Supersonic Man

December 1, 2011

Same Height II: the Rejumpening

Filed under: Hobbyism and Nerdry — Supersonic Man @ 10:52 am

Let’s try to put a little more rigor into the question of “do all animals jump the same height”, as discussed in this previous post.  We saw what appeared to be the same results being produced at different scales under certain assumptions… let’s check if it’s really a mathematical equality.

Our variables:

  • let b be the vertical distance through which the animal’s leg provides upward acceleration
  • let h be the vertical distance that the animal coasts upward after leg acceleration ceases
  • let m be the animal’s total mass
  • let f be the upward force that the animal is capable of exerting
  • let t be the time over which that force is exerted before the feet leave the ground
  • let g be the acceleration of gravity

We can now put together a set of equations that define our model.

Upward acceleration during takeoff:  a = f/mg       …boundary condition: f/m must be > g
Duration of acceleration:  t² = 2b/a
Velocity at takeoff:  v = at
Duration of upward movement after liftoff:  u = v/g
Height of jump after liftoff:  h = u²g/2

Now let’s scale up the size of the animal by some factor s.  The implications of the square-cube law are that:

  • leg length scales as the first power:  b’ = sb
  • mass scales as the cube:  m’ = s³m
  • force available scales as the square:  f’ = s²f

Of course in real life these will all be approximate, as the larger animal’s proportions will differ from the smaller one’s. But for the sake of mathematical inquiry we are assuming an exact copy at a different scale.

New value of acceleration:  a’ = f’/m’g
New duration of acceleration:  t’² = 2b’/a’
New velocity at takeoff:  v’ = a’t’
New duration of upward movement after liftoff:  u’ = v’/g
New height of jump after liftoff:  h’ = u’²g/2

The assertion to be tested:  does b + h = b’ + h’ ?

b + u²g/2  =?  sb + u’²g/2       …expand h and h’ and b’

b + v²/2g  =?  sb + v’²/2g        …expand u and u’

2gb + a²t²  =?  2gbs + a’²t’²        …expand v and v’, scale by 2g

2gb + 2ab  =?  2gbs + 2a’bs        …expand t and t’, and cancel

2gb + 2bf/m – 2bg  =?  2gbs + 2bsf’/m’ – 2gbs        …expand a and a’

f/m  =?  sf’/m’           … hah, most of the crap drops out!

f/m  =?  s³f/s³m        …expand f’ and m’

f/m = f/m        …one last cancel

Yes, that is an identity!  I have proved that all animals jump the same height.

(There is a unicode symbol for equality with question mark, “≟”, but I found it doesn’t show in a lot of browsers, such as the one in my phone.)

Let’s try an additional supposition.  Does leg length really matter?  Instead of saying that b’ = sb, let’s assume that b’ is an independent variable.  What happens then?  b’ + h’ = b’ + u’²g/2 = b’ + v’²/2g, scaled by 2g = 2gb’ + a’²t’² = 2gb’ + 2a’b’ = 2gb’ + 2b’(f’/m’g) = 2gb’ +2b’f’/m’ – 2gb’ = 2b’s²f/s³m = 2b’f/sm.  The same steps on the unscaled case yield 2bf/m, so the scaled case equals the unscaled case times b’/s, which means that proportionally longer legs would help you jump higher, right?

Except there’s a gotcha: by increasing lever length, you effectively decrease f, so instead of f’ = s²f, you have f’ = s³bf/b’.  So our derivation above ends with …2bb’s³f/s³b’m, which cancels down to 2bf/m, which is back to exactly matching the unscaled case.  So: all animals jump the same height, regardless of leg length.  Right?

…Wait a minute.  I don’t think that really accounted for the scaling of f in the first place, before messing with b’.  The correct value of f’ may already have been closer to sf than to s²f.  My proof may be invalid. Is that so?

