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Homework Help: Physics question involving trig.

  1. Oct 29, 2013 #1
    An astronaut arrives on the planet Oceania and climbs to the top of a cliff overlooking the sea. The astronaut’s eye is 100 m above the sea level and he observes that the horizon in all directions appears to be at angle of 5 mrad below the local horizontal.

    What is the radius of the planet Oceania at sea level?

    How far away is the horizon from the astronaut?

    [Hint: the line of sight from the astronaut to the horizon is tangential to surface of the planet at sea level.]

    Having drawn out the problem I have two equations:
    Pythagoras: x^2 + R^2 = (R+100)^2
    Sine Law: x= (R+100)sinA

    x is the distance between the horizon and the astronaut
    R is the radius of the planet
    A is the angle between the two lengths of radius of the planet (sorry might sound a bit confusing, it makes more sense if you draw it out!)

    I then tried to combine them and solve for the radius of the planet R but im not having any luck getting the right answer :(

    Any steps, answer, and/or hints are all greatly appreciated! :)
  2. jcsd
  3. Oct 29, 2013 #2

    Simon Bridge

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    Welcome to PF;

    It helps to be careful about what you call things...

    You should have drawn a triangle OAB.
    O is at the center of the planet.
    A is where the astronaut is standing,
    B is on the surface of Oceania at sea level.

    The angle OBA is ##\frac{\pi}{2}## (working in radians).
    The angle AOB is ##\alpha##
    The angle OAB is ##\frac{\pi}{2}-\alpha##

    Lengths of the sides:
    ##|OB|=R## (radius of Oceania at sea level)
    ##|OA|=R+a## (a=100m) it is best to keep variables.
    ##|AB|=x = R\tan\alpha##

    You will also need another triangle: ABC
    points A and B are as before.
    point C is on the same line as OB, further out from Oceania.
    The line AC is perpendicular to the line OA.
    The angle CAB is given to you - from the drawing you should be able to see how it is related to ##\alpha##.

    You now have three right-angled triangles.
    ... but I'm afraid you'll probably have to attach a diagram.
  4. Oct 29, 2013 #3
    Thank you, I followed the steps you have given me and I drew the first triangle, however im having trouble understanding where the line AC should go, I made the AC line red in the diagram, im not sure if I drew it correctly. I attached the diagram which I drew in a word doc., I don't know any other way of drawing diagrams online, I hope its not an inconvenience.

    Attached Files:

  5. Oct 29, 2013 #4

    Simon Bridge

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    The line AC is supposed to be the local horizontal to A (the Astronaut).
    The line AB is the line to the horizon as seen from A.

    It is poor netiquette to post attachments in poorly documented formats like docx - or any MS Office format.
    Not everyone can read them.

    For diagrams, please use png or gif format.
    For photographs: jpg is usually good enough.
    For general documents, please use an open document format or pdf

    The attachment function of PF lets you upload from your computer - another approach is to use a service like google-docs.
    Last edited: Oct 29, 2013
  6. Oct 29, 2013 #5
    From your figure, R/(R+a) = cosα. You know α, so you can calculate R.
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