Trampoline question

I gave a talk yesterday, and did not have time to get to this question, from Eric S., which we discussed during the dinner hour:

Who could launch themselves higher on a trampoline? LeBron James or Simone Biles?

She is a world class female gymnast, and much lighter (and less strong) than LeBron.  The question is assuming that both parties are motivated to win the competition, and have sufficient time to train to achieve their maximum potential in the contest.

Ultimately I settled on Simone as the better answer, mumbling something about small ants being very powerful for their size, and that magnification and extension of muscle spans ends up producing problematic results.  The power gain from the extra weight might be more than offset by the “drag” loss on the way up.  But I genuinely do not know.  Your view?

Addendum: This is an interesting article on animals and elastic springs.  And Jason Kottke adds comment, amazing photo too.

Comments

By highest do you mean how far the bottom of their feet rise above the floor or are you measuring from the top of each contestant's head?

Well, that is an interesting point. And to add to it - how strong/elastic is the trampoline in terms of the amount of energy that can be transferred and accumulated over succeeding jumps, as possibly compared to a 'launch?'

A single jump up and down and then measuring the height of a following launch tends not to take advantage of what makes a trampoline interesting, after all.

If multiple jumps were involved with a trampoline that could effectively store energy, the answer is fairly obvious. Though with careful timing if both were on the trampoline simultaneously, the gymnast could be launched with his help in a way that would be impressive.

Yea. Very sloppy or trolling by Tyler.
Are we talking about at sea level or atop Pikes Peak?
Does the air density matter? If it does, then relative humidity, as well as altitude, should also be specified for the test. Do the minor gravity differences in different locations matter?

The question, considering the difference in height between the two - .58m or almost 2 feet - is considerably more relevant than all of the factors you list combined, likely by an order of magnitude - or a couple.

Things like this are won and lost by millimeters. Typically the difference between an excellent athlete and a outstanding athlete is genetic and not training. Presumably both of these athletes would have the opportunity to train and prepare for such a contest. So I think it would be difficult to predict without sufficient competition to see who is better.

How about without a trampoline? Could Simone dunk?

It seems clear that LeBron James could jump higher. Isn't what people do on trampolines just jumping, magnified by what the trampoline does? I'm far from sure of this, having tried hard to avoid trampolines and jumping for most of my life.

Why wouldn’t it just be the person that can jump higher?

That would be true if one were to take Prof. Cowen's 'launch' literally. In which case, no need for a trampoline to begin with.

Trampoline's store energy. It's going to come down to which person can effectively store the most energy per mass. And frankly I don't know the answer.

This is a power to weight ratio question, modified by the persons ability to efficiently use the trampoline and how you measure the height.

Assumeably, we are measuring how high they can jump by how far their feet rise. Otherwise, the difference in stature is overwhelming. It still would favor LeBron, but to a lesser extent than how high they could reach with their hand.

The power to weight ratio is likely to favor LeBron (since males have thicker muscle fibers than females) but the efficient use of the trampoline would likely favor Simone (since as a gymnast, her skill would likely be greater).

Exactly.

If both sides have enough time, it's a matter of stored energy per unit mass.

I think the physical limiting factor is likely to be the trampoline, not the athlete. If both sides can push the trampoline close to its mechanical/elastic limits, then this should favour Simone. If not....LeBron.

Power to weight ratio; Simone.

That seems questionable. Males tend to be stronger than females at the same weight. Male muscles have larger fibers than female muscles.

Easy win for LeBron in absolute terms (he can jump higher than Simone). So we already know he packs more power to weight, at least in the vertical dimension. Trampoline just adds energy recovery from one jump to the next. Eventually you stabilize at the height such that friction losses equal your power, and that height will be higher for LeBron since he has more power.

This seems like the correct answer.

Yes, but trampolines are not limitlessly elastic. What if either or both Athletes can push the trampoline to its elastic limit?

