How can cats jump so high?

The answer to the question is all about mechanics and muscles. The domestic cat and indeed the wild cats have lots of fast-twitch fatiguing muscle cells. These cells are designed to propel the animal quickly; to allow the animal to accelerate quickly, but they're not very good at prolonged effort because they tire. The cat is a sprinter rather than a long-distance runner such as the African wild dog, mule or horse.

The puma is perhaps the best jumper of the large wild cat species. Image: PoC.

The second reason is all about the mechanics, the leg bones, particularly the hind leg bones of the domestic cat which are long. The muscles working to move these long levers allow the cat to jump so high. The concept of long levers applying a stronger force than short levers can be seen in bolt cutters. They have long handles. And if you want to unscrew a nut which is rusted in or very stiff, if you use a long spanner you will be more successful than if you use a short one.

And of course the hindquarters of a domestic cat and the other cats such as the mountain lion are very powerful. When these muscles contract quickly combined with long levers the cat is propelled forward either to attack an animal or leap upwards. 

There is also the matter of power-to-weight ratio. The heavier cats will be less good at jumping than the lighter species. The lion, for instance, is not a great jumper relative to the leopard which is considerably lighter.

The lion is built to grab hold of large prey and subdue them. This is why their arms (forelegs) are immensely strong much like the tiger's. This power-to-weight ratio is best seen in the caracal which is a medium-sized wild cat species. This is the cat that jumps the highest. They have an ideal power-to-weight ratio. The picture below is of a caracal catching or trying to catch a bird (or an object thrown by someone to simulate a bird) in flight:

Caracal leaping to catch a bird in flight. Picture in public domain.

All the medium-sized cats are probably more adept at jumping vertically than the big cats. There is too much inertia to overcome in a big cat and the power-to-weight ratio is not as good as for the smaller species.

And one of the factor which may play a role as to why cats can jump so high is the floating shoulder. The forelimbs are connected to the rest of the body by muscle. The cats clavicle floats and is anchored by muscle. This allows cats to lengthen their stride and it enhances the range of motions that cats have. This indirectly helps a cat to jump well.

The cat's foot is elongated and it looks like a leg bone but it isn't. At the end are the toes and cats walk on their toes because they are digitigrades. I would argue that this also adds to the leverage that a domestic cat has which once again supports the ability to jump high and horizontally over long distances.

The video on this page is of a female F1 Savannah cat whose name is Magic. I don't know whether she is alive still but at the time she was the tallest domestic cat to the shoulder in the world. Savannah cats of this type i.e. first filials have a serval father and the serval has the longest legs to body size of all the wild cats. That's why the F1 Savannah cat is such a fantastic upwards jumper. Once again it's about leverage using long levers.