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The following problem is from my textbook:

THE QUESTION

A car is moving at a constant speed of $40 km/h$ along a straight road which heads towards a large vertical wall and makes a sharp $90^\circ$ turn by the side if the wall. A fly flying at a constant speed of $100 km/h$, starts from the wall towards the car at an instant when the car is $20 km$ away, flies until it reaches the glasspane of the car and returns to the wall at the same speed. It continues to fly between the car and the wall till the car makes the $90^\circ$ turn.

(a) What is the total distance the fly has travelled during this period?

(b) How many trips has it made between the car and the wall?

WHAT I ASK

I am able to calculate the part $(a)$ which is $50 km$ but didn't solve the second part. I looked at the solution and it said the fly makes infinte trips.

The solution derives a formula for distance of the car from the wall at the beginning of the $n^{th}$ trip of the fly and it was this

$(\frac{3}{7})^{n-1} \times 20 km$

and it said "Trips will go on till the car reaches the turn that is the distance reduces to zero. This will be the case when $n$ becomes infinity. Hence the fly makes infinite trips."

Well from the nature of the problem and from mathematics the answer is infinity but how on earth is it physically possible for the fly to make infinite trips. The physical answer must be a finite number. Is it possible to work it out from the given facts?

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  • $\begingroup$ I guess you need the size of the fly and the car and how close the car can actually get to the wall. Does the fly land on the windshield? $\endgroup$ Jul 23, 2017 at 20:15
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    $\begingroup$ The answer to this question (which is very very old) is all over the internet, seriously. mathforum.org/dr.math/faq/faq.fly.trains.html $\endgroup$
    – user163104
    Jul 23, 2017 at 20:18
  • $\begingroup$ @Countt010 can you enlighten me with some resources? $\endgroup$
    – anjanik012
    Jul 23, 2017 at 20:23
  • $\begingroup$ Also, you might look up the solution to Zeno's paradox, which is probably also in your textbook as a " insoluble" problem. The answer is in terms of the time taken , not about the distance, which is the trick in the question. $\endgroup$
    – user163104
    Jul 23, 2017 at 20:24
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    $\begingroup$ Whenever math gives results like "infinite", in reality it just means "many". Of course you are right that this will not be infinite, because in reality the fly decelerates and accelerates when turning each time and it also is not a single point but has a tiny spatial volume. All such things are irrelevant and negligible for the normal real-life tasks you could encounter - but this task you have here is physically useless. It brings no insight into anything physical, but only trains mathematical logic. When that is the case, don't overthink it to be simulating reality. $\endgroup$
    – Steeven
    Jul 23, 2017 at 20:50

1 Answer 1

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There are two main reason to the fact that the fly wouldn't really make infinite trips:

  • First of all, the question assumes that it can travel at $100\,km/h$ in one direction, and then instantly change its direction. This means that the fly's acceleration is infinite during a small interval of time, or, in other words, that it experienced an infinite force. This is not physically possible, yet on the beginning of the experience, it is a reasonnable assumption, but as the car gets closer to the wall this hypothesis becomes more and more irrealistic, since the fly has not the time to perform the U-turn.

  • Secondly, a real fly has a non-null size, while the answer assumes that its size is null. For a real fly, when the car will get to, say, $5\,mm$ of the wall, the fly will be smashed, and the experience ends.

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