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Two Snakes eating each other
By Anonymous on Tuesday, December 14, 1999 - 10:23 pm:

Two snakes of the same size are eating each other at the same rate and they started at the same time. What will happen?

It was in a maths quiz thing but nobody got it perfectly right.

CN

By Graham Lee (P1021) on Monday, December 27, 1999 - 07:37 pm:

We will assume that a snake is a perfectly uniform flexible (though inextensible) rod, and that it does not die when the heart gets swallowed!
Also, the two snakes are equal (though some are more equal than others for all lovers of a good book)

What should happen is that the two snakes should, in eating each other, make each other shorter, so that the radius of the shape (could be a circle, an ellipse, or some other, non-conic shape) made by the two snakes decreases continually. When the radius of the shape is equal to the thickness of the snake, they will be able to consume each other no longer. Of course, if the snakes are modelled as having negligible thickness, then they will perfectly devour each other.

Happy New Year (the millennium's next year),

GL.

By Richard Dwight (Rpd25) on Monday, December 27, 1999 - 11:23 pm:

Graham's answer seems about right to me, but you say that nobody in the quiz got it "perfectly right". So what was the perfect answer? I'm sure someone must have guessed that the two thin snakes completely devour each other.

Richard.

By Clare Nicholson (P1732) on Thursday, December 30, 1999 - 08:23 pm:

Well, Yes of course some people gave that answer but another way of looking at it is that what one snake loses from itself it gains from the other. If Graham's answer was perfectly right then the points in the quiz proberly would have been lost for the not full explanation of the radius of the circle or ellipse etc. getting smaller

CN.

By Clare Nicholson (P1732) on Friday, December 31, 1999 - 02:11 pm:

And also they must be left with at least two mouths cause they can't eat each other at exactly the same time, one must eat the other or neither get totally eatten.

By Graham Lee (P1021) on Sunday, January 2, 2000 - 12:23 pm:

Clare is correct, if you model a snake as a snake.
Of course, being a physicist I didn't do that!!!!

If a snake actually "consumes" that which it is fed by the other snake, then their thickness will gradually increase, call the thickness of one snake T, and say that it increases by a scalar amount with respect to time, say k. Therefore we get T=kt. The radius of the shape made by the two snakes, r, decreases as a function of time, say r=r0/t, where r0is the original radius. The two snakes will consume each other until T=r, i.e. kt=r0/t or t²=r0/k.

By Graham Lee (P1021) on Thursday, January 6, 2000 - 02:17 pm:

I've just been trying to work out whether the fact that there are two solutions two the time t is relevant, as it is in things like projectile equations and sometimes can infer complex answers (i.e. those that involve i, the square root of -1), but I think that in this case the negative answer can just be discarded.
Anyone want to add anything that I haven't thought of, such as elasticity (and would one snake's teeth fall out before the other snake snapped?), but please don't go around setting up different parameters for each snake, as this thing is hard enough as it is!!!

In fact, I'm going to set you lot that particular problem. There's two elastic snakes eating each other, and when they bite in to each other they also pull back with a force F. Find the radius of the circle the two snakes create as a function of time, taking the original length of each snake to be l0 and the moduli of elasticity as M (I would use lambda, but I can't have HTML tags and Discus tags in the same message!) in each case.

Have Fun!!!

GL.

By Michael Doré (P904) on Thursday, January 6, 2000 - 08:28 pm:

I think that there could be problems when you let r = k/t as this implies the radius was unbounded...

Anyway here is my attempt at your rather tricky question.

I'm going to assume for the moment that we can say the tension is given by Mx/l where x is the extension of the snake. In fact I'm not sure at all about this: what forces hold the snake in a circle?

Anyway, if the radius is r then the acceleration of the COM is the second derivative of -2r/pi (using the formula for the COM of a semi-circle). So the radius displays SHM give or take a constant with period root 8m lo/M I hope.

But still I would be interested to hear how the circle is maintained.

