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This probably includes a little of biology, but I believe it's mainly physics, so I hope it's ok to ask here:

Imagine 2 persons: Person A weights 120 pounds and person B weights 180 pounds.

Imagine they both fall from a big height:
a) landing on their knees (a good jump)
b) falling on their back (a bad fall)

For both of those persons, in both situations, for which of them the fall is easier to handle?

I reckon:

  1. when falling down, the speed is not relative to weight, therefore both those persons will hit the ground at the same speed
  2. but, I assume, as the heavier person will create bigger force to the ground, the ground will hit him back with the same force, therefore a heavier person should get a bigger blow when hitting the ground
  3. now, this one I'm not sure about, but I assume that a heavier person (considering it's not an unhealthy, fat person, but a healthy one, with muscles, etc) is able to withstand a greater force/blow

Assuming the last assumption is correct, how are assumptions 2 and 3 related? Is there anything I missed?

Overall, which person can withstand falls from greater heights: a lighter one, or a heavier one?

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"landing on their knees (a good jump)" This is probably not what you meant. In gymnastics/martial arts/pankour you take a fall by absorbing energy and impulse smoothly by bending joints (say ankles, knees, and hips) after you land (say on your feet), by transferring downward motion into sideways motion that you can absorb by rolling, and/or by taking the hit across a big areas (flat falls are no fun, but better than not knowing how to do a flat fall). –  dmckee May 25 '11 at 2:04
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6 Answers 6

up vote 4 down vote accepted

It is difficult to determine which will fare better.

Small mammals can survive a fall from arbitrary distances. Here's one article I found talking about cats. A chief contributor to small mammals' survival is that they have a lower terminal velocity due to the way wind resistance scales. Wind resistance scales with the area of the animal, while weight scales with the volume, so large animals fall faster because they have a higher volume-surface area ration. On a long fall (hundreds of times the animal's body length), body size influences impact velocity

Even without this effect, small animals may have an advantage. In humans, we can study the statistics of plane crash survivors. According to Wikipedia "Since 1970, two-thirds of lone survivors of airline crashes have been children or flight crew." Children make up a small percentage of passengers, so it follows they have a better chance of surviving. It's no great leap to attribute this to their body size. However, a BBC news story has an expert saying there is no physiological advantage to being a child when it comes to surviving a plane crash.

I have occasionally seen this sort of question addressed with dimensional analysis, but the difficulty is that it's difficult to pin down what you want to try to scale. The peak force or pressure, the energy dissipated per unit mass, the peak power dissipation per unit mass?

Here's an example argument:

If we assume the two people are impacting at the same speed, they need to dissipate the same amount of energy per unit mass. Assume that people can dissipate a certain amount of energy per unit mass in a given time without harm. Then whoever can make the impact last for a longer time will fare better. A taller person can bend their legs through a longer distance, and therefore can make the impact take a longer time, and therefore can fare better.

However, the assumptions in this argument would need to be verified before we can take it very seriously.

Here's another argument:

When two people hit the ground at the same speed, the time it takes them to stop is proportional to their linear dimension because this time is roughly their height divided by the speed that mechanical waves move through their body. Their acceleration is inversely proportional to height. Their mass is proportional to the cube of their height, so the force is proportional to the square of their height. That makes the pressure independent of height, so large and small people will fare equally well.

Here's another:

Same as above, but the deceleration time is proportional to the square root of height because they're flexing their knees, and so the stopping distance is proportional to height. This now favors short people.

Another:

Same as above, but mass scales with the square of height, because people are not scale invariant (the BMI uses an exponent of two). This favors tall people.

Another:

Same as above, but mass is independent of height because we're considering a skinny twerp and a muscular jock. This now favors light people.

My conclusion is that the problem is indeterminate. It depends on whether we're talking about a scaled-down version of the same person, or a single guy who starts taking steroids to prepare for a parachute jump. It also depends on various material properties of the human body, and on what sorts of things cause injury. Ultimately I think it's an empirical question, or at least one that requires extensive computer modeling.

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"Children make up a small percentage of passengers, so it follows they have a better chance of surviving." What? –  yayu May 25 '11 at 3:23
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@yayu The context is that children are also a large percentage of survivors. If they are both a small percentage of passengers and a large percentage of survivors, they must have a better chance of surviving. –  Mark Eichenlaub May 25 '11 at 3:25
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Let's proceed step by step:

Assuming you drop two balls of the same material and density but different radii (hence, different mass) from the same height. Which one is more likely to break?

