How can cats jump so high?

Cats can jump high because their hind legs are made up of long levers and they are driven by fast-twitch fatiguing muscles. Long levers create speed. Think of those plastic devices in which you placed a tennis ball at one end which are used to throw the ball a long distance for your dog to chase. This plastic extension to your arm with a clamp at the end to hold a tennis ball extends the leverage of the arm. In extending the leverage the ball is thrown further and with less effort. Long levers increase the mechanical advantage.

Caracal jumping vertically. They are perhaps the best vertical jumper of all the cats. Photo: public domain.

As for the muscles, they mainly consist of fast-twitch fatiguing cells. These muscles propel cats to leap several times their own length in a single jump. They use up all their energy in that moment. These levers and fast twitch muscles also allow the domestic cat to reach impressive speeds of about 48 km/h or 30 mph, maximum. 

Running at speed is enhanced by the fact that the domestic cat has a floating shoulder. Unlike a human's collarbone which connects the shoulder and breastbone the domestic cat's vestigial clavicle floats and is only anchored by muscle. This allows a freedom of movement which lengthens the cat's stride and range of motion.

Cats need to be good jumpers because they like to be high up for safety. They live in a vertical world and many wild cat species like to climb trees. I guess you know all that. The leopard takes large prey items into trees which reminds us of this cat's enormous strength. The small margay lives in trees and has a double jointed ankle to allow them to hang onto branches like monkeys. Yes, the cat is a beautifully engineered predator and their muscles and levers maximise their jumping ability. Finally, the medium-sized wild cat species, the caracal can jump the highest of all cats and certainly relative to their size vertically from a standing start (see picture).