1) A molecule moving vertically towards the plate with velocity -u in the ground frame has a velocity -u-v in the reference frame of the plate and hence bounces back from the front side with the velocity u+v ( the plate (prop) is more massive than the air molecules). so in the ground frame the velocity of the molecule is now u+2v.
2) At the backside of the plate, a molecule with velocity u in the ground frame has a velocity u-v in the reference frame of the plate and thus after reflection, a velocity -u+v
In the ground frame the velocity is now -u+2v. T
3) This means in the ground frame, the kinetic energy of the molecules being reflected from the front has increased from m/2*u² to m/2*(u+2v)² =m/2*u² +2*m*u*v + 2*m*v², i.e. an increase of +2*m*u*v + 2*m*v².
4) Molecules reflected from the back side, have changed their energy from m/2*u² to m/2*(-u+2v)² = m/2*u² -2*m*u*v + 2*m*v²
i.e. a change of -2*m*u*v + 2*m*v².
Adding and averaging (3) and (4) gives the average increase of the kinetic energy of the molecules with mass m hitting the plate with velocity v as
ΔK = 2*m*v².
A plate traveling at v wrt the air, will lose KE. So will the cart.
To suggest otherwise is to claim the cart can be pushed in still air and spontaneously accelerate.
http://blog.makezine.com/archive/2010/11/what-ive-learned-about-wind-carts.html
I wondered what a physics professor would make of all this, so I emailed Dr. Paul J. Camp, a professor in the physics department at Spelman College in Atlanta, Ga. Camp had helped me with other physics brain-teasers in the past and I appreciated his clear explanations. I asked him to take a look at Goodman’s video, as well as Charles Platt’s article, and the treadmill cart videos. “What do you say?” I asked.
“Impossible,” replied Camp. “Would violate conservation of momentum and conservation of energy …
In fact, we can state this in pretty bare terms — for a car moving downwind at wind speed to go a little faster is physically indistinguishable from a car at rest on the ground in stationary air to suddenly leap into motion. What can be done in one inertial frame can be done in any other inertial frame with the same physical circumstances
A is a stationary cart in still air
B is the cart above WS
C is the above conclusion, based on Spork's claim.