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Yao Ming and the froghopper

May 12, 2009

Just a few days after a tough and inspiring performance against the Lakers, Yao Ming is injured again, with a stress fracture in his foot.  Lakers coach Phil Jackson commented on Yao’s fragility by saying “A person in physics once told me if man was 60 feet tall, the first step he’d take, he’d completely crumble … Gravity’s a bitch.”

But is it true?  Certainly larger people have heavier bodies, but they also have thicker bones and larger muscles.  So why are tall people more prone to injury?  This post examines that question using one of the simplest and most powerful ideas in physics: the scaling argument.  I will also try to make a prediction for the “injury prone-ness” of a person as a function of their height.

Before I get to Yao Ming, allow me to discuss a simple example using what is arguably the best athlete in the animal kingdom: the froghopper.  The froghopper is a little insect, barely half a centimeter long, but it has about a 27″ vertical jump.  That’s about 140 times its own body length, so in a certain sense it would be like me jumping 840 vertical feet.  Pretty impressive.  But if we put the froghopper in an enlarging ray, and blew it up 365 times so that it was the same size as me, would it really be able to jump 840 feet?

The answer is no.  That’s because an object’s weight is proportional to its body volume, which is proportional to the cube of its size.  So making the froghopper 365 times larger would make it 365^3 = 48.6 million times heavier.  The froghopper’s ability to jump depends on the volume of its muscles, which also increase by 365^3 times after it gets put through the enlarging ray.  So the ability of the froghopper to jump remains the same: it gets a lot stronger, but also proportionally heavier.  Therefore, a 6-foot froghopper could jump the same height as a half-centimer froghopper: 27 inches.  It just looks much less impressive.

Now let’s think about Yao Ming, who is sort of like a normal person put through an enlarging ray.  The propensity for one of Yao’s bones to fracture depends on the stress he puts on them.  Stress can be defined as weight divided by cross-sectional area.  So if weight depends on volume (size^3) and the cross-sectional area of his poor foot bones depends on size^2, then the stress grows as (volume / area) = (size^3 / size^2), or in other words, the stress increases directly with size.  You can think of it this way: by virtue of his great height, Yao’s bones are about 1.7 times thicker than the average person’s, but he weighs about 2.2 times more.  Thus, his bones have a harder time than yours do.

So how much more likely is he to get injured than the average man (height 5’8″) ?  Well, there are people out there who break bones for a living and have addressed this very question.  They found that the frequency of stress fracture in bone grows as the stress it is under to the power 0.06 1/.06 = 17.  Putting together their conclusions, along with the observation that Yao gets some kind of stress fracture every year or so, we can estimate how many years it would take for athletes of various heights to come down with a stress fracture:

stress_fracturesOf course, this chart is just a general prediction and is not meant to be completely accurate.  Some people will be hardier than others, and those people tend to be athletes (no stress fractures yet for a 7’1″ Shaquille O’Neal).  But the shape is important.  It suggests that if you’re under 6 feet tall (hooray, most of the world!) there is really not much risk of a stress fracture.  You can have a 20-year career as an athlete without too much worry.  But for those above 7 feet tall, your chance of a fracture is about 10 times greater than for your 6-foot brethren.

So finally, was Phil Jackson right about the collapse of a 60 foot man?  Actually, by my estimate, he was quite conservative.  As far as I can tell, a 16’3″ man would fracture his tibia the first time he took a step.

There is a sad and interesting footnote to this story.  The depressing far-right side of the graph above corresponds to the 8’11” Robert Wadlow, the tallest man in recorded history who suffered from an overactive pituitary gland.  By his late teens he was already incapable of walking without leg braces, and he continued to grow until his death at age 22.

8 Comments leave one →
  1. gravityandlevity permalink*
    May 12, 2009 5:03 pm

    If you haven’t already, please see the TrueHoop page as well for some further discussion:

  2. Scott permalink
    May 12, 2009 5:09 pm

    I agree with the plausibility of the conclusion of the fourth paragraph, but the reasoning is not quite right. From what I have read, strength goes with the cross-sectional area of muscle, which goes with the square of length, so the larger frog would be 365^3 times heavier, but only 365^2 stronger. The distance traveled when jumping, however, is related to the speed at “lift off”, so even though the big frog has much lower acceleration (1/365th, by simple F=ma calculation), he has longer legs, and thus a longer distance (and thus time) to “take off”. I remember working it out once (now I am not 100% sure, but I think one can do it using the kinematic equations) and the added length of the big frog makes up for the lower acceleration, and thus both will achieve roughly the same velocity at “lift off”.

    Hope that makes sense. Regardless, great post.

    • gravityandlevity permalink*
      May 12, 2009 5:20 pm

      Hi Scott.

      You’re right that the force you can output depends on the cross-sectional area of the muscle, so the maximum force output of the froghopper would be 365^2 times greater. The energy that you can get out of a jump, though, depends on the volume of the muscle. Your reasoning was perfectly fine: the work done by the muscle is equal to the force times the distance over which it works, which depends on the length of the muscle. So all together, energy depends on cross-sectional area (from the force) times length. This is the same as volume = area * length

  3. gravityandlevity permalink*
    May 12, 2009 5:52 pm

    For the record, it seems I was wrong about Shaquille O’Neal. Apparently he had some bone fracture in his knee in 1997:

    (sorry for the enormously long link)

  4. May 12, 2009 6:03 pm

    why do you draw a correlation with height when the stress he puts on his feet should correspond to overall mass (weight)?

    there are other 300lb men in the NBA (none 7 foot 5 obviously). shouldn’t their careers also be hampered by nagging injury?

    I tend to think Yao’s injuries have to do with his particular biomechanical makeup (feet structure, ankle structure, his stride when running, stance when landing), bad luck and overusage on the hardwood than rather than pure height

    • gravityandlevity permalink*
      May 12, 2009 6:07 pm

      Hi ed,

      You may be right; Yao’s biomechanics may be sub-optimal. Perhaps there is something about his feet/ankles/running stride that makes him particularly susceptible to injury.

      What is certainly true, though, is that if you could somehow shrink Yao down to 6 feet tall, keeping everything about him exactly the same and in proportion, he would be much less susceptible to injury (at least bone fracture). About 30 times less.

  5. Daniel permalink
    May 12, 2009 7:33 pm

    Shaq’s and Yao’s relative foot sizes could be the reason (bigger bones in the feet). Shaq’s shoe size is 23 while Yao’s is just 18 (seems kind of small for his ridiculous height).


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