That’s probably the right answer. But it does depend on the spring constant and damping coefficient of the trampoline, and whether Simon and Lebron are bouncing on the same trampoline or trampolines that are uniquely tailored to their respective masses and jumping actions. A related issue is that trampolines have a spring constant that varies with displacement, which further complicates the situation.

Slightly more interesting question is who would jump higher in percentage of body height.

A physics question: depends on the elasticity of the trampoline, the weight of the "jumper" nd the energy exerted in the upwards push .... obvious answer

F=ma, and the physics of the trampoline remain constant for both athletes. Plyometric training could improve both athletes' abilities to generate more acceleration, but there's no overcoming the mass factor.

As the bouncer gets more athletic the energy loss per bounce becomes more significant. That will tend to be larger for lighter bouncers, so I'd go with LeBron.

People working in athletics or kinesiology could answer this type of question better than TC.

Who could launch themselves higher on a trampoline? ...and land in a safely manner? I think muscular coordination is more important than raw muscle force, thus I'd go for Simone Biles.

If only maximum height counts, Lebron James. But after max height, expect an outcome similar to Evel Knievel.

James seems pretty coordinated to me when I watch him play basketball. Those finger rolls, bounce passes in traffic, fade-away jumpers, making layups after drawing a foul, etc...they all suggest a pretty high degree of muscular coordination.

One thing not directly mentioned (some may be suggesting this with comments regarding spring rates) that should be part of the answer has to be acceleration. Which person can generate the greatest acceleration wins. At least that is my intuition.

I think the physics work out better for LeBron. Kangaroos jump higher than rabbits which jump higher than Gerbils which jump higher than grass hoppers. Now in relation to their size smaller jumpers jump higher.

I can tell you from past gymnastics experience it probably would be the smaller gymnast. Back in the '70's my coach used to put football players on the tramp in the spotting/twisting belt he invented.

https://www.google.com/search?q=charlie+pond+twisting+belt&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwje4cHstNPbAhVrwVQKHc0FDQQQsAQIOQ&biw=867&bih=709

Many of them could soon do double and even triple back flips in the belt, without other assistance. But they were too heavy and didn't seem to get the proportional rebound as smaller athletes. We had recently moved to the larger 'Goliath' trampoline, and that probably helped. Maybe an even larger tramp today pulled tight enough would allow the larger athlete to go as high. It is not raw strength, but more a combination of coordination, timing and total body movement.

Our strongest gymnast who could vertical jump 48" (same as MJ) could not go as high as most of our trampolinists.

Before me there was Dickie Browning and the world high jump record. Technique over size or strength.

https://trackandfieldnews.com/discussion/showthread.php?110430-tumbling-high-jump-record

The Lilliputian effect is a definite help in gymnastics and why gymnasts tend to be smaller. Strength essentially goes up by the square (muscle cross section), mass by the cube. At least this is what we were told 50 years ago.

No personal experience myself, but it does seem small people are more springy on trampolines than large. The acceleration period from bottom of the trampolines stretch to the top of the rebound seems longer for large people making me think launch velocity would be slower for large people. But I don't know.

Lets recast this as a Trump vs Kim Jong Un contest. Who would win that one? Who would admit to losing it? Does it even work as a diplomatic tool?

I have no idea to your questions, but the images of any two heads of state in suits jumping madly on trampolines to see who can get higher is impressively funny.

Maybe the answer would change depending on how firm or squishy the given trampoline was. Firmer = better for heavier contestant = advantage Lebron, and squishier = better for lighter contestant = advantage Simone?

Final answer: I have no idea.

I sense a viral video in the works.

This is a simple problem, actually. Air resistance will not even enter the picture. Discard any answers based on it. Speeds are too low, and distance is too short. What will make the difference is power to weight ratio. I would put my money on Ms. Biles. LeBron might come out on top for absolute height reached, due to a starting advantage, although I doubt even that. Simone Biles to take the whole enchilada - both relative height and absolute height.

