1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Spiralling around the Earth

  1. Apr 22, 2015 #1
    1. The problem statement, all variables and given/known data
    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

    2. Relevant 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

    3. 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
  2. jcsd
  3. Apr 22, 2015 #2


    User Avatar
    Science Advisor

    I would start with spherical coordinates. With constant radius, R, [itex]x= R cos(\theta)sin(\phi)[/itex], [itex]y= R sin(\theta)sin(\phi)[/itex], [itex]z= R cos(phi)[/itex]. Here we have [itex]z= R cos(\phi)= R- vt[/itex] and [itex]\phi= \omega t[/te]x] so [itex]z= R cos(\omega t)= R- vt[/itex]
  4. Apr 22, 2015 #3
    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.
  5. Apr 22, 2015 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Please post your working as far as you get.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted