Why a man on the Moon can jump 21 times higher than on the Earth

[sophiecentaur]

, I think it goes back to when it was worth a pound of gold.

HAha. I know the Pound Stirling (GBP) has been doing quite well recently and at times in the past but a pound of gold? At the present time, scrap gold (high carrat) is about £20.00 per gram. The Pound (livre) was originally based on a pound (troy) of silver and that's bought for scrap at about £0.25 per gram at the present time.

 

[Art Vanderlay]

The answer is more like a ratio of 10. Here's the analysis:

Equate k.e. at take-off point with p.e.:

1/2 mv^2 = mgh => h = v^2/2g

where m is the mass, h is the peak height.

Assume force applied by legs F is constant and the same in both gravities (will come back to this). Let d me the crouching distance (i.e. the distance over which the upwards force will be applied) and t be the time the force is applied

F - mg = ma

a = v/t and d = at^2 / 2. Using these and the formula for h above, you get a = gh/d (there's probably a quicker way to get to that result)

So F - mg = mgh/d

let F = kmg (i.e. F as the ratio of the weight of the person). Then

(k - 1) = h/d

On the moon, let the gravity g' be rg and h' be the peak height. So on the moon:

(k - r) = rh'/d

so h'/h = (k-r)/r(k-1)

Since k = 1 + h/d (above),

h'/h = (1 + h/d - r)/(rh/d)

Putting in numbers, assume a jump height on Earth of .5m and a crouch distance of .4m and r is about 1/6, then:

h'/h = (1 + 5/4 - 1/6)/(1/6 * 5/4) = 10

The crouch distance, d is going to be between .3 and .5, so at these extremes (but assuming a jump height still of .5 you get 9 and 11 respectively for the ratio of heights.

Coming back to the assumption of the force, let's check the time to execute the jump implied by these results. From d = 1/2 at^2 and a = gh/d:

t^2 = 2d^2 / gh

So assuming d = .4 and h = .5, t = .26s, which seems reasonable. In the reduced gravity situation, the corresponding figure is this divided by sqrt(rh'/h), which is .2s. I would assume therefore that the speed limit of contracting the muscles is not being reached, since we are only asking them to contract slightly faster. If speed of contraction is actually a limiting factor, then the effect would be a reduction in the force applied and therefore slightly reduce the height ratio.

The additional assumption about the force is that it is constant and (implicitly) that the way it is created (the physiology) is not affected by the reduced gravity. That would have to be verified experimentally, but my guess is that it has little effect.

 

[sophiecentaur]

I would assume therefore that the speed limit of contracting the muscles is not being reached,

It would be easy to estimate this effect by seeing how far you can launch a mass of man/6 vertically when laying on your back. This would give a launch speed. It would be useful to do this with a range of masses. I realize that both mg and ma are relevant. The ma could be examined in a similar way with masses on a pendulum and seeing what height they reach. You could find the relationship between ma and mg with this sort of experiment.
Not to denigrate the theoretical approach.

 

[A.T.]

I would assume therefore that the speed limit of contracting the muscles is not being reached, since we are only asking them to contract slightly faster. If speed of contraction is actually a limiting factor, then the effect would be a reduction in the force applied and therefore slightly reduce the height ratio.

Faster contraction speed always reduces the force, even if the speed limit isn't reached.

https://en.wikipedia.org/wiki/Muscle_contraction#Force-velocity_relationships

 

[Art Vanderlay]

Faster contraction speed always reduces the force, even if the speed limit isn't reached.

OK, that's good. To account for this, let's say the force is reduced to x of it's value on Earth, then h'/h = (x(1+h/d)-r)/(rh/d). Putting in some numbers, you get x=.9, h'/h = 8.9 and x = .8, h'/h = 7.8.

The other issue with the model of course is that the contraction force is not constant, so you actually want ∫F over the contraction period. The factor x above is more accurately the reduced impulse due to the different force function.

As a next approximation you could use the force/velocity relationship to introduce a velocity dependent resistive force, i.e. the EoM becomes F - mg - βv where β is effectively a drag coefficient. If you looked at standard drag theory (e.g. sinking object in a viscous fluid) then it should be simple to apply. But given that we don't know how the force varies kinematically due to the way the legs generate it (i.e. both the physiology and the machinery - there are at least 3 pivot points in the joints and the force is generated by effectively pulling cables to straighten the joins... non-trivial) it's probably too much model and too little data.

The main point is, assuming no other advantage for the legs in lower gravity it can't be more than 10 (for d=.4, h=.5) and it can be more than 6 (which is based on flawed analysis that ignores that fact that you are able to apply more net force due to less gravity).

 

[sophiecentaur]

Humans took a million years to get their leg mechanics right for the local g. When (not if) the colonisation of the Moon has produced a big enough enclosure to do the test for real, I think that the record height will be achieved using a lever system to match the load on the legs to something nearer weight on Earth. I'm not suggesting energy storage (trampolining would break the rules) but a simple impedance transformation. (No idea how it could be done, to extend all those levers in the leg.)

