chingel said:
How can the thrust be higher than the drag at the wheel? Aren't they are mechanically connected?
Thrust is a force, and drag is a force. You do know that by using a lever or gears, you can put x amount of force into one end and get 2x, 3x, or 10x force out of the other end right?
Gears are mechanically connected yet change the amount of force put in at one end to a different force at the other; no magic, just leverage or mechanical advantage.
chingel said:
Move air back at v/2 relative to what, the cart or the ground?
Relative to the cart.
chingel said:
If you use gears to make the propeller move faster than the wheels, it will displace more air but also drag the wheels more, because it does more work and needs more energy, if you make it spin slower it provides less acceleration but also causes less drag, at least the way I understand it.
But we use gears to
slow down the propeller, displacing less air but supplying more force for the same amount of work taken from the wheels.
For the sake of clarity, let's take losses out of the picture for a moment.
The wind is blowing at 10 mph. The cart, without the propeller engaged, gets pushed up to 10 mph. The pitch of the prop and the gearing of the cart is set up to push air back relative to the cart at half the speed of the wheels. The prop is engaged; the air around the cart gets pushed back at 5 mph, half the speed of the wheels. Since the force at the wheels is geared down, the force at the prop is twice as much as the drag at the wheels and the cart accelerates. F=MA so if the net force acting on the cart is positive, as it is so far, the cart will accelerate.
The cart gets to 15 mph; the prop is now turning fast enough to push air back at 7.5 mph and the force at the prop is still higher than at the wheels, so the cart will still keep accelerating. About now you're thinking that if that is true,the cart will keep accelerating forever; it won't, because the gear ratio determines the top speed in accordance to the speed of the wind.When the cart gets to 20 mph, the prop is pushing air back at 10 mph and the wind is still blowing at 10 mph, meaning that the prop is no longer exerting any force on the air, it's just freewheeling. With this gear ratio, the absolute best the cart can travel across the ground is twice the speed of the wind.
Now the wind drops to 7.5 mph; what happens to the cart? At 20 mph, it is still pushing air back at 10 mph but with the wind at only 7.5 mph, that means that instead pushing air back, it is actually trying to drag air forward at 2.5 mph. The net force acting on the cart is now negative, so the only thing that can happen is that the cart will slow down. What speed does it stabilize at? Again, twice the speed of the wind, or 15 mph.
Gearing the wheel speed down by half (0.5) to power the propeller gives a theoretical top speed of twice the wind speed. A different ratio will give a different theoretical multiple of the wind speed. Gearing the cart down less means the prop is spinning faster but with less force; it has the potential to faster for the same wind speed but it needs to be more efficient because the prop is supplying less force than before to overcome the real world losses.