Update: Added discussion on launch angle at the end of the post.

Edit: The final numbers in this post went through a few rounds of revision. What is the world coming to, when you have to track down missing factors of 2 in your blog posts?!

This week, I’m looking at the strategies and mechanisms by which different animals solve the problem of getting around. I started off by writing about how birds and aquatic animals conserve energy on-the-go. This post is another spinoff on the theme of locomotion.

Here’s a clip from one of my favorite documentaries, David Attenborough’s Life of Mammals. It shows the incredible sifaka lemur of Madagascar, a primate that has a really remarkable way of getting around. (If the embed doesn’t work, you can watch it here)

As they launch out from the trees, they almost look like they’re defying gravity. And so, taking inspiration from Dot Physics, I thought it might be interesting to put physics to use and analyze the flight of the sifaka.

I loaded the above video into Tracker, a handy open source video analysis software. I can then use Tracker to plot the motion of the sifaka. I chose to analyze the jump at about 21 seconds in. I like this shot because it isn’t in slow motion (that messes up the physics), the camera is perfectly still (we expect no less from Attenborough’s crew), and the lemur is leaping in the plane of the camera (there are no skewed perspective issues that would be a pain to deal with). The whole jump lasts under a second, but at 30 frames per second, there should be plenty of data points.

This is what it looks like when you track the sifaka’s motion:

The red dots are the position of the sifaka at every frame. That’s the data. In order to analyze it, we need to set a scale on the video. I drew this yellow line as a reference for 1 unit of size (call it 1 sifaka long). And how big is that?

If we believe this picture that I found on the National Geographic website, then a sifaka is about half the size of this folded arms dude.

Now, to the physics..

While the sifaka flies through the air, the only force acting on it is gravity, which points downwards. So the acceleration of the lemur should also be downwards. (I’m ignoring air resistance. We’ll find out if this is a good idea.)

If we plot its horizontal motion, it should be moving at a fixed speed, with no acceleration. But its vertical motion will give away its acceleration.

This is what we get if we plot at the horizontal position of all the points with respect to time.

The squares are the data points, and the line is a plot of the equation of a straight line

I was amazed by how well they agree, since I expected air resistance to matter a little more. I guess ignoring air resistance is a pretty good approximation.

We find that there’s a straight line relationship between position and time, which implies that the sifaka moves at a constant speed in the horizontal direction. The slope of this line () has units of meters/second (or in our case sifaka/second) and is the speed of the sifaka.

What about the vertical direction? Well, it certainly can’t be a straight line relationship with time, because at some point the sifaka turns and comes back down. Here is what the plot looks like:

The small squares are the vertical positions of the dots plotted versus time, and the red curve is the plot of an equation for a parabola

Here is the vertical launch speed, is acceleration, and is time.

So, over time, the vertical position traces out a parabola, which is a characteristic shape for motion under a fixed acceleration (in this case, the earth is accelerating the lemur downwards). The nice thing about analyzing motion is that we can analyze the horizontal and vertical motion independently of each other.

The fit to the parabola is not great, but it’s not too shabby either. I suspect the main reason for the discrepancy is that its hard to track the center of mass of the sifaka, and if you choose any other place on the sifaka, you’ll also be tracking the spin of the sifaka about its center of mass.

By solving for the values of , and that best match the data, we get the launch speed and acceleration of the lemur.

To be a little more empirical about things, I did this analysis twice, and averaged the results. Here’s what I got:

Horizontal launch speed:
Vertical launch speed:
Vertical acceleration:

The negative sign on the acceleration indicates that gravity is pulling the sifaka downwards  (in the negative y direction). So far things look good qualitatively, but do the numbers work out?

Well, according to National Geographic, the tail of a sifaka monkey is 46 cm, whereas according to wikipedia it is 50 to 60 cm. Let’s go with 50 cm on average. The length scale I drew in Tracker is about the length of the Sifaka’s tail. So we can set 1 sifaka = 0.5 meters.

That gives us a value of for the acceleration caused by gravity, which is within 16% of the known result of . I think that’s pretty darn good for a first stab at video analysis, especially as the sifaka was a blur in each frame and often obscured by trees.

Next, we can use Pythagoras’ theorem in the above velocity triangle to solve for the total launch speed

where and are the horizontal and vertical components of velocity.

This gives a launch speed of 4.25 meters per second or 9.5 miles per hour (15.3 km/h). This speed sounds reasonable to me, as it’s about how fast your typical bicycle moves. If we include a fudge factor that fixes our acceleration to the known result, then the launch speed is actually faster by 16%.

Update: added discussion on launch angle.

We can also solve for the launch angle of the sifaka, by using some high-school trigonometry on the triangle:

Solving for the angle gives 34.7 degrees.

Is this angle correct? Fortunately, Tracker has a handy built in protractor, so we can check it. Marking out the initial leap for both runs, I get an average launch angle of 34.5 degrees.

Which agrees to within half a percent of our result inferred from the physics!! Eerily accurate..

It’s a bit of a coincidence that the result is as close as it is, given the many possible sources of error. However, one reason why this result is so accurate is that the angle comes from a ratio , and so common sources of error (such as error in estimating the length of a sifaka) end up cancelling out. This is also why physicists prefer to measure ratios, rather than numbers that have units (they call such quantities dimensionless).

