Flight to Infinity

Two trains start 20 miles apart and head toward each other, each going at a steady speed of 10 m.p.h. At the same time, a bird that travels at a steady 15 m.p.h. starts from the front of the first train and flies to the front of the second one, and then turns around and flies to the front of the first train again, and continues in this manner till the trains meet.

Questions:-

(1) How much distance will the bird cover before the trains meet?

(2) How many trips will the bird make?

Question 1 is easy. The relative speed between the two trains is 20 mph. So they meet after $\frac{20}{20} = 1$ hour. The bird would have travelled $15 * 1 = 15$ miles.

Question 2 however is not so simple. Consider that the trains are $x$ miles apart. The bird takes $\frac{x}{15 + 10}$ hours to make a trip. The distance between the trains reduces to $x - \left(2 * 10 * \frac{x}{25}\right) = \frac{x}{5}$ miles. This means that one trip of the bird reduces the distance between the trains to one-fifth. Given that the trains start 20 miles apart, we can say the distance between the trains after $t$ trips of the bird would be $\frac{20}{5^t}$. Thus we get the total number of trips of the bird as $T$:

$\frac{20}{5^T} = 0 \Rightarrow T = \infty$

The answer, though mathematically logical and valid, is a little hard to digest. At least I found it so. How can a bird make infinite trips? Bicyclists, flies, speed.. all seem so real and yet the answer seems so detached from reality. The trains meet in finite time, the bird travels a finite distance (answer of question 1) and yet the bird seems to have made infinite trips... How can infinity exist within things that are seemingly finite?

Infinity is a hard concept to understand and feel. I will not claim to have understood it. I have just made my peace with it. I do however seem to have an explanation as to how this seemingly realistic problem transcends to the unreal. When I say "seemingly realistic", I mean that it is perfectly possible to have trains which travel at constant speed and birds which fly at constant speed. By unreal, I am referring to how hard it is to believe that a bird can make infinite trips.

Although the speeds and situations are realistic, it is impossible to have a bird which touches a train and turns back immediately. The problem assumes that, if the bird is perched on a train at time $t$, it is in flight at times $t - \epsilon$ and $t + \epsilon$, for all $\epsilon > 0$. This is exactly where it moves to the unreal.

However, if we did consider the bird to take even a minimal, but non-zero, time to switch between the trains (say $\lambda$), we would have - bird takes $\frac{x}{25} + \lambda$ hours to make a trip. Thus one trip of the bird would reduce the distance between the trains to $\frac{x}{5} - 20 \times \lambda$. Obviously, then, we would have a finite number of trips.

Coming back to the concept of infinity, it is hard to feel it. The best you can do is have a fleeting image of it when you close your eyes and imagine the universe. A mosquito itch on your knee immediately brings you back to your sad room lit with fluorescent light. I end with this exquisite quote by Jacob Bernoulli

... Even as the finite encloses an infinite series

And in the unlimited limits appear,

So the soul of immensity dwells in minuta

And in the narrowest limits, no limits inhere

What joy to discern the minute in infinity!

The vast to perceive in the small, what Divinity ...