Cats are uniquely adapted for jumping and running. Unlike the spinal columns of humans which possess vertebrae that are held together by ligaments, the vertebrae of the spinal columns of cats are held together by muscles. This allows cats to bend their spinal columns into a perfect U shape. Cats can flex and extend their spines in a manner that allows their bodies to act like a spring when running or jumping. Their powerful hindleg muscles also allow them to generate large forces, and the length of their hind limbs relative to their lean body mass maximizes their takeoff velocity when jumping.
In the video below, Science Cat performs a cat jump.