Strange Word Problem Two

[TromboneAl]

OK, now for the more difficult problem.

An aircraft starts at the north pole and proceeds to the south pole via a spiral pattern. Each spiral path is separated from the previous one by 150 miles. To clarify, each time the aircraft passes the prime meridian, for example, it is 150 miles further south.

Approximately how far will the aircraft have traveled when it reaches the south pole?

I figure that the distance between the poles is about 12,500 miles, and the craft would have to make only 83 spiral laps around the planet (12,500 / 150). I'd approximate the length of each spiral by calculating the circumference of the earth along a latitude line at the midpoint of that lap. I ignore the fact that the aircraft is a few miles above the surface.

Perhaps there's an easier way?

 

[braumeister]

Geographic pole or magnetic pole?

Regardless, at the completion of the 45th lap, the plane will be directly overhead of the train John is traveling on, which left City A at 8:30 am and is approaching City B, from which Sally left on a train heading in the opposite direction at 9:15 am.

I used to really hate these problems.

 

[Meadbh]

What kind of a book is this, Al? Are you venturing into science fiction, space, etc?

I did read your recent travel book and enjoyed it.

 

[FUEGO]

There may be an "elegant solution" using some sort of integral in a spherical polar coordinate system. It's been a long time since high school calculus though.

 

[NW-Bound]

The approximation described by T-Al is a reasonable one. However, I asked myself how an exact length could be obtained. Well, before an analytical solution can be obtained or even attempted, one must first define this spiral curve in an analytical form.

I first thought of using a rhumb line or loxodrome (line of constant azimuth), but such a line will not intersect the same meridian with a constant 150-mi spacing. This brings forth a more fundamental problem: the above description of the spiral is inadequate as it does not fully or uniquely define the curve.

Hmm... This is not going to be that simple, unless I miss something.

Now, to throw more complications into the problem, suppose we want to use the ellipsoidal model of the earth instead of the spherical one.

 

[ls99]

Um, what altitude is the plane flying at?

Not that I actually am interested in solving the problem. Flying at just above sea level the flight will be abruptly terminated by the first land mass or mountain range. The greater the altiude the longer the distance, just sayin'

 

[38Chevy454]

Plane will run out of fuel and can't complete the flight.

I do think the approximation proposed is close though if you ignore fuel.

 

[REattempt]

How far away...


http://contrailscience.com/how-far-away-is-that-contrail/

 

[KingB]

After an internet search, the answer is 42.

 

[scrinch]

After an internet search, the answer is 42.

Partial credit. You must specify your answer with units...furlongs, parsecs, or angstroms?

 

[scrinch]

Al, if 12,500 miles is accurate enough for your estimate of earth's pole-to-pole semi-circumference, then your method of calculating the spiral length is plenty accurate. As NW-Bound says, the actual detailed geometry of such a flight path is a little complicated, as the as the angle between the flight path and lines of latitude changes throughout the flight. This path traces an "Archimedean Spherical Spiral" and you'd have to set up an integral to find its true length. Integral calculus was the stop where I got off the college math train.

 

[BTravlin]

After an internet search, the answer is 42.

Yes, only partial credit. You must show your work!

 

[gromit]

You can find some info on this pattern by Googling "Archemedian spherical spiral" or "spherical spiral".

 

[NW-Bound]

Good find!

Here's a link from Wikipedia, but the referenced article is in German. Note that an Archimedian spiral on a sphere is more complicated than the original 2-D one.

File:KUGSPI-5 Archimedische Kugelspirale.gif - Wikipedia, the free encyclopedia

 

[Bestwifeever]

I am pretty sure I won't be able to read this book.

 

[REWahoo]

I am pretty sure I won't be able to read this book.

Sure you can read it. Understanding it OTOH, could be problematic.

 

[aja8888]

I am pretty sure I won't be able to read this book.

I'm pretty sure I won't want to read this book...

 

[TromboneAl]

I am pretty sure I won't be able to read this book.

I'm pretty sure I won't be able to write this book.

Yes, science fiction. Craft will be at 40,000 feet, and fuel won't be an issue.

I only need a ballpark figure, so I'll use my latitude approximation. I was hoping there would be an elegant solution as in my favorite word problem:

Two trains 1000 miles away are traveling towards one another. The first is going 30 MPH and the second, 20 MPH. A fly going 40 MPH starts at one train, flies to the other, and turns around and goes back. He continues this until the trains crash. How far does the fly travel?
​

I like it because the tendency is to try to figure out how far the fly travels for each leg of his trip and add them up. Very difficult. That's the kind of solution I'm planning on using in my problem.

But there's an elegant solution that makes it trivial: The trains are closing at 50 MPH so they will travel for 20 hours before crashing. The fly is traveling at 40 MPH, so in 20 hours, he will travel 800 miles.

I was hoping for a similar elegant. solution to my problem.

 

[NW-Bound]

...I was hoping for a similar elegant. solution to my problem.

In this case, I doubt that one exists, but I hope someone will prove me wrong.

Regarding the Archimedian spiral on a sphere (that people have already done and I need to catch up with that), I now wonder if anyone has extended that to the ellipsoid model of the earth (GPS uses the WGS-84 ellipsoid model). The complexity would increase greatly when one goes from a sphere to an ellipsoid. Following is an example.

The shortest distance between two points on a sphere is the great circle path. Given latitude and longitude coordinates of two end points, the path can be computed using spherical trigonometry. In general, the shortest path on a 3-D surface between two points is called a geodesic. Geodesics on a sphere are relatively simple, but for geodesics on an ellipsoid see the following link.

Geodesics on an ellipsoid - Wikipedia, the free encyclopedia

 

[Bestwifeever]

Will Lena be on the plane?

 

[HFWR]

Is it wetter on water after a rain?

 

[NW-Bound]

Regarding the problem T-Al recounted that has a simple solution, I remember reading the following anecdote somewhere.

When this problem was posed to John von Neumann, he thought about it for a short time, then gave the correct answer. The asker then said that Neumann must have heard this problem before, and knew the trick.

Neumann replied "What's so tricky about computing the convergence limit of an infinite series?"

I did not know what to think of that. Was Neumann so good that he could compute the limit of the series in his head, or was he turning the table by pulling the leg of the asker?

 

[devans0]

Hmmm. quick and very dirty approximation. 150 mi means 41.43 circumference trips to go from pole to equator. A latitude circumference is distance at equator * cos (lat) or in our case, the fraction (45/42)*n using degrees and rounding up the fraction trip. run 42 iterations with n from 1 to 42. double that.

 

[gromit]

I think the "elegant solution" to the spiral distance problem is going to involve renting a plane and then writing an additional book about the adventure.

 

[splitwdw]

I'm pretty sure I won't be able to write this book.

Yes, science fiction. Craft will be at 40,000 feet, and fuel won't be an issue......I was hoping for a similar elegant. solution to my problem.

The elegant solution is, this is fiction.......the solution is whatever you invent.