You may be aware that some types of caterpillars travel as a group and in that group some caterpillars climb on the backs of other caterpillars and walk forward over the seething mass of caterpillars underneath them. At the front of the pack they drop off onto the ground and are in turn walked over.
Various experts have said that there are several reasons for this including appearing to be a bigger animal than just a bunch of caterpillars.
Of interest is the theory that the group of caterpillars travels at twice the speed of an individial caterpillar, so the group moves from a place of safety to another place of safety in half the time and presumably with reduced risk of being attacked.
The problem is the math. Sure the bottom guys are travelling at unit speed, and the top guys are travelling at 2x unit speed, but surely the average speed of the ensemble is only 1.5 x unit speed rather than 2x unit speed?
I assure you the problem is far more difficult than at first glance which is why I raise it here.
(Possibly unrelated is the group speed of waves compared to the individual speed of waves.)
It seems to depend in a complicated way on how the caterpillars enter/exit the top layer.
The simplest way I can think of modeling it is that a caterpillar jumps layers as soon as a slot is open. So something like this sequence will keep repeating:
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Assuming that we measure speed in body lengths/time unit, this sequence takes two time units to repeat and adds a body length each time. So 0.5, which added to the baseline 1.0 get the 1.5 speed that you mentioned.
This adds two lengths every two time units, for a total speed of 2. But it’s even less realistic in the transfer of caterpillars between layers. More layers adds more speed but it gets ridiculous after a while, and I don’t know how to model it.
You’re right. Think of caterpillar tracks (heh) on a tank, the top of the track goes at 2 x the tank’s speed while the bottom of the track is stationary. The tank is therefore travelling at the average of the top and bottom of the track. Caterpillars would be the same but because the bottom caterpillars are moving, the group speed is 1.5 x the unit speed.
An individual caterpillar might be able to make the trip at twice the normal speed only if the trip is short enough and the caterpillar is positioned so that he makes the entire journey as a top caterpillar and none of it on the bottom.
This relates to something that I was trying to explain to some friends in the pub recently. At 60mph, a point on a car tyre will change speed from zero to 120mph, about 14 times every second. (A 15" rim will be rotating at around 812rpm).
This is exactly the same as the tank track analogy (which I wish I had thought of) above.
That’s a good observation. It might be a little less than 1.5x, depending on the number of caterpillars on each layer (and assuming only one layer). If, say, 60 percent of them are typically on the first layer, the speed would be 1.4x. We’d need a caterpillar behaviorist to know for sure.
The burden on a caterpillar carrying another caterpillar on his shoulders would slow it down, don’cha think? Especially if caterpillars were piled ten layers deep. “He ain’t heavy. He’s my brother!”
When you think about it, the caterpillars at the back are leaping up at 2x unit speed and at the front leaping down at 2x unit speed, so the front and back edges must be moving at 2x unit speed.
So how is it that the average caterpillar speed is 1.5 units while the group speed is 2 units?
How in the world can a caterpillar in the back get on top of another caterpillar if they’re all moving the same speed? If Henry runs all the way to the front and plops down in front of Bob, then Bob has to look at Henry’s caterbutt the rest of the trip cuz nobody is moving faster than Henry. That it to say, Henry’s just as fast as anyone else.
The only way to get on top of the pack is to temporarily move faster than them. Then when you end up at the back again, you have to slow down and let your mates climb over you.
Jump to about the 3 minute mark to see a simulation done in Lego stop motion that absolutely and unequivocally proves the swarm can move faster than an individual.
Lets do the math. Assume each caterpillar is 1 cm and can crawl its own length in 1 second: speed is 1 cm /s.
Imagine the top and bottom rows of caterpillars are perfectly in line, so the first one on top is the front edge. Now start the clock. In 1 sec, that caterpillar on top will move 2 cm. He now drops to the bottom but* continues to be the front edge* for 1 more second, only advancing 1 centimeter, until the one crawling on his back also makes it to the front and replaces him as front edge.
So, the front edge moves 2 centimeters in the first second, but only 1 centimeter in the 2nd second, or 3 centimeters / 2 seconds = 1.5 cm /s.
Piggybacking on **newme **to answer JezzOZ’s objection …
As a caterpillar becomes the rearmost they must jump upwards *and forwards * relative to the pack to land on the one just ahead. So he/she is moving faster than the pack at the moment.
Can we pretend we are talking about cats instead? It’s quicker to write.
It can only work if the cats aren’t walking at their top speed. They need to hold something reserve so they can get on top of the cat in front once they reach the rare.
[QUOTE=LSLGuy;18000956 they must jump upwards *and forwards * relative to the pack to land on the one just ahead. So he/she is moving faster than the pack at the moment.[/QUOTE]
Not so. Consider the situation of James bond driving onto the back of a moving truck. He approaches the truck at close to zero difference in velocity - then assuming he has front wheel drive he instantly accellerates when his wheels land on the truck tray and he then has to brake fast to stop hitting the end. (This also works with rear-wheel drive but the transition occurs at a later moment.
Similarly a caterpillar transitions from 1 unit to 2 unit velocity in minimal time. At no stage does it exceed 2x velocity (elongation of caterpillar etc ignored for the purposes.)
If the overtaking caterpillar must necessarily move faster to overtake, it just shrinks in the direction of its travel and its time slows down, exactly counteracting the effects of its greater speed. (See: Einstein.) This becomes especially noticeable when you have a whole swarm of them doing this, in which case nobody gets anywhere. This is just the same phenomenon we’ve all observed while trying to pass other cars on the freeway at rush hour.
No, we can’t just pretend we are talking about cats. Cats, like caterpillars and everyone else, are just large macroscopic collections of interacting quantum probability wave functions (or particles), BUT cats have developed the skill of manipulating the probabilities of their own superposed states. This was all discussed and analyzed in this thread last October; see Post #8et seq. for my quantum analysis and others’ commentary in more detail.
