This is a continuation of my attempt to solve the following problem: A snail crawls along a twig of length L0 at 1 cm/d, and the 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 part 2 where I did manage to solve it.
At the end of the last post, I had solved for the equation that describes the position of the snail over time,
Also, recall that the length of the twig is
since it grows at 2 cm/d. We want to find the time t when these two are equal.
[Edit: I missed a very obvious step at this point, so the next few steps will be unhelpful. Free free to skip ahead to the next edit.] We need to isolate the natural logarithm.
Exponentiating both sides gives
Well, the left side is great, but the right side is a mess.
If I bring the 2t – L0 term out, I could split it up.
This was unhelpful and I have no idea how to simplify this.
[Edit: At this point I get back on track.] Let’s go back to the last simple equation.
I should get rid of this complex fraction, even if it means making the numerator more complex.
Oh, it’s so obvious now. I don’t know how I missed realizing that this fraction would simplify.
I feel a bit foolish. Now, let’s exponentiate both sides.
That’s much simpler.
And finally, the long-awaited answer,
Here are my initial observations: It seems “elegant” enough to be a solution to this problem. I like that it depends on e. I also notice that the time it takes is directly proportional to the initial length of the twig. I’m a bit surprised, as the snail’s net speed is not constant but rather increases as it progresses. Also, it’s apparent in retrospect that the first factor, in the equation for the snail’s position over time,
is half the length of the twig at that time, L(t):
I don’t have an intuitive understanding of what the other quantity could represent, nor do I know if the 2 and 1 have any connection to the twig growth and snail inherent velocity. It would be interesting to repeat this but leave all the quantities unspecified.