BTW - debunking some previous comments. LeBron would have the LESSER power to weight ratio, as he is heavier and larger. Note the one (former) gymnast's answer preceding, re football players. And the trampoline springiness and "storing" energy are nonsense. A trampoline amplifies (in returning to its original shape) the energy put into it. One would assume that the trampoline used by both parties would be the same, or the "contest" would be null. Changing the trampoline dimensions should make no difference in the final outcome, unless the dimensions were made so large that the lighter athlete was unable to deform the trampoline at all. The individual weight advantage should only come into play on the first bounce or two, but would quickly be nullified. Even if the contest were limited to a one bounce flying leap - such as a gymnast would take on approach to the vaulting horse, I would put my money on Biles. The lighter weighted individual would have an advantage, both in going up, and coming down (undamaged). A big fella has a harder time landing. Less power to weight again.

'And the trampoline springiness and "storing" energy are nonsense'

Technically, you are correct that the trampoline does not store energy. It is the person bouncing where the energy is being stored - the trampoline merely allows the energy originally expended in a jump to be recovered to a significant degree.

However, a trampoline does not amplify the energy - it does not add additional energy to the jumping person, it simply conserves it.

This is correct. Anything else would go against basic physics.

I also makes no sense to anybody who has ever been on a trampoline. the first jump is more, not less, difficult than a normal jump. At the same time, if you don't put any energy into the system, you would assume that it would immediately stop if it would only amplify energy. However, if you throw f.e. a ball on a trampoline, it will bounce up and down for a time, but always less high than before. This is consistent with the model of the trampoline 'storing' some percentage of the energy.

And it actually is the trampoline that stores the energy - namely, in the moment of maximum deformation, the trampoline has stored the energy. After that, it is re-applying this energy to the person, which than stores it in the air, applies it to the trampoline on impact, and so on.

'And it actually is the trampoline that stores the energy'

Yes - and no. The energy is transferred when being recovered, but sure, for a brief moment, the energy is in the trampoline.

In my very first comment, the term 'dynamic system' was removed when talking about energy storage.

Debunking the debunking - no, men have significantly more muscle mass per kilogram than women given similar training. Adding to that, male muscles are significantly stronger per kilogram than female muscles due to larger muscle fibers. Both taken together mean that men have a MUCH higher power to weight ratio.

Let's assume that LeBron weighs 275 pounds and Simone weighs 100.

If LeBron were scaled down to 100 pounds, who wins? I'm not sure, but I'm guessing LeBron.

If Simone were scaled up to 275 pounds, who wins? I'm pretty sure this would be LeBron.

Would the 100 pound LeBron outjump the 275 pound LeBron? My guess is no. Would the 275 pound Simone outjump the 100 pound Simone? Pretty sure the answer is no.

linvega - problem with that - men are stronger at the same weight. When you increase size, you don't proportionally increase output by the same amount. Given Simone Biles level of athletic accomplishment, one can confidently say she has a higher power to weight ratio than LeBron. As John points out in another comment (following) - * . . .smaller people tend to be stronger per pound (more force/mass) - a 300lb Olympic weightlifter does not lift twice as much weight as a 150lb Olympic weightlifter - so my guess would be Simone.*

good point but
mr. rogers or zardos in a beige buttondown
has anybody ever really taken a good look at the results of
the sociology experiment where you put 2 year olds in front of
a tv for a couplehours a day (the same time of day)when they were designed by evolution
to be outside empirically eating bugs
it looks like some of them might get sorta hypnotized by the screen
like on a tedtalk where everbody talks like chickens,
at this point in lifes rich pagent shouldn't we be teaching
a little more skepticism in regards what slides down
onto the screen like a squishy
zebra muscle?
the squish
that is mucous

Probably Simone because the trampoline would have less mass to propel upward against the pull of gravity.... unless LeBron has enough additional strength in his jump to overcome that advantage and combined with his height advantage, especially if you include arm length, he might reach a higher apex. The potential energy in the springs ie strength of the springs would matter.

http://aplusphysics.com/community/index.php?/blogs/entry/985-physics-of-trampolines/

After thinking about it, I decided that LeBron might have a harder time jumping off a trampoline. He would require more energy to control his vertical ascent given his mass and height (He would have a harder time maintaining a straight vertical position.). Simone's compact shape would be more of a bullet and her gymnastic training is better for this type of takeoff. So some of LeBron's physical strength advantage would be lost. In any case, Simone would look cuter doing it

Peak Spudd Webb beats both.

