In conclusion, part of the power needed to do the (156 N)(9 m/s) of work per second which is being done on the cart can indeed come from the propeller, just as part of the power needed to run the car's motor can come from the generator. It doesn't all have to come from the wind. So, it is in fact correct to add the energy loss of the wind to the power provided by the propeller, to get the total work done on the cart by the 156 N of force on the propeller.
Yes indeed, and if you followed my calculations you will see where I have done that. I added the 350 Watts at the propeller to the MAX amount of power that can be extracted from the wind (given the parameters of the problem) and the total was about 1,350 Watts , rounded UP, and it was well short of the 1,440 Watts needed.
In fact, the calculation can be made even simpler, and the result for the cart is much worse!
All we need to do in order to prove
or disprove this claim, is examine the two fundamental laws of conservation, The Law of conservation of Momentum, and the Law of conservation of Energy.
We can do that quite easily with the example we have been examining, with the 295 Kg cart moving at 9 m/s directly down wind in a wind of 7 m/s. We already calculated the air mass to be 72 kg, and the amount of acceleration given to the air by the propeller to be 2.2 m/s.
Now it is a simple matter to calculate the initial and final velocities, momentum and energies to see if it all agrees with the two great conservation laws:
Initially, the cart is moving at 295 kg cart is moving at 9 m/s and has a momentum of
2,655 kg x m/s
Initially, the 72 kg block of wind is moving at 7 m/s and has a momentum of
504 kg x m/s.
The Total momentum of this system is
3,160 kg x m/s and this must be conserved.
After being worked on by the propeller, the air mass is slowed by 2.2 m/s to a velocity of 4.8 m/s
Finally, the 72 kg block of wind, moving at 4.8 m/s has a momentum of
346 kg x m/s.
To conserve momentum, the cart must have a final momentum of 3,160 – 346 = 2,814 kg x m/s.
The 295 kg cart’s final velocity must be 9.54 m/s having a momentum of
2,814 kg x m/s.
So far, the law of conservation of momentum has been observed, but in order for this to be valid, the law of conservation of energy must also be shown to be conserved:
Initially, the 295 kg cart, moving at 9 m/s has kinetic energy of
11,950 Joules
Initially, the 72 kg block of wind, moving at 7 m/s, has kinetic energy of
1,765 Joules.
The total energy of the system is
13,715 Joules,
and this too must be conserved.
After being worked on by the propeller, the air mass is slowed by 2.2 m/s to a velocity of 4.8 m/s.
Finally, the 72 kg block of wind, moving at 4.8 m/s, has a kinetic energy of
830 Joules.
Finally, the 295 kg cart, moving at 9.54 m/s, has a kinetic energy of
13,425 Joules.
The Total kinetic energy of the system is
14,255 Joules,
and that is 540 Joules more than the system started out with.
There has been no input of energy from any other source, as all calculations are done in the frame where the ground is at rest and has no energy. The cart has gained more energy than the wind lost, based on the relative velocities and conservation of momentum.
Conservation of energy has been violated, and that is not possible.
I invite anyone here to provide a similar method of calculation that shows the cart can go directly down wind, faster than the wind, and not be in violation of the two great laws of conservation. I have shown you the cart, if it did as claimed, is in violation of these fundamental laws of physics. Now the burden of proof is on the carteers, to show me that it isn’t. Use whatever method you like for propeller thrust and whatever efficiencies strike your fancy, but please be good enough to show your calculations!