Suppose I had two trampolines both 48" in diameter and both 1 foot off the ground. The only difference was the spring tension.
The weaker of the two will bottom out if a 100# ball is dropped from 5 ft and the stiffer will bottom out with a 10 ft drop of the same ball.
Will the two send the ball back up to the exact same height if dropped from the same distance?
How would the stiffness affect a jumper? More jumps to reach a given height? Same height attainable with both if not bottomed out?
Unlikely. There are frictional losses involved in stretching the fabric and springs and in moving air out of the way. These are likely to be meaningfully greater for the “softer” trampoline.
Assume Zero hysteresis
And zero air resistance? No energy lost to friction?
Yes Basically just a load in load out type question. I am pretty sure the two would be about equal with something dropped but not so sure about how it would react to a jumper.
Assuming zero hysterisis and other dissipative effects, then there will be no dissipation in either. But that’s a trivial enough answer that I suspect that it’s not actually what you were asking.
There is one actual difference … Consider something passively bouncing up and down to the same height , because there is no friction sound or destruction of the trampoline or anything.
The less stiff trampoline will give the jumper more time in negative range - (where +ve means in flight… not touching ) - so that extra time will be added to time each jump takes. The less stiff tramp gives a longer period, lower frequency.