Re: Spring stiffness versus maximum height in a trampoline

According to Hooke's law, the magnitude of the restoring force is equal to 
the spring constant times the displacement and in the opposite direction 
(F = -K * dX).  But does a stronger/stiffer trampoline mean a higher jump?
That is a pretty good question because it is subtle.  Let me simplify it 
considerably.

Basically, we are trying to figure out what property of the trampoline and 
its motion that gives the biggest boost to the jumper's leap.  Once the 
jumper is in the air, we can think of him as a simple ballistic object.  
So I suspect, the most important quantity is his initial velocity (or 
speed in the upward direction).

Imagine a very rigid trampoline with a large spring constant.  You might 
imagine taking it to the absurd limit of almost rigid like a slab of 
concrete or wood floor.  In this case, the jumpers maximum height is 
entirely provided by his natural leaping ability.

At the other end of the spectrum is something like a rubber sheet which 
stretches a bunch but has very little restoring force.  That is pretty 
difficult to leap from as well.

So, the answer must be somewhere in between the extremes.  In practice, 
the highest jumps comes from good timing between jumping and the bounce-
back of the trampoline.  The jumper leaps when the trampoline is adding 
the greatest speed to his jump, which occurs right around the middle of 
the oscillation of the trampoline, just like other oscillating devices 
like a pendulum or a spring, or a spring loaded firing device like a 
crossbow, slingshot, or bow and arrow.  Oscillators have their greatest 
speed at the center of their range of motion.  For example, a pendulum is 
moving the fastest at the center of its swing and slowest (zero speed) 
when turning around.  A slingshot or bow works pretty much the same way 
because once the elastic or bowstring passes the center point, the 
restoring forces act to slow it down.

The difference between that and the jumper is that the jumper is active
and pushes off the rebounding surface, unlike the projectile from a crossbow,
slingshot, or bow.

I hope this helps.

Sincerely,

Tom "Big Bouncer" Cull