There's nothing like sitting on a sharp tack to make you spring out of your chair. The verb spring means to leap or bounce up suddenly.
When you spring, you jump abruptly. In fact, your body acts like another definition of spring, "a coil of metal that bounces back when pushed." The Old English springan, "to leap, burst forth, or fly up," is the root of spring, and most uses of the word are clearly connected to that origin. Even when an idea springs to mind, or you spring bad news on someone, the thoughts seem to leap forward.