Does using a trampoline conserve energy? In a thought experiment, we have a trampoline on the ground (well it's fancier than a spring) and a ball which is made to undergo free-fall from a height $h$.
In a non ideal scenario, will the ball bounce back to the same height $h$
?Its definitely not higher, so does the strength of the trampoline(or spring force constant $k$) change the max height? That is, can it account for the drag loss due to air?
Possible eqns:
$$mgh+\frac{1}{2}kx^2=F_{drag}(h+x)+mgh'$$
Will the energy loss due to this air drag be compensated by the trampoline? Is $h=h'$ ?
This is done in a real setting, not an ideal one.
 A: Energy losses:

*

*The single largest one is the air resistance of the mat.  Each bounce (of a person sized mass) pushes a volume of air, down, then up.  Competition grade mats are over 50% space. Instead of moving a parcel of air feet, you are moving it fractions of a cm.


*The second source of loss:  On the rebound when the mat is nearing it's flat position, it stops accelerating.  At this point the mass leaves the mat, but the mat is still moving.  A competition mat is about 10 Kg.  A garden trampoline mat about twice this.  The springs are heavy, but they don't move much.  Some people claim they can feel the difference using a shorter, fatter, but lighter spring. Such a spring has less mass, and being shorter, less movement.  Preliminary back of the envelope calcs suggest this is a VERY small effect.


*On a cheap trampoline, the fabric of the mat is stretchy.  Most stretchy fabrics have a high hysteresis, so a lot of the energy put in  isn't recovered.
I have a good garden tramp.  With a reasonable jump it takes about 7 bounces with straight locked legs to damp to the point where I'm not leaving the trampoline.
I also use an olympic tramp at my gym.  The height I can get at maximum effort at home, I can get just using my calf muscles at the gym.  It feels like I'm getting more than 90% of my height back, but this may be an illusion.  I do know that it takes a lot longer to come to rest on a crash and burn.
A: First think about the ideal system:
State 1) the ball is at a height h, the trampoline is at rest. So the ball has potential energy and the trampoline has none.
State 2) The ball just hit the trampoline, the ball has kinetic energy and potential energy (yes the ball will still travel a bit until maximal deformation). the trampoline is still at rest so it has no potential energy.
State 3) the ball has deformed the trampoline to its maximal deformation, the ball has no kinetic energy nor potential energy. All the energy is "stored" in the trampoline deformation so the trampoline has some potential energy.
State 4) The trampoline has bounce back the ball and the ball is just about leaving the trampoline. So the ball has some kinetic energy, but the trampoline is now also moving so it also have a bit of kinetic energy!
State 5) the ball goes to its higher point so it has now only potential energy, but the trampoline is now bouncing so it also have some energy...
Therefore even in the ideal case (without friction and without heat transfer due to the deformation of the trampoline) the ball will never be able to reach back the same height... 
