This is a widely circulated interesting problem. Perhaps many readers have already heard of it, but being well-known does not necessarily mean being “well-solved.” Here, I will present the problem, write down my own answer, and discuss my views on it. The problem is as follows:
The Ant Racing the Rubber Rope
An ant crawls at a constant speed of v_0 = 1\text{ cm/s} from one end to the other along a rubber rope with an initial length of l = 100\text{ meters}. Every second, the rubber rope stretches by 100\text{ meters}. For example, after 10\text{ seconds}, the rope has stretched by 1000\text{ meters}. Assume the rubber rope can be stretched indefinitely and the stretching is uniform. The ant’s position naturally moves forward relative to the uniform stretching of the rope. The question is: if this continues, can the ant eventually reach the other end of the rubber rope?
Solution Process
If one has a basic foundation in calculus, I believe this problem is not difficult. Can it be solved within the scope of high school mathematics? I don’t think so, because although the following process can be discretized to obtain an approximate answer, estimating the time requires calculating the approximate value of the series: S(n) = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} This also requires the use of calculus theory.
Suppose the ant is at position x at time t, and its velocity at this moment is \dot{x}. Its velocity is its own crawling speed (v_0 = 1\text{ cm/s}) plus the stretching speed of the rope. The total stretching speed of the rope is v_1 = 100\text{ m/s}. At this time, the length of the rope is l + v_1 t. The stretching speed at position x is v_2 = \frac{v_1 x}{l + v_1 t} (distributed proportionally), so we have: \dot{x} = \frac{v_1 x}{l + v_1 t} + v_0 Let v_1 x = y and l + v_1 t = \tau. The above equation becomes: \frac{dy}{d\tau} = \frac{y}{\tau} + v_0 Further let \frac{y}{\tau} = u and \ln \tau = v. Substituting these, we get: \frac{du}{dv} = v_0 So: u = v_0 v - v_0 \ln l; \quad \frac{y}{\tau} = v_0 \ln \tau - v_0 \ln l Finally, we get: \ln\left(1 + \frac{v_1 t}{l}\right) = \frac{v_1 x}{v_0 (l + v_1 t)} To crawl across the rubber rope, at least x = l + v_1 t, so: \ln\left(1 + \frac{v_1 t}{l}\right) = \frac{v_1}{v_0} For the problem in this article, this means: \ln(1 + t) = 10000 Solving this gives t \approx 10^{4343}\text{ s}, which is 10^{4335}\text{ years}!
My Answer
The answer shows that in a finite amount of time, the ant can crawl across. However, the time required is 10^{4335}\text{ years}! This not only exceeds the lifespan of an ant but also exceeds the age of the universe as we currently know it! From this perspective, or rather, from a physical perspective, this is impossible to achieve. That is to say, the answer given by a physicist should be that it is impossible to crawl across!
Physics is a discipline that combines experiment and theory, and experiments inevitably involve errors. Within the precision of an experiment, if an error is smaller than a certain value, it is considered equal; if it is larger than a certain value, it is considered infinite, and so on. For example, given a square with a side length of 1.0, its diagonal length is 1.4, not \sqrt{2}. Regarding the answer in this article, 10^{4335}\text{ years} is considered an infinite amount of time by (current) physicists, and thus it is unachievable.
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