 

[Art Vanderlay]

Humans took a million years to get their leg mechanics right for the local g. When (not if) the colonisation of the Moon has produced a big enough enclosure to do the test for real, I think that the record height will be achieved using a lever system to match the load on the legs to something nearer weight on Earth. I'm not suggesting energy storage (trampolining would break the rules) but a simple impedance transformation. (No idea how it could be done, to extend all those levers in the leg.)

Just because the mechanics may be optional for Earth, it doesn't mean that it is not also optimal for lower g. I say may, because that's not how evolution works - it doesn't optimise everything, only the things that provide a selective advantage. That's why we aren't as strong as most of the primate family even though we are descended from a common ancestor.

That aside, here's a much quicker way of deriving the result (the first one was just writing down stuff as I thought of it!):

Work done in jumping phase = Fd (=kmg where k in units of weight of person)
mgd of this is used against gravity, so net remaining energy = (k-1)mgd
This is in the form of k.e. and converted to p.e., so mgh = (k-1)mgd => h = (k-1)d
On moon: g'h' = (k-g'/g)d
So h'/h = (k-r)/g(k-1)

The rest is the same as above

 

[sophiecentaur]

Just because the mechanics may be optional for Earth, it doesn't mean that it is not also optimal for lower g. I say may, because that's not how evolution works - it doesn't optimise everything, only the things that provide a selective advantage.

I would have said that efficient use of muscles in running and lifting could be a massive evolutionary advantage. But I have no direct evidence one way or another, of course. It's not the sort of thing that can be resolved by Physics. The "why"s about this sort of thing are impossible to be certain about but humans have been making use of tools for a long time and intelligent use of tools reduces the need for sheer strength, compared with the other apes. As the astronauts have shown, muscles tend to adopt the appropriate size to match demand. I guess that implies the need to do the moon experiment with recently arrived athletes.

 

[Art Vanderlay]

I would have said that efficient use of muscles in running and lifting could be a massive evolutionary advantage. But I have no direct evidence one way or another, of course. It's not the sort of thing that can be resolved by Physics. The "why"s about this sort of thing are impossible to be certain about but humans have been making use of tools for a long time and intelligent use of tools reduces the need for sheer strength, compared with the other apes. As the astronauts have shown, muscles tend to adopt the appropriate size to match demand. I guess that implies the need to do the moon experiment with recently arrived athletes.

Yep, exactly right, it isn't likely that there has been an advantage to further optimising the muscles since we have relied more heavily on intelligence. Also, the fact that there is no advantage means that there is no disadvantage to it becoming less efficient. It will do so due to natural drift and also if by becoming less efficient it adds another advantage, e.g. in our case, not having fuel-hungry large muscles enables us to survive with less food.

When we were developing in this direction I think it is highly likely we were evolving for speed and stamina rather than jump height.

 

[A.T.]

The "why"s about this sort of thing are impossible to be certain about but humans have been making use of tools for a long time and intelligent use of tools reduces the need for sheer strength, compared with the other apes.

Here a study about vertical jump performance of apes, suggesting that it requires muscle properties significantly different than those of human muscles.

http://rspb.royalsocietypublishing.org/content/273/1598/2177

 

[sophiecentaur]

Here a study about vertical jump performance of apes, suggesting that it requires muscle properties significantly different than those of human muscles.

http://rspb.royalsocietypublishing.org/content/273/1598/2177

That's interesting and appears to be a pretty thorough bit of investigation. I am surprised that I couldn't find a comment on the obvious difference between ape and early human lifestyle - humans were runners and not arboreal - probably some of the most effective running hunters ever and seemed to have managed to bring down massive prey by simply exhausting them by running them into the ground. I heard (unspecified radio programme) that it was assumed that both sexes would have needed the same abilities in order for the mums with children could be present at the kill in order to eat the stuff before other predators arrived to steal it.
But I am talking well above my pay grade on this topic, of course. (Standard PF practice, so no apology. )
On the topic of evolution and gravity, whatever the lifestyle of an Earth organism, it will have developed with a fixed value of g and, just as with all other abilities, there is every reason to suspect that some degree of optimisation of all abilities must have taken place. 'Nature' seems to do no more than absolutely necessary, when it comes to abilities. We really don't preform well outside a narrow range of temperatures (without clothes etc.), in non-standard proportions of atmospheric gases or in the presence of unfamiliar microbes. A local g of g/6 is not a trivial difference and I don't see how it can be assumed that we would be well adapted well to it.

 

[Art Vanderlay]

A local g of g/6 is not a trivial difference and I don't see how it can be assumed that we would be well adapted well to it.

Because there is nothing significant about the physics to suggest otherwise. Go and check out a gym and see people doing leg press (any g) or squat (greater g). there are many examples of where we would use our legs to push less than our own weight. A simple experiment on Earth would be to suspend someone from a tether providing a constant upwards force of 5/6g... that's an identical environment to the moon. I'd be surprised if this has not already been done, e.g. in preparing for the moon landings.

Also, I think it is a pretty trivial difference. Do the mechanics of jumping change when you have a heavy bag on you bag? Why should it be so much different in lower g?

 

[sophiecentaur]

Because there is nothing significant about the physics to suggest otherwise.

