Forces when you hit the ground

[Gavroy]

hi

i have been thinking about this scaling problem: Let me say a man and a cat jump from the top of a high building. both hit the ground, but as we all know the cat has a much higher chance to be alive after the collision. so is this caused by the biological or physical differences between the two bodies?

or let me ask it in a different way, would this phenomenon also occur, if you would let a small cube and a big(and heavier) cube fall from a building? intuitively, i would say so. but where is the physical explanation for this?

 

[gsal]

do you know F=ma?

a is gravity

and because the man has a lot more mass than the cat...well, it simply takes a lot more force to stop the man than the cat; in other words, the man is subjected to a much larger force when hitting the ground, even though they both hit the ground at the same velocity

 

[fluidistic]

do you know F=ma?

a is gravity

and because the man has a lot more mass than the cat...well, it simply takes a lot more force to stop the man than the cat; in other words, the man is subjected to a much larger force when hitting the ground, even though they both hit the ground at the same velocity

True but the bones of a man are bigger and thus should be able to endure bigger forces before breaking, compared to the ones of a cat.
Interesting question. Maybe cats have some system of suspension that is much more efficient to the one of a man.

 

[afreiden]

True but the bones of a man are bigger and thus should be able to endure bigger forces before breaking, compared to the ones of a cat.
Interesting question. Maybe cats have some system of suspension that is much more efficient to the one of a man.

Watch from 11:15-22:15.

Professor Walter Lewin predicts that an animal's bone diameter (femur), d, should be proportional to L^1.5, where L is the length of the femur. I.e. d = $\alpha$*L^1.5 for any given animal. This result essentially comes from assuming that the diameter, d, that is required (the diameter that would be optimal from an evolutionary/survival of the fittest standpoint) is a function of the stress demand in the femur, which is ultimately a function of the static weight of the animal.

Then, he shows that in real life, d = $\beta$*L^1.0 (directly proportional!). This part is really funny (he's a very entertaining lecturer) so you should watch the video.

Basically, cats (and smaller animals in general) have disproportionately thick bones -- disproportionate to, say, the static weight of a cat. This is what Walter Lewin proved. And it is for this reason, that a cat is more likely to survive a fall compared to a man.Now, WHY?? Why do cats have disproportionately thick bones??

I was disappointed that the prof didn't provide an explanation (perhaps he does in a later lecture).
My first thought was that his argument was flawed because he didn't consider that bones can buckle. This key simplification in the derivation is his "breaking argument" at about 14:45. He basically says that if you double the area of a bone, it can take twice as much force. This simple concept doesn't consider length at all. Intuitively, buckling force, for example, clearly depends on both area AND length. There was a time when I redid this part of his derivation, using formulas for buckling, and got d is proportional to L^1.25 or something like that..

Still not quite 1:1 as his 'experimental' results suggest..

So...

My theory: one has to consider that smaller animals evolved to be able to withstand falling from great heights without injury (i.e. cats). Perhaps these small animals evolved this ability in order to escape predators. Elephants, for example, do not have predators and so do not need bones that can withstand such demands. Just a theory.

 

[gsal]

a man's bones may be bigger but nevertheless made of the same material...

also, it is clear that the strength to weight ratio is much larger in a cat that in a man...think about it, most cats can launch themselves from the ground up quite a few times their own height...how many men can do this?

the same about landing on your feet and be able to absorb the impact...cats do it like nothing from very high places...

 

[Gavroy]

okay maybe this biological stuff leads not to the underlying cause.

what do you think would be the result if i regard a cube that has the volume 1 and density 1 and a cube with volume 8, so where each length is doubled but the density stays the same.

would there be also this effect? what would you say?

 

[modarn]

do you know F=ma?

a is gravity

and because the man has a lot more mass than the cat...well, it simply takes a lot more force to stop the man than the cat; in other words, the man is subjected to a much larger force when hitting the ground, even though they both hit the ground at the same velocity

I think I understand. But then (as the acceleration is always a=g) the force that hits you is the same no mater if you fall 1m or 100m?.

 

[gsal]

good point, modarn...I think we are forgetting about http://en.wikipedia.org/wiki/Momentum" ...visit this page to see the raltionship between how F=ma can be deduced from momentum.

Another way we can look at this is simply from the kinetic energy point of view...mv²/2 ...

the man and the cat hit the ground with the same velocity, sqrt(2gh), yet, one has clearly a lot more kinetic energy than the other and hence it is going to take a lot more work to stop the man...and if work is force time distance but the ground does not give much or if the ground gives the same for both, the cat and the man, then the force to stop the man is much larger for the man as his mass is to the cat's

 

[DaveC426913]

Read up on Galileo's Square-Cube Law.

Take a cube one foot on a side; its mass is 1kg.
Now take a cube 2 feet on a side; this cube is not twice as large; it is twice as long, twice as tall and twice as deep. And its mass is not 2kg, it is 8kg.

If this cube has legs, those legs are twice as wide and twice as deep. The area of the limbs is what holds the weight - and the area of the bones has quadrupled.

So: double the size of an object, and you quadruple the area of its supporting structures, but octuple the mass those structures need to support.

This is why there are no giants.

 

[chrisbaird]

Physics is not scale-invariant. http://books.google.com/books?id=fdnPAAAAMAAJ&dq=inauthor:galileo&pg=PA1#v=onepage&q&f=false"