I skip the run-up part, because that's in fact not what interests us. What interests us is that there is a steady-state solution DWFTTW.
schroder said:
Now I would like you to consider the steady state of the cart running in the faster than TT state (motor driven) The cart is already moving in the opposite direction to the TT. The wheel is in constant contact with the TT and is therefore constantly able to extract drive power from the TT and of course the motor which is driving the TT. As long as the wheel does not slip or slide, as long as it continues to roll against the TT, the cart has a continuous source of energy being supplied to it.
You should ask yourself WHY the wheel is able to extract energy from the turntable. Imagine that the system was in a vacuum. Would it now also be able to go against the movement of the turntable ? How does it do so ? (it is correct that the wheel is extracting energy from the turntable in the lab frame, but you should understand that the air is playing a crucial role here).
There is now no blunt force Tailwind, as the cart is moving forward. There is a continuous headwind, which the propeller can make efficient use of to screw into and this also provides a driving force for the cart to continue to move forward indefinitely for as long as the motor is running. It is a true steady state condition.
Because you think it is the headwind which is driving the propeller ? No, the propeller is being driven by the wheel.
Finally, let us now consider the cart in the outdoor (wind powered) situation at DDWFTTW. The cart is already moving downwind, having been accelerated to be going faster than the wind. The cart has lost contact with the Tailwind that was responsible for pushing it up to this state.
Can you tell me why in this case, as you think, the headwind is not driving the wheel as it was (according to you) on the turntable ?
It does not have a continuous source of power being provided to it completely unlike the cart on the motorized TT.
But this is your fundamental error. Because here you don't SEE a motor, you suddenly jump to an argument which *should* make you arrive at a conclusion. It is here that you refuse to make a transformation of one reference frame to another (in another post you even claim that the reference frame *doesn't exist* - that's a bit a strange argument: any reference frame exists if it can be reached by adding a velocity to an existing reference frame).
The wind is there because of a power source. In a wind tunnel, that is the ventilator which provides the wind, and outside, there are natural phenomena mainly driven by the sun who make the wind blow and cost energy. So there IS a power source all right.
The wheel is in constant contact with the ground, and if the cart can be kept moving forwards, the wheel would drive the propeller which would provide a forward acceleration to keep the steady state going.
Yes, exactly as it does on the turntable.
However, without the continuous source of power from the wind, the only thing the wheel/ground interface provides is a source of rolling friction.
Again, that's your error. I pointed it out to zooby also: you have to make your energy balance in one and the same reference frame, and not switch frames when doing so.
And here you did: you first went to the reference frame of the ground, in which the wind has available energy, but not the ground (because it is static), from which you conclude that the ground cannot be a source of energy (IN THIS FRAME). THEN, you switched to a frame that had the same velocity as the wind, and in THAT frame, of course the wind doesn't have any energy (because it is static). However in *that* frame, the ground can give you energy (because the ground moves). But because you did your energy analysis in two different frames (ground frame -> ground no energy ; wind frame -> wind no energy) you conclude that no energy is available and hence the cart cannot have any energy.
Now, the harder part to understand is why the ground is a source of energy in the wind (or cart) frame. And to understand that, you have to consider the system that powers the velocity difference between the two (ground and air). On the TT, that's easy to see, it is the motor of the turntable. In a wind tunnel, it is the ventilator. You can just as well say that the ventilator "drives the wind", but you could also say that the ventilator "drives the tunnel" (in the frame of the air).
I cannot see any way that you can justifiably say that the two steady states are equivalent.
Because you simply need to switch reference frames. You make a galilean transformation of a reference frame attached to the ground, to a reference frame attached to the cart. That is, you express all forces and motions and so on in a coordinate system attached to these things, and the velocities then change by adding the velocity vector of the difference of the two frames to the velocities of one, to obtain the other. And when you make a force diagram and so on in one frame, and then in the other, you see that both are equivalent. As such, the motions will be equivalent, and hence, the situations are.
And there is a property in Newtonian mechanics which tells you that the forces remain the same (if both are inertial frames), and that the properties of energy conservation and momentum conservation, if they hold in one, hold in the other, and that the equations of motion are the same. BUT, don't think that the individual energy contributions in the energy balance are the same !
If you shoot a gun, in the frame of the gun, the bullet gains kinetic energy. In the frame of teh bullet, the gun gains kinetic energy and the bullet loses all of it.