The terminal velocity of the smaller ball will be lower and hence the acceleration it will face on impact will be lower than the bigger ball. Everything else is the same (ball material, strength etc.) and hence the bigger ball is more likely to break.

Now what if I add elasticity to the equation?

Imagine you are dropping an ice ball and a tennis ball. Even if both hit the ground with the same velocity, the ice ball is more likely to shatter into pieces. Why? Because, ice is more inelastic than fiber. The tennis ball is more likely to bounce back. More elasticity increases the time it takes to come to a complete halt (thereby reducing the acceleration). For the same reason, insects and other invertebrates are more likely to sustain fewer injuries than vertebrates of comparable size (bone vs flesh).

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This is the partial derivative approach by the way. I do realize that combining and extrapolating from here is not trivial. But if we assume that the subjects are composed of the same materials and are of roughly the same size and shape then their ability to break the fall (landing as a gymnast does for example), achieve large drag by eagle spreading etc. will play a large role. –  Monster Truck May 25 '11 at 8:18
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Simple impact damage models of humans (crash-test dummies) are usually based upon peak acceleration. Average acceleration during impact will scale inversely with collision time. If the distance for force absorption scales with height, the impact is self-similar, then peak acceleration scales as the inverse square-root of height. Note a thicker body/bone/muscle should be stronger in proportion to area, so force may scale with surface area, but this is irrelevant. So the taller person wins.

Arguments about insects don't apply, the terminal velocity of insects is low because of their small size.

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I agree with Mark. If I were betting, weight would not be enough information.

I think the better physically trained person has a better chance of surviving. It is analogous to picking up somebody who weighs 30 kilos. A child, and children are mostly muscle, hangs on. A sack of dead weight might break your back. In effect a well muscled child seems lighter than its dead weight.

So the datum muscle/flab should be given.

Well muscled people will break a fall in various instinctive ways, aiding survival. Now of course heavier folks tend to be badly trained and the muscle to flab ratio is smaller, so in that case the light one wins. For two equally muscled people with the same terminal velocity, it will be a tie.

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"Better physically trained"? This is the physics stack exchange, not the physical training stack exchange :) But aside from that joke, I disagree that it's a tie even if the different weights have the same terminal velocity, due to the force of gravity at the surface (see my answer for more details). –  Anonymous Coward May 26 '11 at 18:44
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I'm assuming the two subjects are made out of the same material(s), and have the same proportions. I will also assume that the barrier they collide with is unyielding.

With these assumptions the lighter person is better off. Here's why:

First consider them in zero g, moving at the same velocity and colliding with the rigid barrier. If the length of a person is L, then their momentum scales as L^3. The distance over which they can stop scales as L, so the force of the collision scales as L^2. The strength of each person scales as L^2 (cross-section), so they're equivalently well off.

As a fun aside: this is the same reasoning that explains why all animals pretty much jump the same height. Note that a jumping 10-pound cat raises its center of mass by about 1 meter, a half-ton horse can raise its center of mass by about 1 meter.

So in the zero-g, constant velocity case, the big person and little person are equally well off. However, for the "falling from big heights" problem, the little person has two advantages:

1) Air resistance will result in a smaller velocity for the smaller person at impact.

2) With gravity, at impact, the force on the person include both the force from deceleration (which scales as L^2 for a given velocity, see above) but also the force from counteracting static gravity at the surface, which scales as L^3. This is probably not a big deal for the two masses mentioned in the problem, but starts to become important as the animal becomes very large.

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The lighter person. The degree of injury is related to surface area. This is why insects don't mind being dropped and tumbling babies just smile. I wonder what the dinosaurs were like in action?

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I just realized that the "surface area" argument leads to the opposite conclusion! I don't mind downvotes but I hope someone can clarify. –  Dan Brumleve May 25 '11 at 1:50
    
The scaling argument you want here relates the mass (which is roughly proportional to volume) to the cross-sectional area of the limb/member/part taking the impact (think stress and strain, here). And it requires one additional element: you must assume that the people are built to the same relative plan, one simply scaled up relative the other. And scaling is important in this business. –  dmckee May 25 '11 at 2:00
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The scaling argument that involves surface area is the one relating to heat loss/retention. –  dmckee May 25 '11 at 2:06
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