Monday, October 06, 2008

The North-East Paradox

This is a hypothetical problem I invented while my mind was idling for some time.


Suppose that Mr. Flight Pilot, a very capable pilot, decided to take an adventurous journey. This is what he planned:

From a point on equator, he would start flying north-east. Should he continue his journey for some time, he would have circumnavigated the world and emerged at the same location (a trivial solution). However, he had a wackier plan. Instead of going straight, he decided to direct his flight to the instantaneous north-east at each moment.

Take a moment to let that settle in and take one such imaginary flight in your mind to understand how this will turn out to be. You must have deduced by now that instead of going around the earth, his journey will be directed towards the north pole (only slightly misdirected by a mere 45°). It should appear quite logical that he would start spiraling into the north pole whizzing around the world at a faster rate as he closes in to the north pole.

Bonus Question: How long will he take to reach the north pole?

Bonus Answer: Never. Actually after infinite amount of time should he survive. As you can see from the image below (which is supposed to be taken from the top of north pole, like Stereographic projection), the direction of motion of the flight is always inclined with respect to the north pole. So although it would continue to get progressively closer to the north pole, he will never reach there.

(Pardon my poor illustrations)

I know you must have seen many problem like this, where you get progressively closer to the destination but never reach there. Problems that challenge your notions of a well-behaved world by throwing in infinities. But these problems no longer challenge us. In fact, these are so out-of-date, that they have become boring. The only reason I invented this problem is that I wanted to test my mathematical skills. The problem I gave myself to solve was: Solve for position of flight as a function of time, given the Radius of Earth (R), and speed of flight (v). You may want to try it yourself.

But it soon became clear that my mathematical skills aren't what they used to be. So after trying for a couple of days, I gave up on the problem. But since an idle mind is a devil's workshop, I had to invent a new problem.

Meet Mr. Polar Tourist. He wanted to travel to the north pole, but did not want to undertake the long, tedious journey . He heard about Mr. Flight Pilot, and his way of reaching close to the north pole. As he was a good mathematician himself, Mr. Polar Tourist decided to take help from Mr. Flight Pilot. His plan was simple. He knew that being in flight, he will never reach the north pole. So he decided that he will calculate the optimum location of jumping off the flight (using a parachute, of course) and walk the rest of the way. The way to calculate the optimum location was simple. As soon as walking (with his walking speed of w) will begin to take him closer to the north pole faster as compared to the flight, he will get down. Since he will walk directly to the north pole, and not at a weird angle, he will surely reach the north pole in finite amount of time.

But there was a big problem once he started doing the calculations. He couldn't figure out what would be the best time to exit the aircraft. You see, again following from the figure above, the instantaneous component of flight's speed towards the north pole is v/√2, which is faster than normal walking speed as v»w. Thus, the flight would always take him closer to the north pole faster than walking. The trouble is, this implies that he will always have to prefer sitting in the flight (although we have shown that it will never reach the north pole) over getting down and walking to the north pole (which obviously will complete the journey in finite time). This leads to a paradox. Can you explain why this paradox occurs, and what is the solution to this paradox?

As expected, I know the answer and will post it soon. Just try to solve this paradox for the time being.

Things to keep in mind while trying to solve this problem:
  • This is a kinematics problem. So please do not include the solution of paradox as impossibility to undertake this flight (non-existence of ideal compass, delays in changing directions, fuel, longevity of pilot).
  • Please also do not include aspects related to dynamics (role of centrifugal force)
  • Arguments relating to quantum mechanics are also strictly forbidden (calculating the uncertainty in position for the given speed and stating the impossibility to undertake the exercise once the distance of flight from north pole goes below this number: via Heisenberg' uncertainty principle)
  • Assume all ideal things for simplicity if you are trying to solve it by mathematically formulating the location in terms of time (spherical earth, flying at near ground level)
  • It may make more sense to try thinking of other (similar) paradoxes like Achilles and the tortoise. The given example, however, is clearly not a solution to the problem as although the tortoise problem is infinite in steps, it is finite in time. The given problem is (apparently) infinite in both time and distance.

Update: Solution provided in the comments.


Ravi said...

Hey Bhagwaan, JEE ka paper set kar rahe ho kya [:o]

Minnesotastan said...

This is reminiscent of the killer rabbit puzzle.

Ambuj Saxena said...

@Minnesotastan, Interesting puzzle, but I believe they are fundamentally different. Glad you reminded me that I haven't posted the solution yet. Will post it in a day or two.

Ambuj Saxena said...

Solution to the puzzle:

The instantaneous velocity of the flight towards the North Pole is constant (|v|/√2). Thus, the flight will reach the pole in finite time, T=(πR/2)/(|v|/√2).

Go figure your infinites!