How Do You Calculate the Shortest Flight Path Between Two Points on the Globe?

[TheBestMiller]

Homework Statement

To attend the 2000 Summer Olympics, a fan flew from Mosselbaai, South Africa (34S, 22E) to Sydney, Australia (34S, 151E). (a) What is the smallest angular distance the fan has to travel: (1) 34 degrees (2) 12 degrees (3) 117 degrees (4) 129 degrees? Why?
(b) Determine the approximate shortest flight distance, in kilometers.

Homework Equations

The Attempt at a Solution

Am I just adding the vectors of the east directions? I assumed that when it asked for the quantity in angular distance that I needed to change the Cartesian coordinates to polar coordinates, and I do realize I will need to convert to radians and use S=r(theta) to solve part b, correct?

Any help or guidance at all would be greatly appreciated. Thanks in advance!

 

[SteamKing]

Homework Statement

To attend the 2000 Summer Olympics, a fan flew from Mosselbaai, South Africa (34S, 22E) to Sydney, Australia (34S, 151E). (a) What is the smallest angular distance the fan has to travel: (1) 34 degrees (2) 12 degrees (3) 117 degrees (4) 129 degrees? Why?
(b) Determine the approximate shortest flight distance, in kilometers.

Homework Equations

The Attempt at a Solution

Am I just adding the vectors of the east directions?

Why are you adding?

I assumed that when it asked for the quantity in angular distance that I needed to change the Cartesian coordinates to polar coordinates, and I do realize I will need to convert to radians and use S=r(theta) to solve part b, correct?

Strictly speaking, geographic positions (i.e., latitude and longitude) are not cartesian coordinates. They are a special means of specifying a location on the surface of a sphere.

In order to determine the distance flown, you will need to find the circumference of the Earth at the parallel of latitude on which the flight was taken, in this case 34 degrees South.

 

[TheBestMiller]

In order to determine the distance flown, you will need to find the circumference of the Earth at the parallel of latitude on which the flight was taken, in this case 34 degrees South.

This wasn't provided in the prompt, but outside resources tell me it is 33258 kilometers.

Why are you adding?

Ahhh I meant subtraction, silly mistake.

 

[haruspex]

In order to determine the distance flown, you will need to find the circumference of the Earth at the parallel of latitude on which the flight was taken, in this case 34 degrees South.

Not so. That would simply produce the difference of the longitudes. The shortest distance is along a great circle. That will dip further south than 34S along the way.
TheBestMiller, consider a slice through the Earth, perpendicular to the NS axis, at 34S. You know the angle the endpoints subtend to the centre of that circle. What is the radius of the circle? How far apart, in a straight line through the earth, are the endpoints?

 

[SteamKing]

This wasn't provided in the prompt, but outside resources tell me it is 33258 kilometers.

That's approximately correct. (How could you verify this figure, knowing only the average radius of the earth, or 6371 km?)

What's the distance of the flight taken from SA to Oz then?

 

[TheBestMiller]

Not so. That would simply produce the difference of the longitudes. The shortest distance is along a great circle. That will dip further south than 34S along the way.
TheBestMiller, consider a slice through the Earth, perpendicular to the NS axis, at 34S. You know the angle the endpoints subtend to the centre of that circle. What is the radius of the circle? How far apart, in a straight line through the earth, are the endpoints?

Well R=√x^2 + y^2 correct? In that case the radius for point 1 would be approximately 40.5 and point 2 would be 154.8. Would I then set up a triangle with the radii and use Pythagorean theorem for the distance, in a straight line, between the two points?

 

[TheBestMiller]

That's approximately correct. (How could you verify this figure, knowing only the average radius of the earth, or 6371 km?)

What's the distance of the flight taken from SA to Oz then?

It would have to be an arc length because the Earth is a sphere. So I would use S=r(theta)?

 

[SteamKing]

Well R=√x^2 + y^2 correct? In that case the radius for point 1 would be approximately 40.5 and point 2 would be 154.8. Would I then set up a triangle with the radii and use Pythagorean theorem for the distance, in a straight line, between the two points?

It's not clear what you are calculating here. The geographic positions of the two cities in SA and Oz are given in latitude, longitude format, both of which parts are measured in degrees. You can't treat these geographic coordinates like they were cartesian coordinates.

 

[TheBestMiller]

It's not clear what you are calculating here. The geographic positions of the two cities in SA and Oz are given in latitude, longitude format, both of which parts are measured in degrees. You can't treat these geographic coordinates like they were cartesian coordinates.

Ahhhh alright, that makes things much more clear.

 

[haruspex]

Well R=√x^2 + y^2 correct? In that case the radius for point 1 would be approximately 40.5 and point 2 would be 154.8. Would I then set up a triangle with the radii and use Pythagorean theorem for the distance, in a straight line, between the two points?

I'm not sure what you're doing there, maybe you misunderstood.
Let the endpoints be A, B and the centre of the Earth be O. Poles are N and S. Let P be the point on the line NOS at the latitude of A and B, i.e. APO and BPO are right angles.
Draw a circle containing NOPS and A. What is the distance AP? What is the angle APB? So what is the distance AB?

 

[CWatters]

Haruspex mentioned Great Circle. Google it and make sure you understand the concept. It will make answering the problem a lot easier!