Reaching the Unreachable

December 15, 2022

Share this article:

Have you ever felt like giving up? Wanting to quit something because it seems impossible to reach your goal (for example passing Difference- and Differential Equations)? Of course you have, but don’t give up too fast! Sometimes, it is possible to succeed in what seemed like a hopeless mission (like passing Difference- and Differential Equations). Today we will discuss the “Ant on the rubber band” paradox which perfectly displays such a situation.  

The ant on the rubber band 

The backstory of the “Ant on the rubber band” paradox is as follows. Imagine a rubber band which is 1 km long, and an ant on this rubber band at one of the end points. The ant starts crawling along the rubber band with a speed of 1 cm per second and wants to reach the other end point of the rubber band. At the same time, the rubber band begins streching at a constant rate of 1 km per second, so that after 1 second the band is 2 km long, after 2 seconds it is 3 km long, etcetera. Then, the question is: will the ant ever reach the other side of the rubber band?  

At first you might think that the answer is obvious. After all, the rubber band is stretching way faster than the ant is crawling . But the key to understanding this paradox is noticing that when the band stretches, it will also move the ant. We can see this by looking at the proportion of the band that the ant has covered after n seconds. For example, after 1 second the ant has travelled 1 cm which is equal to \frac{1}{100} of the length of the rubber band. But don’t forget that the band also stretches at the same time. After 1 second, the rubber band stretches from 1 km to 2 km. The crucial part is that the band stretches evenly across its full length, thus the 0,99 km in front of the ant becomes 1,98 km and the 0,01 km behind the ant doubles to 0,02 km.  Then, the ant has covered \frac{2}{200} = \frac{1}{100} of the length of the rubber band. Thus despite the band stretches, the proportion of the length of the band covered by the ant has not changed. Let us formalize this mathematically.  

Mathematical proof 

Consider a rubber band on the x-axis, with starting point x=0 and end point x=a+bt where a, b>0 are constants and t denotes the time. The ant starts moving along the rope at t=0 with a constant velocity v>0 relative to the rope at its current position. Assume the rope stretches suddenly after each second, so that the end point moves from x=a to x=a+b at t=1, and from x=a+b to x=a+2b at t=2, etc.  

Let \theta(t) be the proportion of the distance from the starting point to the end point which the ant has covered at time t. After 1 second, the ant has travelled distance v but the rope has also streched and because of that the end point moved to a+b.  Then,

    \[\theta(1) = \frac{v}{a+b}.\]

In the next second, the ant has again travelled distance v but the length of the rope also stretched to a+2b , thus the ant travelled \frac{v}{a+2b} of the distance from the starting point to the end point.  Then, 

    \[\theta(2) = \theta(1) + \frac{v}{a+2b} = \frac{v}{a+b} + \frac{v}{a+2b}.\]

In general, for any n \in N, we have

    \[\theta(n) = \frac{v}{a+b} + \frac{v}{a+2b} + ….+ \frac{v}{a+nb} = \sum \frac{v}{a+nb}.\]

Note that for any m \in N, we have

    \[\frac{v}{a+mb} \geq \frac{v}{ma+mb} = \frac{1}{m} * \frac{v}{a+b}.\]

Thus 

    \[\theta(n) = \sum \frac{v}{a+nb} \ge  \frac{v}{a+b} * \sum \frac{1}{k}.\]

 

We know

    \[\sum \frac{1}{k} = 1 + \frac{1}{2} + … + \frac{1}{n}\]

is a harmonic series which diverges. Consequently we can find an Ñ \in N such that

    \[\sum \frac{1}{k} = 1 + \frac{1}{2} + … + \frac{1}{n} \geq \frac{a+b}{v}.\]

which implies that

    \[\theta(\tilde{N}) = \sum \frac{v}{a+nb} \geq \frac{v}{a+b} * \sum \frac{1}{k} \geq \frac{v}{a+b} * \frac{a+b}{v} = 1.\]

 Recall that \theta(t) is the proportion of the length of the rope travelled in total by the ant at time t. We have just shown that there exists an Ñ such that \theta(\tilde{N}) \geq 1 and therefore the proportion of the length covered by the ant will eventually be 1. Hence, given enougn time, the ant will always reach the other side of the rubber band! 

To return to the question discussed in the introdution of this article, yes, it is possible to succeed in what seemed like an impossible mission. However, that does not mean this is always easy. Let us take a look at our ant crawling on the rubber band. It is to be expected that, with the given speed of the ant of 1 cm per second, the speed of 1 km per second by which the rubber band is stretching and the starting length of the rubber band of 1 km, the ant will need a really long time to reach the other side of the rubber band. To be precise, in this case it will take approximately 2.8 * 10^43429 seconds until the ant has reached the other side. Nevertheless, if the ant really has the ambition of reaching the other side, it is possible. The moral of this story is that you can succeed at things which appeared to be undoable. Just like the ant, do not give up too fast because you are capable of a lot more than you think!  

Read more

Van Gogh – not just a painter

Van Gogh – not just a painter

Van Gogh is undoubtedly one of the most famous painters of all time, but perhaps it’s time to consider him a mathematician. Starry Night is one of Van Gogh’s most renowned paintings, depicting just that: a starry night. With a bright moon on the right, and Venus in...

The Ewing Theory

The Ewing Theory

What happens if your team’s best player goes down with an injury, leaves for another team or retires? Your team should be less successful, right? Well, as it turns out, this is not necessarily always true. Sometimes, a team can inexplicably flourish without their...

Why your Dobble cards always match

Why your Dobble cards always match

Dobble: a game played by kids, but still very popular among adults. In the game, you have to draw two random cards and place them face-up on the table between all the players. Then, you have to look for the identical symbol between the two cards. Between every two...