Do trampolines conserve energy?
Unless Emmy Noether got it wrong, energy is conserved, so I suppose the easy answer is "yes".
I suppose you want me to ask myself if a trampoline is an exception to the law.
If I drop a bowling ball from a given height onto the centre of a trampoline it will bounce for a while, but will soon stop. Hopefully, I can account for all the energy in terms of friction; including ball/air friction, friction between fibres and molecular friction.
An athlete does not come to a stop involuntarily because he/she inputs energy at appropriate junctures. More energy is being put into the system, but there is the additional factor of the rising temperature of the athlete, as well as sustained bouncing.
I see no law breaking here, but if you expect me to do the maths to prove it you're barking up the wrong tree.
