Why do cats always land on their feet? 

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Just adding some details on how the cat manages to turn its body while preserving its angular momentum.

Rotating different parts of his body at different times in the right direction isn't the reason why the cat manages to preserve its angular momentum. The reason for so is that there is no external torque acting on the cat! So, no matter what the cat does, as long as he doesn't touch anything (let us consider the fall to be such that the air resistance is negligible here), his angular momentum will be constant.

What the cat really does is use this angular momentum conservation law. Cats are able to turn their "upper" and "lower" body in opposite directions with no difficulty at all (they are very flexible). To understand how it all happens, let's see what a cat does while falling upside down.

First, the cat notices he is upside down, and he does so with his eyes or with the gyroscope in his ears. Then, he will divide his body (front and back) into two separate rotation axes that are tilted from one another. While falling down, he starts turning his front legs to the ground (it doesn't matter which direction he chooses to do so) and the back legs will turn in the opposite direction due to the conservation of momentum angular. Problem is, if he does so, he might end up with his body twisted and this could be a problem! So, to avoid this, when turning to the front, he will extend his back legs while tucking in his front legs. By doing so, he increases the moment of inertia of the back and decreases that of the front. This causes the front to spin faster than the back! When the front is aligned with the ground, he extends his front legs to increase the angular momentum and stop the rotation. Finally, he twists the back legs along the rotation axis, returning them to the original position and bracing for impact (all legs extended)!

You can watch the awesome video listed as a source for a really good explanation along with a real cat falling in slow motion!

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Well, they don't always---sometimes cats do fall on their backs. But they definitely land on their feet more often than most animals (including humans) would under similar conditions.

This is because cats have a reflex that allows them to rotate their bodies in mid-air extremely fast; we actually didn't understand how this works until we invented slow-motion cameras that let us watch it happen.

What's really amazing about this reflex is that the cat does not "push" against anything; there is zero net force on the cat's body and thus zero net change in angular momentum. By rotating different parts of their body at different times in the right direction, the cat is able to rotate without ever changing their angular momentum or the position of their center of mass. A cat could rotate in outer space if they wanted to (though without gravity, they wouldn't know which way to point!).

Why and how did they get such a weird reflex? Our best guess is that they evolved this reflex because it helped them survive falls during a period in which their ancestors primarily lived and hunted in trees. Cats that were a bit quicker at rotating were more likely to land on their feet, more likely to survive, and thus more likely to reproduce. Over perhaps 10,000 years or so, the ability to rotate in this way became fixed in the cat population.

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