# Homework Help: Infinite number of turns in finite time

1. Feb 2, 2013

### paras02

1. The problem statement, all variables and given/known data

A car is moving with a constant speed of 40 km/h along a straight road which heads towards a large vertical wall and makes a sharp 90° turn by side of the wall . A fly flying at a constant speed of 100 km/h , start from the wall towards the car at an instant when the car is 20km away, flies until it reaches the glass pane of the car and return to the wall at same speed. It continues until the car makes the 90° turn. how many trips has it made between the car and the wall ?

2. Relevant equations
distance = speed * time

3. The attempt at a solution
Suppose the car is at a distance 'x' away(at A) when the fly is at the wall(at O ) . The fly hit the glass pane at B, taking a time 't'. Then
AB = (40km/h)t ,
OB = (100km/h)t
Thus x = AB + OB
= (140km/h)t
t =x / 140 km/h
OB = 5x / 7
The fly returns to the wall and during this time period car moves BC. The time taken by the fly in this return path is
5x /7 / 100 or x / 140
BC = 40x /140 or 2x / 7
OC = OB - OC
= 3x / 7
Distance of the car at beginning of 2nd trip
= 3*20 / 7
Distance of the car at beginning of 3rd trip
= 32*20 / 72
Distance of the car at beginning of nth trip
= 3n-1*20 / 7n-1
trips will go on till the distance is reduced to zero . this will be the case when n approaches to infinity .hence the fly make infinite number of turns.

But in my view it is not possible as car will take only 30 minutes to reach the wall. Thus fly can't make infinite number of turns.

2. Feb 2, 2013

### clamtrox

Well, it depends on what assumption you make. If you assume the fly is pointlike, and the glass can go arbitrarily close to the wall, then the fly certainly can make an infinite number of turns.

In normal cars though, the glass really cannot touch the wall. Perhaps you can use that as a cutoff in your sum.

3. Feb 2, 2013

### tiny-tim

hi paras02!

∑ (1/7)n is finite

so the total distance is finite

and (since the fly's speed is constant) the total time is finite

4. Feb 2, 2013

### paras02

How Ʃ (1/7)n is finite? pls clarify.

5. Feb 2, 2013

### tiny-tim

1/2 + 1/4 + 1/8 + 1/16 + … = 1

similarly, ∑ (1/7)n = 1/6

6. Feb 2, 2013

### haruspex

And in that observation you came very close to finding the easiest way to solve the problem.
The period under consideration is 30 minutes, and the fly flies constantly at 100kph, so how far does it fly?
No, but allow that it can make some huge number of turns, N (it's a theoretical fly). That produces an answer which depends on N. As you allow N to increase, the answer gets closer and closer to the value, X say, that you get by allowing N to be infinite. So while you can argue that N is never infinite, the answer can be made arbitrarily close to X, so to all intents and purposes the answer is X.

7. Feb 3, 2013

### paras02

Suppose a man standing near by observes all this incident. He will notice that at the end of 30th minute or in 31st minute the motion of the fly has stopped. Thus he can count the exact number of trips provided that the man has this ability, but theoretical it is not possible as the distance between fly and car will never reduce to zero. Will you explain this paradox.

8. Feb 3, 2013

### haruspex

If we are dealing with a theoretical model rather than the real world, it will be an infinity of trips.
You have to be careful with what you mean by 'never'. With the usual meaning, that the event will not occur in a finite time, your statement is wrong: the distance will reduce to zero. The Achilles + Tortoise paradox arises because 'never' is taken to mean not within the scope of a given iterative analysis (which is structured so as not to be able to reach the moment in time when A catches T).

9. Feb 4, 2013

### paras02

ok! thankyou guys