Michael

By Graham Lee (P1021) on Tuesday, January 11, 2000 - 06:56 pm:

The circle is created as a panderance to those among us who like to draw big diagrams (as my Mechanics teacher alawys insists). Basically, two pieces of string connected at each end to each other can be modelled as a circle, so I can talk about the radius decreasing rather than having a straight line that actually contains two snakes. It's just easier to visualise like that.
Anyhow, snakes could presumably use their lateral muscles, scales and friction between themselves and the ground (ok, now I introduce m as well) to contain themselves within a cricle, if they so desired. Anyway, as they eat each other up, they might start as two collinear snakes but would have to bend round as their thickness approached their length, therefore tending towards a circular shape.
If the radius experiences SHM (I assume you mean that each point on the snake experiences SHM about the center of the "circle"), then surely having eaten each other, the snakes then grow back to "radius" Ro, but having flipped sides?
No, because you define the motion as having stopped when they completely consume each other (or can eat no more). But, if I defined the radius of the circle as R->Ro- as t->0+, then I would not have the problem of infinite snakes. Of course, time could start at t=1, in which case there is no problem at all.

GL. (a.k.a. "Casanova", but G.K.Y.)

By Michael Doré (P904) on Wednesday, January 12, 2000 - 01:46 pm:

Aha - I think I understand what you're saying now. So ignore my last reply (I was answering a different question).

When you say the snake pulls with a force F then is this constant? Do the snakes eat each other at a uniform rate?

Thanks,

Michael

By Graham Lee (P1021) on Friday, January 14, 2000 - 06:35 pm:

Hello everyone! (Espesh Michael)

When I introduced forces I was actually talking about those that would be required to maintain a semicircular snake (try saying that with your teeth out). But now that you mention it, there is no reason to assume that snakes don't exert a constant force F upon their prey. At least, over a particular amount of time. You see, if we start having elastic snakes, though of course all tensions cancel while the snakes are in equilibrium, as soon as one opens its mouth to take a bite - PING!!!! snake goes flying all over the place.
Now, imagine a man eating a single, 20ft strand of spaghetti, and the sort of strategy he would employ were he in a Laurel and Hardy film. This is how two snakes would eat each other to avoid becoming a ballistics question.

GL.

By Graham Lee (P1021) on Friday, January 14, 2000 - 06:41 pm:

P.S. Why should the center of mass move at all? Whether the two snakes are annular or collinear, when they eat each other at constant rate, the same amount of length disappears from each side, so they either become a smaller annulus or a shorter line, but their centroid (and they are uniform I have decided, so C.O.M.=centroid) isn't going anywhere. Obviously their density increases (as they must have to put all that snake somewhere), and in fact their thickness should increase as their length decreases (conservation of volume).

By Graham Lee (P1021) on Saturday, January 15, 2000 - 02:39 pm:

D'oh!
If their density can change, their volume doesn't need to be conserved at all! If anyone ever understood my thought processes, they could do a Scientific American Presents all on their own!

(a confused)GL.

By GRANT on Sunday, February 6, 2000 - 02:34 pm:

This is really confusing because don't you get to the point where neither snake can eat any more else they eat each others mouth, and then they wouldn't be eating, or something weird like that?

Please somebody explain this simply to me. My brain's boggled by the though of a snake eating a snake while another snake eats that snake.

By Clare Nicholson (P1732) on Thursday, February 10, 2000 - 09:45 pm:

That is exactly what I thought when I first put the problem on the page, I'm only 13 and I understand less now than before!

Clare Nicholson (Problem writer!)

By Charles Wray (M354) on Saturday, February 26, 2000 - 04:27 am:

Graham while not a herpotologist I do know that snakes don't "bite" but ingest the victim by peristalsis. As a result the snake being swallowed and the swallower snake decrease in length uniformly until only a half a snake remains (of each) hence the circle or ellipse(whatever remaining would be one snake long.