On further thought, maybe not. I just realized that there’s a variable I didn’t account for, which is how closely the muscle’s tendon is attached to the joint as a proportion of bone length.  (Not that our math really models a bone as a lever… because of its simplicity, it describes something more like a vertical hydraulic piston.)  It’s this tendon position that determines the leverage ratio that affects f’, not the total leg length. If it scales uniformly with the leg, then f’ = s²f remains valid.  This means that my pre-gotcha conclusion looks valid again: animals of different sizes jump the same height, but proportionally longer legs do help, as long as you preserve the same lever ratio by also scaling the tendon anchorage.

Intuitively that seems dubious. Making the legs too long and thin would make jumping impossible.  But I think I see how it’s valid: the longer leg simply requires a larger muscle to power it with the same force, as it has to do more work by exerting the same force through a longer range of motion.  Specifically, the muscle has to be elongated.  It has to accommodate the longer range of motion, while it can remain the same width; its work output goes up linearly but its force remains the same.  This increases its mass.  Longer legs for jumping means that the leg as a whole has to be a larger portion of body weight.  Grasshoppers are an example of this strategy: their back legs are both long and bulky.  Kangaroos also have quite heavy back legs, and people are kind of that way too.

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5 Comments »

  1. I see that someone cited this in an argument about basketball, because some assumed big players could naturally outjump small ones, which is false. And that made me realize that my conclusion may be valid for grasshoppers and kangaroos, but is wrong for humans. Why? Because we stand with straight legs, meaning we have to crouch before leaping. In the math above, the height travelled, b + h, includes the crouch. That doesn’t count in real life. For humans you need to compare the h value only, and it’s smaller for large people.

    For that matter, most leaping animals are bound to crouch at least a little. Most don’t have to crouch the full distance b; various critters might range over different fractions of b that they crouch by, from grasshoppers and frogs not needing to bother with any crouch, to humans needing almost 100%. Cats and dogs might crouch 50% or so, rats much less.

    Comment by Supersonic Man — August 31, 2015 @ 7:03 am | Reply

  2. I half remember lectures on this topic from my first year university physics days. I don’t recall the formula or the exact multiple but I think it was either 2 or 2.5.
    They demonstrated that the approximate limit for all animals was 2 (or 2.5 or some other value) times the centre of gravity/mass for the animal. That is an animal’s upper limit is to raise it’s centre of gravity/mass to 2 (or 2.5) times higher than it was before leaving the ground. They showed this held approximately true for all jumping animals – frogs and rabbits to gazelles and kangaroos and to Olympic athletes. Of course it helps to be designed to jump, people aren’t designed to jump but manage to approach the theoretical limit by contorting and twisting around their center of mass as they go over the bar backwards. Rabbits and frogs just jump – no contortion required. Elephants just can’t.

    Having longer legs helps because it raises your and of gravity – your centre of gravity is higher when your feet finally leave the ground.

    That I think was using direct muscle power excluding the tricks used by some animals, life the fleas you mentioned, which my lectures considered not to be true jumping.

    Comment by Greg — June 5, 2016 @ 3:06 am | Reply

  3. That 2x or 2.5x conclusion falls completely apart for insects and spiders, even if you disqualify fleas, which I’m not convinced you should. Heck, I’ve seen a rat do far more than that when jumping out of a garbage can. Frogs too.

    Comment by Supersonic Man — June 5, 2016 @ 3:16 am | Reply

  4. Are you sure? Note I wrote “before leaving they ground”. By rising up in their hind legs as they jump many animals raise their center of gravity before leaving the ground and thus manage to jump higher.

    Comment by Greg — June 6, 2016 @ 12:55 am | Reply

    • Grasshoppers can jump like ten times the height that their legs can raise their CoG. Crickets, five times or so, that I’ve seen. Don’t offhand know about spiders or leafhoppers or springtails. The rat in the garbage can raised its CoG between three and four inches, maybe, and jumped at least four times that height without it seeming like any great effort. The frogs I recall most vividly were pacific tree frogs, and they had only about one inch of CoG lift, which they multiplied by… I’d say around ten.

      Comment by Supersonic Man — June 6, 2016 @ 4:22 am | Reply


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