The 'bounce' of a trampoline is generated by the movement of the trampoline springs, situated around the frame of the trampoline and connected to the trampoline bed.
The more trampoline springs that a trampoline has, the better bounce the trampoline will generate. The quality and effectiveness of a trampolines' bounce is also impacted by the amount of movement there is on the bed, more weight again generates more bounce.
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It’s the springs that give a trampoline its bounce. The mat simply connects all the springs together and gives a soft surface to play on.
The springs store your kinetic energy as potential energy by stretching, just like a rubber band. The further you stretch a spring the harder it is. Without the springs it would just be a mat without bounce.
Trampolines as an energy converter
Fundamentally a trampoline is an energy converter. When you jump your muscles are converting your stored potential energy into kinetic energy. As you land the trampoline gets tighter and the springs stretch, safely converting your kinetic energy (jump) into potential energy ready for use.
Once the trampoline has converted and stored all of your kinetic energy it gives it back to you by converting the potential energy from the springs back into kinetic energy and springing you up into the air.