A revolution(?) – rather, a lesson in physics

About a week ago the latest issue of the venerable and patriarchal journal Palaeontographica, Abt. A: Palaeozoology – Stratigraphy fell into my hands. The first thing I noticed was the decidedly odd journal title: why on earth, when they decided to translate the title from original German to English, did they not translate the “Abteilung”? Check out what the first page of a paper shows:
Why not call it “Section”? They do so on the web page.
That tiny irritation aside I was thrilled to see the first of the two papers in this issue:

Carpenter, K.,Hayashi, S., Kobayashi, Y., Maryanska, T., Barsbold, R., Sato, K. & Obata, I. 2011: Saichania chulsanensis (Ornithischia, Ankylosauridae) from the Upper Cretaceous of Mongolia. Palaeontographica A 294:1-61

Flipping through the pages I discovered a detailed description of the osteology of a wonderful, fairly complete and articulated new specimen, which has most of the armor in place! There are plenty of figures (as usual, I want more!), including several views of the mounted skeleton and several views of a reconstruction, and there even is a section on tail strikes to boot! Yeeeeha! I flipped to it immediately, and…..

and I realized that I was reading a sensational breakthrough in Newtonian Physics! On page 32 I found this:

quite obviously, acceleration is equal to velocity! WOW!

OK, /sarcasm. Mistakes happen to the best of us, and I know only too well how much nonsense can creep into a multi-author paper just because everybody edits all over the place in the word processor file. What makes this a bit special is the fact that in a previous paper, Ken and his then co-authors (others than this time) made a similar mistake. When I had emailed Ken about this, because I wanted to compare my modeling work to their methods, he had replied very kindly, referring me to his co-authors, the two physicists. That was really a great, magnanimous, kind thing he did: instead of being pissed, he took the implied criticism in stride! Before I had already had high regards for him, but that day he rose even more in my esteem. Ken, if you’re reading this: many thanks again for being a model colleague!

OK, so what does this slip-up mean?

Without going into any other details, let’s just add the step that is missing in Carpenter et al. 2011’s calculations (all linked to wikipedia for those physics-challenged like me):
We start with data from the paper on acceleration a = 195 m/s². Because we want to calculate the kinetic energy later on, using this equation:

K = 1/2 · m · v²    (1)

we’re missing velocity v. Luckily, we can calculate it without too much trouble because we have acceleration. The velocity v of a body is the acceleration a it is experiencing times the time t for which it is experiencing it (assuming the starting speed is 0 m/s).:

v = a · t (2)

In this case, we need to check for how long the tail club will get accelerated at a rate of 195 m/s². Here, the ugly ∫ sign, denoting an integral, quickly raises its head , but we can adroitly side-step it by assuming uniform acceleration (again: we’re also assuming that starting velocity is 0 m/s). The time it takes the club to complete a swing isn’t given in Carpenter et al. 2011, but we can calculate it from the acceleration and the travel distance. The travel distance s is:

s = 1/2 a · t² (3)

We can now re-arrange the above equation (2):

v = a · t  –>  t = v/a (4)

and insert this into equation (3):

s = 1/2 a · v²/a² = 1/2 v²/a (5)

Now we have one term with one unknown variable – all we have to do is solve for it:

v² = 2 s · a (6)

Wait, that’s not “solving for v“, that’s only solving for v²! but we can stop here, because  equation (1) for the kinetic energy has v² in it, and not v:

K = 1/2 · m · v² = 1/2 m · 2 s ·a = m · s · a (7)

Now, all we have to do is insert he values for m, a and s. Carpenter et al. 2011 calculate a value for club motion distance based on how far a muscle can contract ( 59.7 cm = 0.597 m) – but is that the full distance for acceleration? It is not, because muscles produce force not only by contracting from rest length, but also by being stretched and shortening back to rest length. The force produced is roughly the same for both motions (very roughly), so we need to double the distance: 2 · 0.597 m = 1.194 m

OK, all is ready now for equation (7):

K = 51 kg · 1.194 m · 195 m/s² = 11874.33 kg m²/s² = 11874.33 Nm

So accidentally using the acceleration instead of the velocity gave Carpenter et al. 2011 a kinetic energy that is too high by a factor of 48763.6/11854.33 = 4,106640122011095, roughly 4! The pressure created at impact is thus not roughly 83 MPa, but only about 20 MPa. That’s about a third to a quarter of the value for  shear stress of cortical bone cited by Carpenter et al. 2011.

EDIT 30. Sept. 2011: I just noticed that the Abstract says 13631 N could be created. How that number was arrived at I have no clue whatsoever.

Now, does that mean that Saichania had a big tail club only to pet other animals with? I don’t believe so, for a number of reasons. Some of them have to do with what I perceive to be erroneous assumptions in Carpenter et all.’s calculations, but it also has to do with the fact that the shear stress of cortical bone is often given at much lower values than the one chosen by Carpenter et al. Also, we can calculated the velocity v at impact, for which we now have to really solve equation (6) for v. We simply need to extract the square root:

v = = 21,6 m/s.

That’s going to hurt a lot, likely enough to break ribs etc. If you check out my paper on tail strikes in Kentrosaurus (open access) you’ll see that I cite some research on human cadavers that use impactors weighing less, at much lower speeds, and still manage to break very many ribs in the bodies:

In the interest of traffic safety, a number of studies have been performed in which impactors of various weights were used to cause blunt trauma in human cadavers at speeds typical for auto accidents, usually under 10 m/s. Viano et al. (1989) used a circular 23.4 kg impactor with a 177 cm² impacting surface at speeds of 4.5 m/s, 6.7 m/s and 9.4 m/s in lateral impacts, causing rib fractures and occasionally pubic ramus fractures. Talantikite et al. (1998) used smaller impactors (12 kg and 16 kg) of the same size and a narrower speed range (6 m/s to 8.5 m/s). During 11 tests on human cadavers they recorded between three and eight broken ribs, with between three and 16 separate fractures (Talantikite et al. 1998, table 5). Both studies also recorded occasional liver ruptures (Viano et al. 1989; Talantikite et al. 1998).

So a club strike was bad news for whoever experienced it – just how bad needs to be clarified by further work. Give me a few weeks to model Saichania tail club strikes, write it all up, and submit to a good journal. We’ll see what the reviewers say!

I’ll leave you with a beautiful picture of Saichania stolen from wikipedia under license mentioned there. Carpenter et al. do a nice job of figuring the beast, but this one is in color 😉

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References:

Talantikite, Y., Bouguet, R., Rarnet, M., Guillernot, H., Robin, S., and Voiglio, E. 1998. Human thorax behaviour for side impact – Influence of impact mass and velocities. Proceedings of the Conference on the Enhanced Safety of Vehicles, 1998:98-S7-O-03.

Viano, D.C., Lau, I.V., Asbury, C., King, A.I., and Begeman, P. 1989. Biomechanics of the human chest, abdomen, and pelvis in lateral impact. Accident Analysis and Prevention, 21:553-574. doi:10.1016/0001-4575(89)90070-5.

EDIT 01.10.2011: fixed typo in equation.