A friend mentioned this problem a while back, and while I was able to use a computer to get an accurate numerical solution, I really want to find an analytical solution using math. The problem is that it will require differential equations, which I no longer remember. **Edit: I haven’t yet been able to solve this problem, but I’m preserving my initial attempts below. See Part 2 where I made more progress.
**

The problem is this: a snail starts crawling along a twig, from one end towards the other. The snail can crawl at 1 cm/d. But the twig is growing at 2 cm/d. The twist is that the twig is adding material along its entire length, so the snail will be carried forward a bit, too, depending on where it is. Will the snail ever reach the end? Let’s let *L _{0}* be the initial length of the twig.

Let’s say the snail crawls at 1 cm/d and the twig grows at 2 cm/d. The length of the twig at time *t* would be

.

For any point along the twig, the velocity will be proportional to the distance along the length of the twig, so at a distance *x*

Let’s double-check this: the beginning of the twig is :

.

And at the end of the twig, :

.

The velocity of the snail will include both its movement along the twig (at 1 cm/d) and the growth of the twig. It will be therefore

Since the velocity is the derivative of its position,

and we want to know if and when . In other words, solve for *t*:

Taking the derivative with respect to *t*,

Note that this makes sense, at the time *t* when the snail reaches the end, … wait, no it doesn’t, that’s not correct. Since the functions are only equal at a specific time *t*, I can’t differentiate them and expect them to be equal.

Going back:

Is this an linear differential equation? I hope so, since I’m hoping that will make it easier to solve (though at this point, I don’t remember how to solve linear or nonlinear differential equations.

.

I think I got it into the correct form. I don’t yet know what to do from here, so I’ll read on and also refresh my memory of differentiation rules.

I actually don’t think we’ll need complex differential equation techniques. We should be able to just integrate this entire equation with respect to *t*. I think I see why linear differential equations are easier to solve.

By the sum rule,

.

I spoke too soon. The first term is easy to integrate and the last is trivial, but I don’t know how to integrate the second.

I can rearrange the terms and combine the constants

and I’m now stuck.

OK, I think we might be able to use integration by parts, using the formula

.

If I set

and set

then to find *v*, we need to integrate:

.

And to find *du*, we need to take the derivative of *u*. I’m just going to use the product rule, since I’m rusty at this and I don’t remember the quotient rule.

I’m going to keep going, but I just realized that I am not going to be able to integrate .

Shoot.

.

This is a disaster.

I will have to study this more and return to this problem later.

[…] differential equations textbook and I think I am ready to further tackle the snail problem (see yesterday’s post). In short, a snail crawls along a twig of length L0 at 1 cm/d, and the twig grows 2 cm/d. Will the […]

Pingback by Snail Problem, Part 2 | Ancora Imparo — 22 February 2014 @ 23:57 UTC

[…] twig grows (along its entire length) at 2 cm/d. Will the snail reach the end, and when? You can see part 1 where I successfully set up the differential equation and unsuccessfully tried to solve it, and […]

Pingback by Snail Problem, Part 3 | Ancora Imparo — 23 February 2014 @ 20:28 UTC

[…] the last three posts (see Snail Problem parts 1, 2, and 3), I tackled the following problem: a snail crawls along a twig of length L0 at 1 cm/d, […]

Pingback by Snail Problem, Part 4 | Ancora Imparo — 27 February 2014 @ 06:29 UTC

[…] twig grows (along its entire length) at 2 cm/d. Will the snail reach the end, and when? In parts 1, 2, and 3, I solved this problem, and then in part 4 I turned to the general case, leaving the […]

Pingback by Snail Problem, Part 5 | Ancora Imparo — 28 February 2014 @ 03:04 UTC

[…] http://imparo.wordpress.com/2014/02/22/snail-problem/ […]

Pingback by Snail Problem, continued | Logic, Astronomy, Science, and Ideas Too — 1 March 2014 @ 14:41 UTC