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.