I would bet on Ms. Biles based on my own experience as a child watching both children and adults bounce on trampolines- the children always bounced higher.

Deeper, though, it seems to me that the potential energy stored in the springs is the ultimate factor here- is there reason to believe that one or the other of the competitors could reach that maximum and the other could not? If both eventually max out the springs' energy capacity, then Ms. Biles wins by being the less massive object.

That's easy. Answer is LaBron. That is because jumping on the trampoline is just a derivative of jumping wit no trampoline. Provided the trampoline is efficient (and in that case you will probably need a stiffer trampoline for LaBron because he is heavier) LaBron will win

Just look at the average size of champion trampoline jumpers every year, they tend to be small and muscular, even the men, thus I think Simone takes it.

They are smaller so that can rotate better not so they can bounce higher.

Simone. Greatly superior form/control of her body in flight. LeBron might get higher on initial jumps (she would have to jump a lot higher just catch up to the top of his head) but will lose more energy stabilizing himself at the bottom, and won't be able to reliably redirect energy along a straight vertical path. So she'll convert a greater % of the energy of the fall into vertical distance, gaining more on each subsequent jump before she maxes out the potential of the tramps elasticity. He's just way way more likely to wipe out, wobble, flail, or bounce back at angles.

Going back to the OT and animal spring article. There are 3 springs involved. The trampoline bed, the actual trampoline springs, and the gymnast's body. IMO within some boundaries, it is the ability to create the greatest body spring that will result in the highest jump.

In trampoline it takes at least few prepping jumps to gain maximum height. Absolute leg strength does not seem to come into play. Your arms do a lot. Properly trained I would bet on Spud over LeBron, assuming we're measuring distance above the tramp. Now for absolute height, it might be the taller LeBron touching the highest point.

The height will be determined by whoever has the greatest acceleration - once the jumper leaves the trampoline, they are stuck with the physics of ballistic motion. Since Force = Mass x Acceleration, that means Acceleration = Force / Mass. I know that smaller people tend to be stronger per pound (more force/mass) - a 300lb Olympic weightlifter does not lift twice as much weight as a 150lb Olympic weightlifter - so my guess would be Simone.

Research:

Simone Biles, during the peak of her floor routine, can jump to a height of 9'4" (112 in) where her head is at the highest point (per Mashable: https://mashable.com/2016/08/17/how-high-does-simone-biles-jump/#9dkAaSkXK8qr). Simone Biles is 4'8" (56 inches) tall, meaning that her feet must be at minimum of 56 inches off the ground.

LeBron James has a 40" vertical (measured distance from feet off the ground during a running start) per ESPN.com (http://www.espn.com/espn/thelife/bodyshots/?guide=lebronjames).

Research Analysis:

1) Both athletes are assumed to be measured at a running start given a "floor routine" and the "activity of playing basketball".
2) Simone Biles conducts her floor routine on a trampolined surface, meaning that she has been given extra spring for that specific measurement as it stands today probably by utilizing a preparatory move prior to her marquee move.
3) LeBron James jumps off a hardwood floor, meaning that his surface is much less forgiving.

Back to the numbers.

Simone: 56 inches on a trampoline mid routine.
LeBron: 40 inches on a hardwood floor.