If you were doing any other experiment and suddenly introduced a factor of 1/6 into the variables then would it be good practice to 'assume' that it would would make no difference? You would just have to consider this as a major factor until you could prove otherwise.
We evolved with our weight and mass being relegated by a constant reducing one but not the other by such a large factor is not trivial until it's proved to be.

a tether providing a constant upwards force of 5/6g.

Why not find out about it and prove me wrong then? Unfortunately, I can't imagine NASA having built a test rig tall enough to cope with high jump records but you never know.
Actually, you wouldn't need NASA facilities. If you could find a high bridge with access top and bottom you could attach a pulley with a 5/6 body weight mass on a rope . .etc.

 

[Art Vanderlay]

If you were doing any other experiment and suddenly introduced a factor of 1/6 into the variables then would it be good practice to 'assume' that it would would make no difference? You would just have to consider this as a major factor until you could prove otherwise.
We evolved with our weight and mass being relegated by a constant reducing one but not the other by such a large factor is not trivial until it's proved to be.

Why not find out about it and prove me wrong then? Unfortunately, I can't imagine NASA having built a test rig tall enough to cope with high jump records but you never know.
Actually, you wouldn't need NASA facilities. If you could find a high bridge with access top and bottom you could attach a pulley with a 5/6 body weight mass on a rope . .etc.

High enough? The amount someone can jump from a standing start is about half a meter. Do you think this is high?

This has become a pointless debate with no new physics. The only unknown is whether the mechanics of the human legs dramatically changes in lower g. Well people have different power to weight ratios here on Earth also - do they jump differently?

Just saying some assumption is not valid with no reasoning is not science. There's no reason to believe it is either more or less efficient to jump on the moon - but it could be either way.

Reply if you wish, but I'm no longer responding on this thread since it appears to be just point-scoring based on blind objections with no physical reasoning.

I've given a reasonable model for the system which I hope others will find valuable.

 

[sophiecentaur]

Just saying some assumption is not valid with no reasoning is not science.

It's the other way round, surely. If you make an assumption then you should validate it. I was introducing a note of caution into making such a simple assumption - after all, there have been more than one approach to the theory, even on this thread.
Of course different people perform differently, under the same conditions. The issue is how differently one particular individual would perform under different conditions. Unless you can justify ignoring some parameter then you should really include it - that's a good scientific practice, isn't it?
I think the scientific reasoning behind my doubt would probably come from reference to the Power / Force graph, which doesn't show a simple relationship. I should have thought that evolution would have optimised the way the legs propel and lift the individual (running and hunting - not in a gym exercise).
It would be an interesting experiment to do before actually going all the way to the Moon.

 

[Art Vanderlay]

It's the other way round, surely. If you make an assumption then you should validate it. I was introducing a note of caution into making such a simple assumption - after all, there have been more than one approach to the theory, even on this thread.
Of course different people perform differently, under the same conditions. The issue is how differently one particular individual would perform under different conditions. Unless you can justify ignoring some parameter then you should really include it - that's a good scientific practice, isn't it?
I think the scientific reasoning behind my doubt would probably come from reference to the Power / Force graph, which doesn't show a simple relationship. I should have thought that evolution would have optimised the way the legs propel and lift the individual (running and hunting - not in a gym exercise).
It would be an interesting experiment to do before actually going all the way to the Moon.

You are just providing reasons why others' analysis is "wrong", rather than offering any yourself. Making assumptions is the way we simplify a problem enough to solve it. You could argue that g isn't constant in either case due to height, you could argue that Newtonian physics is invalid since general relativity more correctly describes reality.

I assume the reason why one might ask this question is to understand how the mechanics of the problem affect the outcome, not to understand if the action of jumping itself is changed. So if you want to continue to speculate about this point, carry on, I'll split the problem into 2:

1/ What height will a mass propelled with the same force over the same distance vertically reach on the moon vs the Earth?
2/ Is the way a person jumps affected on the moon?

Answers:

1/ h'/h = (1 + h/d - r)/(rh/d) where h' is the height on the moon, h is the height on the Earth, d is the distance the force is applied over and r is the ratio of the moon's to Earth's gravity. Anyone looking on this forum for the answer to that question, use the parameters you want for d, h and r and you have h'
2/ Feel free to answer yourself or continue to speculate. It doesn't interest me because it can't be tackled theoretically, it requires empirical evidence. My guess is that the only significant factor is the fact the the contraction speed would need to increase (as I have already analysed). This could in principal change the total force integral. I can't see that the contraction mechanism itself would be affected (you should be able to work out why that is yourself)

Maybe someone else can pick up the debate with you on 2. Good luck.

 

[sophiecentaur]

I assume the reason why one might ask this question is to understand how the mechanics of the problem affect the outcome, not to understand if the action of jumping itself is changed. So if you want to continue to speculate about this point, carry on,

My speculation is totally based on the fact that humans who use machines of any sort tend to use Gears. If you have ever tried to cycle fast in a low gear, you will remember that you are speed limited. The same thing applies to motor cars. Engines and (from the graph in that previous post) muscles have a definite optimum speed for operation. Why should this not apply when trying to jump as high as possible in low g?