Did you know that it is possible to make a wind-driven vehicle that goes down-wind faster than the wind? Here is an interesting discussion. Some skeptics can never be convinced. They sense intuitively that it violates some conservation law. Conservation of motion is not just a scientific theory, it is something built into our mechanical intuitions. There are actually two different conservation laws that correspond roughly to our intuition: conservation of momentum and conservation of energy. Neither one is really violated by DDWFTTW, they only appear so to the intuition. Our intuitions can lead us astray when considering situations that are well outside of normal experience, and this is one of those cases.
So let me try my hand at making this reasonable. Take a cart with a propeller that is physically connected to the wheels with a gear so that when the wheels turn, so does the propeller. Lets say that for any speed that you move the wheels at, the propeller turns fast enough that it would, by itself, push the cart at half that speed. This does not violate any conservation laws. Suppose that you are pushing the cart at 10 mph in still air. The propeller turns fast enough that it would, by itself be pushing the cart at 5 mph. But the cart is already going faster than 5 mph, so the propeller is not adding any speed to the cart, in fact the drag of the propeller may actually be trying to slow the cart down.
What happens if you stop pushing the cart? Well the only thing driving the cart is the propeller which is trying to drive the cart at 5 mph, but the cart is going 10 mph. So it gradually slows down to 5 mph. But when it is going 5 mph, the propeller is only trying to drive it at 2.5 mph, so it slows down to 2.5 mph, and the cycle continues until the cart halts.
Now suppose the cart is being pushed at 10 mph and there is a 5 mph tail wind. The propeller is now trying to push the cart at ten mph because the speed of the propeller through the air is 5 mph and the propeller is turning fast enough that it wants to go 5 mph through the air. So what happens if you stop pushing the cart now? The cart is going ten mph along the ground. This causes the wheels to turn fast enough to turn the propeller fast enough that it is pushing the cart along at 5 mph through the air --which is 10 mph on the ground. With a 5 mph tail wind, this is a steady state: once you stop pushing the cart, it will continue to move along at 10 mph which is faster than the 5 mph wind that is pushing it.
There is no violation of conservation of energy here: the propeller is slowing down the wind, taking energy from the wind to move the cart. There will be friction losses of course, but those are figured into my premises. That is, if not for friction losses, maybe the propeller could move the cart at .65 times the wheel speed rather than .5 times the wheel speed.
And of course these numbers are just made up. Maybe it's not possible to have the propeller move the cart at one half of the wheel speed. (But if not, then you can do it at 1/3 or 1/4 or something. For any speed you pick, there is some wind speed that will make the cart move under wind power faster than the wind.) And of course I am neglecting the complex interactions of friction and thrust which vary with wind speed and ground speed, but the point of this thought experiment is not that that DDWFTTW is physically possible, but only that it does not violate any conservation laws.
To find out if it is physically possible, you do an experiment.
The very strong skepticism towards DDWFTTW that you see on various forums is caused by the fact that the skeptics think of this as an energy problem when it is really just a leverage problem. There is plenty of energy in the wind to move the cart faster than the wind, you just have to find a way to apply the leverage to make it happen. One way to do that (maybe not the only way!) is to gear the relationship between the wheels and the prop in such a way that the air speed of the propeller for a given ground speed of the wheels will give you a limiting speed faster than a tail wind.