The experimental procedure:

Using the following trampoline: https://www.gymsupply.com/trampmasterelitetrampoline-clickonproductforpriceinformation.aspx
1) Each participant enters the center of the trampoline.
2) Once the participant has reached the center and the trampoline stabilized, each participant reaches up overhead and the scientist measures the distance from the hand to the ground as well as the head to the ground for each participant.
3) The test would allow for each participant to get 3 prep bounces to build up to their 4th where they could reach as high as they could go.
4) As they reach their apex, the participant must reach up as high as possible with one arm.
5) Measurements will be recorded at the tip of the finger AND at the crown of the head. That way, both participants reach their absolute highest point for documentation.

Analysis:
Based on clear fact, LeBron will physically get higher. He is taller and has a longer wingspan. What would be calculated for fairness is:

(highest point of the hand in air - highest point of hand standing on trampoline) = Highest Hand Jump

and

(Highest pint of the head in air - highest point of head standing on trampoline) = Highest Head Jump

Using these two numbers, we can determine who jumped higher.

Hypothesis:
LeBron would have the highest hand jump and highest head jump, but not by much. His weight and leverage would literally propel him the few inches necessary to win.

So, let's first look into *what* a trampoline actually does physically.

In the moment of impact after a jump, the kinetic energy of the person is transformed into potential energy in the trampoline. Afterwards, that energy is transformed back into kinetic energy. However, due to friction losses, this transformation isn't perfect, so a significant amount of energy is lost. Under this 'naive' model, you would be able to jump higher and higher until the loss is as high as the energy you put into the system. For example if the system loses 50% of the energy of every jump, you would jump 200% of your normal height, because 50%*200%=100%. In other words, it's just a linear function of your initial jumping height. In this case, Lebron wins, because of his higher power to weight ratio which allows his initial jump to be higher.

But in reality, any spring can only store a limited number of energy. So with every jump, you get closer and closer to the maximum energy. At that point, only your weight matters, because your jumping height is a function of energy/weight. Because of that Simone would jump higher, given enough time. However, if either the loss or the maximum energy of the trampoline is sufficiently high, you can't actually reach the maximum energy.

So to conclude, it depends on the size and efficiency of the trampoline. The bigger or less efficient the trampoline, the higher Lebron can reach relative to Simone, and vice versa with a very small and/or efficient trampoline, Simone wins out. Also note that any time limitation works out in favor of Lebron because it doesn't allow Simone to actually reach the maximum energy storable by the springs.

What if they both get trampolines that are optimized for them?

From a theoretic pov, that would be the same trampoline - infinite maximum energy, no loss. With that trampoline though secondary forces would come into play which are otherwise neglible - loss from the person itself, air friction, even survivability given extremely high acceleration. Taking a guess, I'd say that the loss from the person itself will be the biggest effect, which gives Lebron the edge because this means jumping height will still be a linear function of initial height.

From a practical pov this is probably more of an engineering question than a physical one - what are the optimal ways of building a trampoline? Again taking a guess, I'd say there is probably some tradeoff between efficiency and maximum energy - bigger trampolines have more moving parts and will lose more energy, smaller trampolines obviously can't store as much energy. So the answer would depend one the shape of this tradeoff, which I don't know.

The problem is that they may not have the same inefficiency. LeBron, being a giant might exceed the springs elasticity far sooner. Eventually he might push the springs to the floor. Simone, being rather petite may stay closer to the elastic range somewhat longer.

Physically, the energy of the tramp is measured by the amount of spring deflection. At max deflection, both would have stored the same energy, so the one with smaller mass would be propelled higher, in proportion to their mass ratio. This should overwhelm any difference in intrinsic vertical leap ability. Moreover, the smaller gymnast is more likely to have an advantage in the precision timing necessary match the non-stationary resonant frequency.