And there you have it folks, SCIENCE being put to use to answer the burning questions that keep you up at night.

If you want to read more about how the sifakas glide, Darren Naish has a detailed post describing research on the physics of this.

It feels good to be an animal. Unlike trees that are tethered to the ground, we animals have the incredible ability to travel. And we do so in a variety of ways. Some like to walk, others run. Others get around by swimming or flying. There are climbers, leapers, and hoppers, and others that prefer to roll and tumble.

Locomotion certainly affords us a great deal of freedom, but it comes at a considerable energy cost. Through countless generations of incremental evolution, our bodies have arrived at many solutions to balancing our energy budget. Fish have streamlined profiles, birds have hollow bones to stay light, and kangaroos have spring loaded hind legs that seamlessly capture and release the energy needed for flight. In the African savannah, predators chase down their prey using long, muscular legs that give them an efficient stride.

In addition to changes in form, animals can also use strategies to travel more efficiently. Birds that need to fly a long distance often make use of a curious technique. They flap their wings to gain height, and once they builds up enough height, the wings stop moving and they glide back downwards. Many birds repeat this wave-like motion in flight, instead of flying at a fixed altitude.

It’s like the difference between cycling on flat terrain or on an undulating, hilly road. In one case you pedal at a steady pace, in the other you alternately pedal hard and don’t pedal at all. The reason that birds adopt this undulating flight strategy is that it saves them energy.

But what’s special about air? What about animals that live in water? In the ocean, swimming is the equivalent of flying. So do marine animals adopt similar swimming strategies to conserve energy? To answer this question, an international group of researchers led by Adrian Gleiss attached sensors onto sharks and seals. They monitored the swimming motion of the whale shark, the white shark, the northern fur seal, and the southern elephant seal.

Here is an animation that they made directly from their recordings, that shows a whale shark swimming.

It’s as if they’re climbing an imagined hill – they work on the way up, and glide back down. In fact, all four species adopted this undulating swimming strategy. This figure, from their paper, summarizes the authors’ point.

For each animal, you see two plots. The first is a plot of its acceleration, and the second is a plot of its depth. By comparing the two, you can see that all the animals are swimming on the upslope to gain height, and then gliding back down effortlessly, just like a bird, or a cyclist on a hilly road.

The authors emphasize that this is especially remarkable, as these species have distinct evolutionary histories, and very different modes of propulsion. Elephant seals swim using hind limbs modified into flippers, fur seals use their pectoral muscles, and sharks use their tail fin. And yet, we find that in the ocean and the sky, species that are separated by millions of years of evolution are united in their solutions to one of life’s basic problems – how to get around effectively.

But there’s still a puzzle: if wavy swimming is more energy efficient, why don’t all fish do it? Why do some fish swim in this fashion but others chose to swim continuously? The authors claim that it’s all got to do with whether you naturally float or sink. They support this idea with an interesting observation: seals that swim in shallow water do so continuously, but those swimming at greater depths undulate and swim intermittently. The difference is stark – once they exceed a certain depth (15 meters, in the case of the elephant seal), they suddenly become wavy swimmers.

Here’s why the scientists think this happens. If you take a person and (very temporarily) submerge them in water, odds are that they will neither sink nor rise. That’s because humans are what is known as neutrally buoyant, meaning the density of our body exactly matches that of the surrounding water. (This is not quite true. I have a few friends who swear that they sink in water, and they’re probably right. As with any average quantity, there are some that exceed the mean, and others that don’t.)

Seals are in the same boat as us, they don’t need to work to stay afloat. They are naturally buoyant, and so swimming in the wavy way is unnecessary. However, as they dive deeper, things begin to change. As the pressure of the water increases, it squeezes their bodies into a smaller space. The same mass is now packed into a smaller volume, so the seal has become denser than water. Instead of floating, it now sinks. The authors argue that in such a situation, it makes more sense to swim wavy.

And the sharks support this idea. Unlike the seals, they don’t have lungs, or gas bladders like many other fish have. There’s nothing particularly squishy in a shark, and so their body has the same density regardless of depth. And this density exceeds that of water. This means that sharks have to swim to stay afloat. When a shark dies, it sinks like a rock.

According to the authors, this is why sharks always swim in this wavy fashion, irrespective of depth. They suggest that the more likely an animal is to sink, the more of an energy boost it gets from swimming in this interrupted manner. In this way, these heavy animals (or more accurately, dense animals) have all hit on the same clever strategy for getting the best mileage.

Reference:

Gleiss AC, Jorgensen SJ, Liebsch N, Sala JE, Norman B, Hays GC, Quintana F, Grundy E, Campagna C, Trites AW, Block BA, & Wilson RP (2011). Convergent evolution in locomotory patterns of flying and swimming animals. Nature communications, 2 PMID: 21673673

Image References:

All figures are from the paper. The youtube video is an upload of the supplementary video attached with the paper.

Opening image: There is no E.T. around, by Camil Tulcan. Creative Commons licensed.

The image of a floating woman was taken in Weeki Wachi Springs, Florida (1947) by Toni Frissell. Public Domain.