a constant Limiting Velocity, not a constant force. You can't have a system where you have constant N/kW and it just keeps accelerating. If N/kW is held constant, then it will reach a limiting velocity. Therefore, the photon rocket equation F = P/c is wrong! No such equation can exist when you are accelerating a mass. Nature won't allow it. Given a constant input power, eventually the mass will reach a constant velocity < c.

...Call me slow, but I just see new paradoxes arising. Don't see that this resolves anything.If I am on a spacecraft on a parallel course with the emdrive craft coasting along at the "limiting velocity" (having reached that velocity by conventional means), I now see energy DISAPPEARING from the emdrive - it's power plant is producing power (in my reference frame), but kinetic energy is not changing (in my reference frame). And, as has been asked many times many posts ago, what happens, having achieved "limiting velocity" if I turn the emdrive off?

Quote from: frobnicat on 06/06/2015 04:25 PM...Why do you think those results raise such suspicion about their validity (as a "true" effect) ?I think that they raise suspicion because of conservation of momentum

...Why do you think those results raise such suspicion about their validity (as a "true" effect) ?

A number of elaborate explanations have been advanced, ranging from classical electromagnetics to interactions with virtual particles in the quantum vacuum, in order to explain this result and reconcile it with the conservation of momentum. This paper will not address these issues whatsoever, but rather will examine the consequences in relation to the First Law of Thermodynamics if such a device were to exist....The fact that the EM drive, or any other reactionless drive that has a thrust-to-power ratio greater than a photon-emitting device, would enable a perpetual motion machine of the first kind suggests that such a device cannot exist. This objection is not as easily explained away as the conservation of momentum objection to a reactionless drive

Let's discuss this on objective basis, not on what Higgins writes or having a poll on why people have suspicions. It doesn't matter really if most people are more familiar with conservation of energy than conservation of momentum. Nature has its own laws.