I plan to try this test later on today but thought I would ask here first.
Suppose I build a small trampoline with a rubber band in each corner as the spring.
I drop a metal ball on the trampoline from the exact same height and measure how far it bounces back up. I continue to repeat the test but each time stretch the rubber a little tighter to see how it affect the bounce.
The energy going in would be identical in all the tests. How would the tightness of the rubbers affect the energy coming back out? I wish I could eliminate hysteresis in the rubber as I have no idea what it is and I am not sure if it would have more or less affect as the rubbers got tighter. This might skew the results.
Intuitively I feel like the tighter rubber would spring back better but not sure why mathematically.
The rubber band acts as a spring. An ideal spring has a completely uniform spring constant for all loads, and perfectly stores potential energy. So if a rubber band acted like an ideal spring, all your tests would produce the same bounce. But no spring is ideal, and a rubber band is farther from ideal than a steel spring because of the type of deformation it experiences on stretching. A rubber band will give off heat as it is stretched and relaxed, and that heat is your bounce-back potential energy escaping (that’s what hysteresis is). My read of the stress/strain curve in the link is that most heat is released during the middle of the range of the rubber band’s stretch range, so a very slack or very loose rubber band will return the most energy.
I am not a materials engineer so take it with a grain of salt.
It’s not the same energy going in each time, because it’s not just the drop that provides energy - when you tighten the elastic bands you’re putting more energy into the system, stored in the elastics.
That makes good sense and eliminates the need for the next test I was going to try which was if there was a difference start playing with the weight of the ball and the distance of the drop and see if the sweet spot changed.
You could vary one or the other for the sake of science, but changing the weight and changing the distance are equivalent since the net result in both is the amount of kinetic energy the ball has when it makes contact with the trampoline. Note that kinetic energy is 0.5mv[sup]2[/sup], so changing the distance to get a higher velocity is going to give you a lot more bang for the buck than increasing the weight. For example, the velocity of a dropped object is
sqrt(2dg)
where d is the distance dropped and g is the gravitational constant. This can be written as
sqrt(2)sqrt(dg)
If you double the drop distance, you get
sqrt(2x2dg)
or
4sqrt(dg)
The increase in velocity is a factor of 4/sqrt(2) (which is about 2.82). Because velocity is squared to get kinetic energy, doubling the drop distance will multiply the kinetic energy by = 16/2 = 8.
So if you double the distance, it’s equivalent to multiplying the weight times 8.
Good info, I enjoy playing with catapults and things and enjoy attempting to scale things down accurately when building models. More complicated than simple scaling when you are working with mass and velocity as well as KE.
I built a giant bow for a doing Da Vinci Tv series. The bow powered a catapult. When I scaled the model down as best I could it performed infinitely better than the full size model. I must have made a mistake in my scaling LOL.
If the rubber bands were ideal, then both methods of changing the energy would be equivalent. But then, if rubber bands were ideal, then you wouldn’t need to do the tests anyway. I’d do a few tests of mass compared to speed, at the same energy, and check to see if they really do come out the same. If they do, then do the simplified testing regimen for the rest, but they might not.
EDIT: I just looked over CookingWithGas’s post about the energies, and it looks like you made a miscalculation. Doubling the height will have the same effect as doubling the mass. I think you squared somewhere where you should have square-rooted.
What will you find is that as you tighten the “elastic” bands, you move the bands more and more out from their elastic strain to their plastic strain…
So as long as the material stays in the elastic range of the stress strain graph, the bands don’t get damaged, and the energy is returned to the ball. (apart from loss to the air, noise, damage to the surfaces,etc)
But when you get the rubber bands into their plastic area of the stress strain graph, they undergo plastic deformation… they get damaged, they absorb the energy and get hot and don’t return the energy to the ball.
So when the bands are tightened up, the impact does plastic deformation - damage - to the bands, and the bands are loosened in terms of force… but if you consider “tightening them” in terms of strain, you keep on stretching those bands out further and further…
sqrt (2x2) is 2, not 4. But a better way to do it would be to just say that doubling the drop distance gives you a velocity of sqrt(2 x 2dg) = sqrt(2) x sqrt(2dg) = sqrt (2) x the initial velocity.
And of course squaring that gives you two times the square of the original velocity, so twice the kinetic energy.
By the way, doubling the height does give the same increase in kinetic energy as doubling the weight, but it hits at a higher impact speed (and, compared to the larger weight, over a smaller area). For a lot of systems, over not too wide a range of speeds, you’ll likely get similar results, but at some point the details will make a difference. A fired bullet and a thrown football might have similar kinetic energy, but give very different results impacting a trampoline.
Depending on your definition of “better”, there is another interesting point.
With the softer springs, the ball is going to bounce much deaper and softer. Which, at the very least, is going to be more fun to watch.
Compared to rubber bands, steel is not much different than a rock. But a softer thing - a tennis ball, or perhaps even a person – will also bounce. And unless its a perfect tennis ball, or perfect person, the bounce is going to be different on a hard surface and on a soft surface.
Actually, people don’t normally just passively bounce on a trampoline. And if you want to use your legs to bounce higher, how high you can reach is going to depend on the tightness of the rubber bands.
I see your point, and in a problem like this I have taken into account only the rubber band, without considering the properties of the trampoline bed.
For low values of “similar.” The slug from a .357 magnum, for example, carries about 5-6 times the kinetic energy of a thrown football. Same order of magnitude, though (which surprised me), which does make the point.
Comparing to how I figure my energy for archery bows the bed of the trampoline would be considered virtual mass. I would say just for the sake of discussion make the bed as tiny as possible, just something that would connect 4 rubber bands and gibe a target to bounce off of. Or I could just use a balloon stretched over something and increase or decrease the tension on that and forget about the bed.