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The following question and answer is taken from careerbless

A naughty bird is sitting on top of the car. It sees another car approaching it at a distance of 12 km. The speed of the two cars is 60 kmph each. The bird starts flying from the first car and moves towards the second car,reaches the second car and come back to the first car and so on. If the speed at which bird flies is 120kmph then

1)the total distance travelled by the bird before the crash is?

2)the total distance travelled by the bird before it reaches the second car for the second time is?

3)the total number of times that the bird reaches the bonnet of the second car is(theoretically)?

I am clear with the explanation available in the mentioned site for 1 and 2.

For the third part, i.e., the total number of times that the bird reaches the bonnet of the second car is(theoretically), answer given is infinite times with the explanation proving the infinite sequence.

But we know it cannot go on infinite times. It has to be a finite number if we look at it practically. How to explain this?

I am puzzled because the explanation looks convincing (infinite times) whereas we know it cannot be infinite times. Please help in understanding this contradiction.

EDIT: Adding the explanation(below) from careerbless as advised by @DylanSp for clarify and my aim is to understand why it looks as infinite times(theoretically) whereas we know it is finite practically or where my understanding is wrong. (it was a detailed explanation and that is why I have not added initially)

(3) infinite times

As explained for the previous case,

The bird reaches the second car in $\dfrac{12}{180}=\dfrac{1}{15}$ hour for the first time.

In this time, the cars together covers a distance of $\dfrac{1}{15}×120=8$ km and therefore the distance between the cars becomes 12-8=4 km.

The bird reaches back the first car in $\dfrac{4}{180}=\dfrac{1}{45}$ hour.

In this time, the cars together covers a distance of $\dfrac{1}{45}×120=\dfrac{8}{3}$ km and therefore the distance between the cars becomes $4-\dfrac{8}{3}=\dfrac{4}{3}$ km.

Now the bird flies to the second car for the second time. It takes $\dfrac{\left(\dfrac{4}{3}\right)}{180}=\dfrac{1}{135}$ hour for this.

In this time, the cars together covers a distance of $\dfrac{1}{135}×120=\dfrac{8}{ 9}$ km and therefore the distance between the cars becomes $\dfrac{4}{3}-\dfrac{8}{9}=\dfrac{4}{9}$ km.

The bird reaches back the first car in $\dfrac{\left(\dfrac{4}{9}\right)}{180}=\dfrac{1}{405}$ hour.

In this time, the cars together covers a distance of $\dfrac{1}{405}×120=\dfrac{8}{ 27}$ km and therefore the distance between the cars becomes $\dfrac{4}{9}-\dfrac{8}{27}=\dfrac{4}{27}$ km.

Now the bird flies to the second car for the third time. It takes $\dfrac{\left(\dfrac{4}{27}\right)}{180}=\dfrac{1}{1215}$ hour for this.

so on.

Sine this goes on repeatedly, the bird reaches the bonnet of the second car infinite times(theoretically)

Community
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Kiran
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    Can you add the explanation from that site to your question here? Questions should be self-contained as much as possible. – DylanSp Feb 18 '16 at 18:39
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    It is not much different from Zeno's Paradoxes. Theoretically we can talk about an infinite number of iterations of still smaller distances adding up to a finite distance as a total. It makes sense theoretically. – String Feb 18 '16 at 18:43
  • @ DylanSp, added – Kiran Feb 18 '16 at 18:47
  • @String, never thought this will open up a completely new area for me which I was not aware. But, we can find sum of an infinite geometric progression(when r<1) to a finite number, right? so it could be the mathematical explanation? – Kiran Feb 18 '16 at 18:49
  • This is why the question asks for a theoretical answer rather than a practical one. Also, I think you can give a much simpler explanation of why the answer is infinity: in short, the bird flies faster than the cars, and at every step the distance between the cars is positive. Finally, if you wanted to make the scenario slightly more realistic, you could suppose that it takes the bird one second, say, to turn around. There will be a point where the cars are less than one second apart, so the bird can only change direction finitely many times. – Théophile Feb 18 '16 at 18:50
  • ok, but is it possible using any mathematical tool to find the actual number of times , like sum of an infinite geometric progression? – Kiran Feb 18 '16 at 18:55
  • (1) Apply $s=\frac{d}{t}$ and simple algebra. Calculate the time until the crash using the relative speed of trains, 120 km/h. $t=\frac{12 km}{120 km/h}=0.1 h$. Calculate how far the bird will fly in that time. $d=120 km/h \times 0.1=12 km$. – Dan Christensen Feb 20 '16 at 14:54

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As the distance decreases, it takes the bird less each time to cross the distance. Of course, it would take the bird to slow down and change direction, both of which are assumed to take no time at all.

If this seems paradox to you, consider the following spacial paradox (and not a temporal one): You have a pizza and take a knife to cut it in half. You eat one of the two pieces and cut the resulting piece in half and eat one of the pieces again, repeating the process.

Theoretically, is there a maximum number of pieces of pizza which you will have eaten? Just like you can produce an infinite number of slices of pizza from a finite amount of pizza, the bird can travel a finite distance in a finite time involving infinitely many turns.

Roland
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