It's funny to read all the responses from the many people who are overlooking basic Newtonian physics. A lot of years ago I coached the men's volleyball team at an Eastern US university. One of the things we did on the first day of practice was measure vertical jumps of players. These days there are stands that make this easy to do but we did it the old fashion way. 1) You stand next to a wall and mark the highest point of your reach. 2) You then jump (no running or stepping; just standing start) and mark the point at which you can reach on the jump. Subtract 1 from 2 and you have the vertical jump. This automatically corrects for height and arm length and is the most reliable method of measuring.

Lebron has a disadvantage because of his weight and gravitational pull will impact him more than it would Simone. However, his higher weight would mean more potential energy from the trampoline. The critical question is whether it is enough to overcome the gravity deficit he incurs.

I suspect it will be close but Simone might be an inch or two better. Best documented vertical jump I've ever seen was US volleyball player Karch Kiraly whose standing vertical jump was 41 inches.

Our best gymnast did 48 (I witnessed this) as did MJ. That being said, this is not the same as a vertical jump. For instance on the tramp, going for max height, you could never compress the angles and the knees and hips like you would with a vertical jump. Tramp is more like a plyometric rebound.

Is this a standing vertical jump or running. MJ is cited as having a 42 inch jump from standing; the 48 inches is with a running start. Anyway for those interested, all the physics is here: https://www.whatsmyvertical.com/the-physics-of-the-vertical-jump/

Surely it depends on the trampoline. A large man can compress a much larger spring, storing much more energy.

As I and others point out, gravitational pull affects heavier people more. I suspect that it you go through and do the match, the trampoline is probably neutral in effect.

magically "neutral", huh? Or do you mean that since you don't know, your Bayesian prior is 50:50?

'As I and others point out, gravitational pull affects heavier people more'

There is a not exactly obscure Italian who not only disagreed with that formulation, but actually demonstrated why that formulation is incorrect.

(Yes, a heavier object can be seen as having more energy upon impact, but gravitational pull is equal, regardless of mass.)

Jeez, Clockwork, way to miss his point when trying to look like know what you are talking about. All he is doing is pointing out that the mass of the two competitors matters in who is able to get the highest. This is why it took a Saturn V rocket to lift the Apollo missions, but much smaller rockets to send probes to Mars and beyond.

As someone who's coached gymnasts moving to a diving board, they tend to take a while to get anywhere near decent at letting the trampoline throw you. Most of the height is timing and waiting, and gymnasts jump too early most of the time. Yes, Biles is a superhero, and probably doesn't have that problem, but the floor routine spring-floor, or the warmup-track ski-floor are NOTHING like working a trampoline.

After reading these responses, I changed my mind several times as to the winner. Although Lebron has a clear advantage in strength per unit of muscle and despite his awesome coordination for his size, I don't believe he could ever coordinate the jumps as well as Simone. To prove my point, just ask Lebron to execute a jump following a balance beam routine like Simone can almost do in her sleep. If you put a gun to his head and forced him to try (which would likely be required), I doubt seriously he could execute even a simple routine due to his size, let alone make it to the jump. Practically speaking, trampoline requires many of the same skills required in gymnastics (my daughter competed in both sports and broke her ankle practicing for Nationals on the trampoline). That means Simone's execution advantage due to her higher level of fine motor skills used in the sport and relative size/ lower center of gravity would likely overcome LeBron's power/ weight advantage based on pure physics. The outcome cannot be determined for sure, however, since we cannot define precisely the level of relevant skill each athlete is able to bring to the competition. My money would be on Simone, but we may have a new competition for the producers of American Ninja!

I have no idea, but I've solicited advice from people who might know.

The only input I have is that Jason Burnett, one of the best trampoline jumpers in the world, is 5'-5", and 150 pounds. (But obviously, trampoline jumping in the Olympics isn't just about height.)

https://www.youtube.com/watch?v=NJEcMG7wNbg

Why do routinely respond to queries you don't understand with BS answers?

Why not say, 'I don't know'?

Amazing that after the trampolinist tells everyone the correct answer people still keep writing nonsense. As a trampolinist, I would lay any money that Lebron wouldn't even get close to Biles. Biles would probably hit 5-6 meters. If Lebron went 3 meters he'd look down and fall on his face. Form is everything in trampolining, jumping on a solid surface gives no transfer whatsoever.

