I googled for previous discussion, and there was a beginning to my question, but no conclusion. The previous thread started the scaling idea: if you take a human 1/10th the size, what is the effect? But did not reach a clear conclusion.
My question is more realistic: if you take my son, 1/2 my height, and drop him from twice the height (4’ instead of 2’, not far enough to reach terminal velocity), why is he not damaged like I am?
Side factoid: I have clear memories of jumping off the stair landing.as a 3 yr-old. For decades I dismissed this as a false memory, because clearly I couldn’t have jumped that far down: it hurts when I do that. But when I started back at achool again, I see 6yr-old jumping 5’ down, so I’ve revised my ideas.
Because your son is a lot less massive, much less impulse is needed to decelerate him upon hitting the ground. Less impulse = less chance of a broken bone.
All other things being equal, the strength of an object is proportional to the square of the size, while its weight is proportional to the cube. For the same mineshaft:
a mouse walks away,
a rat is stunned,
a dog is injured,
a human is killed, and
a horse splashes.
To add some more math:
If your son were in the same proportions at you but half the height, then he’ll weigh 1/8 of what you do. (1/2 height * 1/2 width * 1/2 depth.) That’s probably not a perfectly accurate assumption, but I doubt it’s far off either.
In falling from twice a distance, assuming that we ignore the air resistance, he’ll have twice the velocity you do when he hits the ground. That means he will have approx one fourth the momentum you do when hitting the ground.
Force, in this situation, equals momentum divided by stopping time. Overall, your son would have a lower stopping time since there’s less of him, but since the important forces in determining injury are those on individual bones and parts of the body, I’m not sure if this is relevant. If we assume that the stopping times are considerable, then the kid is still experiencing about 1/4 the force you do when you land.
Along those lines, when I was maybe 10 to 12, I have two big items.
First, jumping off my back stair landing over concrete steps to land on a concrete patio. Drop was maybe 5 feet. Landed on my feet. I did repeat a few times. No injuries.
Second, jumping off a brick and concrete chimney on that patio, drop of maybe 8 feet onto grassy ground. Did do rollouts on that one.
Parkour practitioners make that look tame, and are older and bigger than I was.
To elaborate, strength is a combination of material property and cross-sectional area. Area is length times length. If the materials are essentially the same - mammal bone and muscle and ligatures - the varying characteristic is area.
Weight is determined by the material property of density combined with the amount of material. For similar materials, the defining characteristic is volume, I.e. 3D size.
Thus, for simplicity, consider a cube of uniform density. As the size increases, the density remains the same, the cross-sectional area doubles, the volume goes up 8-fold. The inherent material strength did not increase, so more energy and momentum are absorbed by a relatively smaller amount of material.
Or you could think of it as the area staying the same, but the block elongates, putting more mass behind each square of surface hitting the ground.
"You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object. Divide an animal’s length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. "
Apart from velocity scaling as the square root of distance fallen, these figures quantify the situation pretty well. Now that we agree on the disparity between the forces experienced by OP and son, it still remains to consider the properties of the material that’ll be absorbing this force.
(Italics mine) Is it a fair assumption to say that the bones of an adult human and those of a growing boy are essentially the same? I would expect more brittleness in the adult skeleton, which might compound the problem of greater forces exerted upon impact with the ground.
Also, kids have not yet suffered “the slings and arrows of outrageous fortune”. They have less arthritis. They are a bit more bendy in general. But they can injure growth plates. I tink the biggest thing is the 1/2 size, 1/4 cross section, 1/8 mass. Actually usually more so, with a little middle aged spread.
Yes, article could’ve been more clearly written. Conceptually a bit scattered. Discussion of terminal velocity was muddled and square-cube law was ignored. I was a little surprised that Haldane was not quoted. Pleased to see it up-thread.
I knew I was paraphrasing someone that I read 50 years ago, but all I could recall was that it might be Asimov, but probably wasn’t. Maybe it was Asimov quoting Haldane, because I’ve read very little of the latter.
Don’t forget, though, that with twice the height there’s twice the distance to decelerate. The child might have only one foot of length to absorb the impact; the adult two. Perhaps twice those numbers if you include tilting the torso down (“Ironman” style landing).
So, if landing velocity for adult and child are roughly same (for falling from a modest height- i.e. NOT a terminal velocity discussion), adult might be able to reduce the impulse of landing twice as well as child. With 8x weight and 4x cross-section of tissues, the stress applied to tissues would be similar?
Huh- doesn’t match my intuition. When I was a kid, we would jump off the garage onto packed earth. I’d never do that now. Maybe I’m just older and wiser.
The thing that’s always bothered me about this article is that Cecil accepts without comment the notion that falling from 6 feet is somehow “proportionally greater” for an insect than a man. That’s bullshit, and rebutting it is the first thing that should have been addressed.
Yes, but it’s meaningless. Acceleration due to gravity is not proportional to body size, and there is absolutely no reason to expect a fall of two feet to harm a 1/8" insect more than it does a six foot human.
The force on the bones is going to be higher in a heavier creature (as F=mass x deceleration). Mass (and thus force) increases roughly as the cube of height whereas the strength of the bones increases roughly as the square of height (as strength is due to cross-sectional area of the bones).
The stress on bones (and joints/ligaments) is force divided by area.
So as you get bigger, the stress on bones from falling a set distance will increase.
Even more so if you get relatively fatter.
It did occur to me though that while it might hurt your legs more, the stress on your head might be less - it doesn’t have the weight of the body to contend with but the increased amount of bendy stuff underneath it means that the distance over which it has to decelerate is greater and thus the deceleration (and thus the force) can be lower. So, while it might hurt an adult more than a child to jump a half-reasonable distance, a child might be more likely to be injured if the distance was high enough for there to be a chance of head injury (this assumes they don’t actually land on their heads).