Projectile motion in biology

In projectile motion without drag, all trajectories are perfectly parabolic and the maximum range is achieved with a launch angle of 45º.

Steven Vogel is a biophysicist who thinks about how principles of physics show up in the design or behavior of biological organisms.  For instance, organisms that jump or throw use projectile motion, which we talk about extensively in introductory mechanics.

Starting from the standard picture of projectile motion that you have learned, are learning, or will learn  (Figure 1), Vogel points out that the smaller the organism, the worse the approximation of neglecting air resistance becomes.  By the time you get to small seeds and spores, trajectories definitely do not look parabolic (Figure 2).  Extensive information about spore ejection (probably more than you want to know), including 50,000 fps video of the process is available in this article and accompanying materials.

Maximum-range trajectories for a jumping flea, a Pilobolus sporangium and a Sordaria eight-spore cluster. Drag causes significant deviations from parabolic behavior, and reduces the launch angle for maximum range below 45º.

See Steven Vogel, “Living in a physical world II. The bio-ballistics of small projectiles” J. Biosci. 30:167-175 (2005) for full details.

A handy manual for destroying opponents' walled cities, if you live in or before the middle ages.

Compare this to the lovingly rendered trajectories of cannonballs bombarding a peaceful hillside town, according to Aristotle’s theory.  Aristotle gets the cannonball‘s trajectory completely wrong, but in the high-air-resistance limit (such as for Sordaria in Figure 2) he’s not so far off.  Of course, Aristotle really is wrong about the physics, but when you add a lot of air resistance to gravity you happen to end up (coincidentally?) with something similar to Aristotle’s prediction.  Lesson: you can safely ignore air resistance if you are a cannonball, but not if you are a spore.

Chronic acceleration research

Ah, the golden years of scientific inquiry.

To quote from Great Mambo Chicken & the Transhuman Experience by Ed Regis (Addison Wesley, 1990), pp. 54-55:

g work done on chickens, for example, by Arthur Hamilton (“Milt”) Smith in the 1970s.  Milt Smith was a gravity specialist at the University of California at Davis who wanted to find out what would happen to humans if they lived in greater-than-normal g-forces.   Naturally, he experimented on animals, and he decided that the animal that most closely resembled man for this specific purpose was the chicken.  Chickens, after all, had a posture similar to man’s: they walked upright on two legs, they had two non-load-bearing limbs (the wings), and so on.  Anyway, Milt Smith and his assistants took a flock of chickens – hundreds of them, in fact – and put them into the two eighteen-foot-long centrifuges in the university’s Chronic Acceleration Research Laboratory, as the place was called.

They spun those chickens up to two-and-a-half gs and let them stay there for a good while.   In fact, they left them spinning like that day and night, for three to six months or more at a time.  The hens went around and around, they clucked and they cackled and they laid their eggs, and as far as those chickens were concerned that was what ordinary life was like: a steady pull of two-and-a-half gs.  Some of those chickens spent the larger portion of their lifetimes in that goddamn accelerator.

Well, it was easy to predict what would happen.  Their bones would get stronger and their muscles would get bigger – because they had all that extra gravity to work against.  A total of twenty-three generations of hens was spun around like this and the same thing happened every time. When the accelerator was turned off, out walked … great Mambo chicken!

These chronically accelerated fowl were paragons of brute strength and endurance.  They’d lost excess body fat, their hearts were pumping out greater-than-normal volumes of blood, and their extensor muscles were bigger than ever.   In consequence of all this, the high-g chickens had developed a three-fold increase in their ability to do work, as measured by wingbeating exercises and treadmill tests.

I haven’t had a chance to track them down (and to be honest I’m afraid I’ll be disappointed), but supposedly these are the references:

Smith, A. H, “Physiological Changes Associated with Long-Term Increases in Acceleration.” In “Life Sciences and Space Research XIV”, edited by P.H.A. Sneath. Berlin:Akademie-Verlag, 1976.
 
Smith, A. H., and C. F. Kelly, “Biological Effects of Chronic Acceleration.” Naval Research Reviews 18:1 (1965)
 
Smith, A. H., and C. F. Kelly,  “Influence of Chronic Acceleration upon Growth and Body Composition.” Annals of the New York Academy of Sciences 110: 413 (1963)

Interactive wave and simple harmonic motion applets

A couple of useful wave simulations, useful if you’re having trouble developing an intuition for the connection between amplitude, wavelength, frequency and speed.

  1. Make your own wave.  Looks like it has some dispersion, so propagating pulses change shape over time.
  2. Several moderately interactive wave simulations.
  3. Reflection as superposition of right- and leftward waves.
  4. Resonance in simple harmonic motion.

Newton’s 3rd law is tricky

Many people have trouble really understanding Newton’s laws.  Including, apparently, the Chinese government, which is building a prototype engine that violates Newton’s third law.  The most straightforward critique describes the problem as a violation of conservation of momentum, but a basic consequence of Newton’s third law (that a closed system cannot produce a net force on itself) is equivalent to conservation of momentum (a closed system cannot spontaneously accelerate).

Swimming: high vs low Reynolds number

The Reynolds number (Re) is a dimensionless quantity that describes whether inertial effects (such as coasting) are important during propulsion in a fluid.  The mathematical description of fluid flow predicts that certain kinds of motion that work perfectly well for high Reynolds number propulsion completely fail at low Reynolds number.  In particular, reciprocal (back-and-forth) motions work for Re>1 but not for Re<1.