I find it most interesting that so many bad answers have been given, so far. Some indicate serious flaws in writer's physics and others might be simply a failure to articulate their understanding convincingly. I make NO claim to know what the answer is. Heck, I don't even know whether the limiting variable is the trampoline or the person (or the interaction between the two). A trampoline is essentially a spring. (or rather a system of springs). In an ideal case, as long as the trampoline returns more velocity/acceleration/momentum/energy than is lost by the person in that jump, the person's kinetic energy will be increased. A 'perfect' trampoline would allow a person to jump until they achieved orbit. Of course a real trampoline will fail long before that. Plus, the amount of →impulse← would eventually break the jumper's bones (legs, back, etc.), long before orbital speeds were attained. My guess is that we can construct something which acts like a trampoline so that the limiting factor is the human frame. James, I'd guess, but do not know ( nor does anyone not intimately (literally) familiar with his and her musculoskeletal system), being much larger than optimum (as far as evolution was involved in his 'design') would fail first. But that's just wild speculation. (We could possibly find out with one of those circus cannons....) So, unless we construct an effectively "ideal" trampoline, then the energy losses will be significant. On the human side, our legs act like springs, too. My guess (I'm no sports physiologist) is that gymnasts rely much, much more on their "springiness" than do hoopsters. So, it could be that someone on top of the gymnast world is standard deviations above the average while a basketball player is not.
My guess is that in a real competition, the outcome would not be predictable without evaluation of their training. I'd guess with real world limitations, James would get his center of gravity higher (and here, I give a pained expression along with an eye-roll to those who suggested feet, hands, or head be used as the point to measure - as if these can't be "gamed"). But to be truthful, I really don't know who has the more powerful →relevant← musculature....and timing. (to benefit from the energy transferred FROM the spring (trampoline) you have to be in contact at the right time. As a % of body weight, I assume James leg press is much greater than Biles, but I could be totally wrong. Her thighs seem to me to be not exceptionally large, but I couldn't find a photograph of Jame's in a pic showing his thighs, which says something culturally, I guess....

Leg press is irrelevant. Biles is trained for the level of spinal compression when jumping on trampolines, James isn't. The upper body loads are critical and completely different to jumping on solid surfaces.

The gymnast wins. Air resistance doesn't matter at these speeds. The physics question is identical to a driven pendulum with two potentials (gravity and spring). Gravity is perfectly conserved, all the losses are in the spring. Both jumpers can add energy to the pendulum indefinitely, so the question comes down to the ratio of power in (LeBron wins) and power lost to deformation of the spring (where I believe the gymnast wins). Total deformation will be higher for the heavier jumper, and power dissipation is higher for increased deformation. The gymnast uses less of the spring's range of motion per unit height, so she wins.

This ignores skill, where I would also assume she has the advantage at these velocities.

Has no one ever seen an adult vs. kid on diving board (adult crushes kid)? How's that different (physics, ignoring any purported skill)? And all those who think LBJ's jumping skills (and related balance, etc.) aren't off the charts, akin to the best ballet dancer/gymnast, obviously don't watch hoops.

Were any of the kids on the divingboard elite gymnasts?

That's kind of the point: Just the fat old dad gets way, way, way more height, on any parameter (feet, center of gravity, etc.) than a very experienced kid. As you might guess, without the weight to make use of the board, kids don't get nearly as high, no matter how skilled. And it's funny you ask about elite gymnasts while ignoring that LBJ is basically the craziest freak athlete ever, in a league that's full of insane athletes (you're not going to convince me that selection for gymnast is higher than selection for NBA level talent, given how much more money there is in the NBA).

How do you explain that the best male Olympic trampolinists include:

Jason Burnett: 5'-5", 150 pounds
Dong Dong: 5'-6", 126 pounds
Gao Lei: 5'-7", 141 pounds

?

