Calculating Distance Traveled on a Winding Trajectory From North to South Pole

[kitsh]

Homework Statement

An airplane flies from the North Pole to the South Pole, following a winding trajectory. Place the center of the Earth at the origin of your coordinate system, and align the south-to-north axis of the Earth with your z axis. The pilot’s trajectory can then be described as follows:
1) The plane’s trajectory is confined to a sphere of radius R centered on the origin.
2) The pilot maintains a constant velocity v in the -z direction, thus the z coordinate can be described as z(t)=R-vt
3) The pilot "winds" around the Earth as she travels south, covering a constant ω radians per second in the azimuthal angle ϕ, thus ϕ(t)=ωt

Calculate the total distance traveled by the pilot. What you will find is that time t is not the best IP with which to parametrize this path. You can start with it, certainly … but then get rid of it in terms of a different choice for your IP: θ, from spherical coordinates

Homework Equations

Spherical coordinates are (r, θ, ϕ)
The Answer should be in the form of ∫A√(1+B^2(sin(θ))^n)dθ where A, B and n are either numerical constants or constants in in the terms of R, v and ω

The Attempt at a Solution

I honestly have no idea how I am supposed to approach this question, it is nothing like anything I have seen in this class or any other

 

[HallsofIvy]

I would start with spherical coordinates. With constant radius, R, $x= R cos(\theta)sin(\phi)$, $y= R sin(\theta)sin(\phi)$, $z= R cos(phi)$. Here we have $z= R cos(\phi)= R- vt$ and [itex]\phi= \omega t[/te]x] so $z= R cos(\omega t)= R- vt$[/itex]

 

[kitsh]

So I converted everything to spherical which helps some and I know the bounds of integration are going to be from 0 to pi and that the integration constant is rdθ but I still can't figure out how to get the integral into the form indicated above.

 

[haruspex]

So I converted everything to spherical which helps some and I know the bounds of integration are going to be from 0 to pi and that the integration constant is rdθ but I still can't figure out how to get the integral into the form indicated above.

Please post your working as far as you get.