The Reynolds number of flow around an object is Re = vd?/?, where v is the speed of the object, d is the size of the object, ? is the density of the fluid and ? is the viscosity of the fluid.  The easiest way to observe the effect of Reynolds number is to move an object from swimming in a tank of water (?~1 cP) to swimming in a tank of corn syrup (?~2000 cP) or silicone oil (? as high as 100,000 cP).  This reduces the Reynolds number of the motion by at least a factor of 2,000 to 100,000.

[youtube]http://www.youtube.com/watch?v=4h079P7qRSw[/youtube]

A classic example of this is a rigid flapping arm or oar.  It propels at high Re, but not at low Re.

[youtube]http://www.youtube.com/watch?v=2kkfHj3LHeE[/youtube]

Since a simple reciprocal motion won’t propel an object at Re<1, the motion needs to be more complicated.  Instead of a back-and-forth motion, bacteria continuously turn a helical flagellum to propel themselves.  As demonstrated by this helical propeller model, this strategy does work at low Re.

[youtube]http://www.youtube.com/watch?v=s_5ygWhcxKk[/youtube]

Kinematics on the football field

New football helmets come with built-in accelerometers to detect the severity of collisions on the football field.  You can play spot-the-physics-misconceptions in this summary article, where the author uses acceleration, force and “g force” (whatever that is) interchangeably.  The risk of concussion must depend on the acceleration of the player’s head (since that’s what causes his brain to strike the inside of his skull); I imagine the force matters if you’re worried about the player getting a skull fracture.

Apparently the threshold for concussion is about 100 g over the short term.  The human body can withstand much smaller sustained accelerations, however.  A few minutes at several g can cause a blackout, though both direction and magnitude matter:

  1. acceleration perpendicular to the spine is relatively well tolerated.
  2. acceleration towards the head causes blood to pool in the lower body, resulting in blackout.
  3. acceleration towards the feet causes overpressure in the brain, leading to a cerebral hemorrhage.  Bad.

No one has managed to sneak human experiments past an institutional review board, but if we extrapolate from the great chicken centrifuge experiments of the 1970s, humans should be able to cope with – and perhaps even thrive under – modest excess gs indefinitely.

Shear-thickening fluid

[youtube]http://www.youtube.com/watch?v=f2XQ97XHjVw[/youtube]

Cornstarch is famous as a shear-thickening fluid, leading to both sober and amusing experiments.  If you were trying to transport a block that continuously excreted a thin layer of cornstarch, the harder you pulled it the more frictional force you would feel.  Snails use the exact opposite strategy: they excrete a shear-thinning fluid that allows them to move even though they’re never actually in direct contact with the ground.

Shear-thickening fluids are also used in prototype bulletproof fabrics.

The normal model of friction at an interface posits that static friction perfectly counteracts any force applied to an object, up to the maximum value fsmax= µs N; then friction jumps to the smaller value  fk= µk N and remains constant as long as the object is moving.  This is equivalent to saying that the interface acts like an infinitely strong shear-thickening film up to the yield force of µs N,  instantly shear thins to µk and thereafter is shear neutral.  This analogy is imperfect, however, because the behavior of a real shear-thinning or -thickening fluid depends strongly on its depth, while there is no equivalent idea of the depth associated with µand µk.

Darwin@Home

[youtube]http://www.youtube.com/watch?v=0RpQ1MsjZJ8[/youtube]

Simulated physics and simulated evolution lead to motile ‘organisms’ in computer code.  Videos on the website are beautiful, but it’s unclear how relevant any of this is to real biophysics.

One of the biggest problems with modeling evolution (or thinking about current organisms from an evolutionary perspective) is that it’s unclear exactly where evolution is going.  It’s easy to claim that evolution operates through “survival of the fittest”, but if you can’t define “fittest” that doesn’t tell you much.

Many biologists I’ve talked to believe that extreme selection pressure (such as on bacterial populations) produces organisms that are “optimal” – but no one seems to know exactly what is being optimized.  A physicist is likely to pick something like energy efficiency (the ratio of power generated to power consumed, which is relatively easy to measure if you’re looking at propulsion), but it seems that many organisms do not operate in an energy-limited regime and are (presumably as a result) not particularly energy efficient.

Michael Phelps is worthless

Michael Phelps apparently consumes a wopping 12,000 calories per day. Since he doesn’t look like he’s gaining wait, he must be doing 12,000 calories’ worth of work during his exercise regimen. If only we could harness that energy instead of wasting it swimming laps, it surely must be worth a pretty penny, right?

Nope.

The nutritional calorie (Cal) is 1000 cal (the physicist’s calorie), which comes to 4,200 J (Joules). Counting all taxes and fees, the retail price that I pay for electricity (one of the most expensive forms of energy) is 20 cents per kW·h, or 20 cents per 3.6 MJ. Altogether, then, one day of Michael Phelps’ physical effort is worth about $2.80.

If Michael Phelps were ever to sell just his physical labor, it wouldn’t even pay for a coffee at Starbucks. Conversely, even an “unskilled laborer” is payed almost entirely for his skills – the value of his labor alone is virtually nil.

This is not economical.

This is why John Henry was doomed, and why the opening scene of Conan the Barbarian is economically implausible.

Free fall without air resistance

We physicists claim that all objects will fall with the same acceleration when acted upon only by gravity (free fall).  Any deviations from this law get blamed on air resistance.  In case you’re skeptical of these assertions, one of the most expensive introductory physics demos in history was conducted during an Apollo landing on the moon.

Astronauts dropped a feather and hammer to show that both experienced the same acceleration in the absence of air.  Though the video quality is not the best, it’s is a neat demonstration.  My favorite part is when the astronauts look like they’re dancing with excitement after the drop.