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Yes. It would be called a helicopter. Full sized ones can have fuel engines, which can carry more energy per weight and can go further. On the other hand, they are heavier and not as durable because of that weight. They cannot change directions like this mid flight.

Can you explain why?

Is it to do with the bulk properties of the materials not varying with scale? Or scaling with some power of length < 3?

If you tried to spec a full sized helicopter engine to make speed changes like those in the video would it tear itself apart?

Yes.

Mass increases with the cube of scale, while structural material cross-section area increases only with the square of scale. So insects can have miniscule legs, while elephants have tree trunks, proportionally.

Right, sometimes called "square-cube law" (http://en.wikipedia.org/wiki/Square-cube_law). Worth knowing about so you can explain to yourself the differences noted above.

Think about it. Ants can carry six times their weight, while larger creatures could never accomplish that. I don't know the physics behind it, but it's not hard to see that small things can have greater strength per mass than large things.

This one has a gasoline engine

It's battery powered, see the battery around 4:25

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