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 L0 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.
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
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 .
This is a disaster.
I will have to study this more and return to this problem later.