Granted, the point isn't simply to get as high as possible, but some very small and light men are at the very top of the sport...

I shouldn't leave out the gold medalist in Rio, Uladzislau Hancharou:

5'-8", 143 pounds.

So every one of those four men is 5'-8" or less, and 150 pounds or less. That seems to indicate pretty strongly that being pretty short and *very* light is a good thing in trampoline.

https://www.youtube.com/watch?v=T3qzymwJEgA

The smaller person has a shorter radius when tucked and can complete revolutions faster.

"The smaller person has a shorter radius when tucked and can complete revolutions faster."

Yes, but they're not just small. They're also thin for their height. Jason Burnett is the "chunkiest" guy of the four, and his BMI is only 24.9. (Dong Dong's BMI is the lowest, at 20.4.)

It seems clear from those four men that any diving board analogy, where the heavier person allegedly gets higher, is not particularly appropriate for trampoline.

P.S. As several comments have mentioned, it's not clear what, "Who could launch themselves higher?" means. Does it mean who could get the lowest part of their body the highest off of the trampoline, or who could reach up to touch the higher point? It's also not clear whether the hypothetical trampoline for the contest would be built to be best for the weight of the contestant, or whether the trampoline would instead be some stock trampoline used for the Olympics, for example.

If the answers to the questions are: "The height would be the height of the lowest part of the body above the trampoline surface" and "The trampoline would be a stock trampoline used in the Olympics"...then my guess would be that Simone Biles would get higher, because her weight and BMI are much closer to the best male trampolinists in the world.

Olympic tramp isn't about max height, with a custom tramp. So it's not just a physics problem. If people want to assume stock tramp, then there's likely a clear answer, given the ability of the material to exert enough force to throw a certain amount of weight. (Appeals to skill make little sense to me, I assume we're assuming LBJ practices some, and given what we know about him, he's going to get very, very good.)

Olympic level trampolinists have been recorded jumping to over 9 meters. Diving boards are not remotely comparable. James is top heavy and simply does not have requisite skills in the air. He jumps with his arms and legs in different directions, leaning forwards or backwards, shoulders not in line with his hips, moving at speed, take off and landing 3 or 4 meters apart. He would not get close to a gymnast in maintaining his line in the air because it is not something he has ever needed to do or even think about, while it is the key to everything the gymnast has trained. Natural balance isn't close to being enough.

I would bet on him being trained. I've seen it myself with my gymnastics coach and football players quickly picking it up back in the '70's. If LeBron had the time, desire, right coaching and equipment, he could eventually touch a higher ceiling than Biles. It might take a larger and firmer tramp than used today in competition. Some one with more recent knowledge and experience might chime in. Back in the '70's we moved to the 'Goliath' tramp. Larger and could accommodate more weight. 250 pounds might be stretching things a bit too far....

Simone by a lot. Why? Well, it turns out that the trampoline helps you out, but the height you reach is dominated by your strength to weight ratio. Your strength is proportional to the cross-sectional area of your muscles, although your muscle composition is important. Both of these people are at the peak of human performance, so I will guess that their muscle makeup is similar with an advantage to LeBron.

Weight, however is proportional to the volume of your body. Hence we have a area (square) relationship to strength and a volume (cube) relationship to weight.

Simone has a huge advantage to LeBron. That is why small creatures appear so amazingly strong for their size, and there is a limit to the maximum size that animals can attain.

Simone would win, and it wouldn't be fair.

I think the skill aspect wins, though one could then ask whether each of them gets a longer period of time to train before the contest.

I'm not going to pretend to know the answer, but I recall that female vaulter McKayla Maroney was able to vault slighty higher than male gymnasts, even though the vault used by gymasts is slightly shorter than the one men use.

Typically it will depend on the strength of the trampoline and Lebron is heavier so case closed.

Well, I think LeBron will have greater reach than Simon...

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