[Physics] Which person can handle falls from big heights better: lighter or heavier

forcesfree fallscaling

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?

Best Answer

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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