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Recursive Projectile To Explicit Formula

  1. Nov 25, 2009 #1
    1. The problem statement, all variables and given/known data

    If a projectile orbits around a planet and lands after x cycles of the equation:
    radius[n+1] = radius[n] - (M* v^2 / radius[n]*G*P/radius[n]^2)
    at a final radius of C. What is the necessary v?

    The only variable that isn't known at the time of solving is v.

    Basically I think I need to solve in terms of v but in order to do that I need to make the recursive formula into an explicit one but I can't figure out how.


    2. Relevant equations

    radius[n+1] = radius[n] - (M* v^2 / radius[n]*G*P/radius[n]^2)

    3. The attempt at a solution
    I tried writing down the sequence to find a simpler pattern with no luck. Especially since even that seems to change drastically if I change the constants such as x (the number of cycles). I tried solving it as if it were a normal equation but couldn't figure out how to get rid of the radius[n+1] term to do that and tried setting it to C (the target radius) to radius[n] (so I could ignore it) and to 0 (seemed a better way of ignoring) all with no success seemingly. Setting to radius[n] led to a cubic explicit equation but that doesn't make as much sense as the quadratic equation from setting it to C since we are dealing with motion that is similar to a conic section which is where I derived the original recursive equation from. So how do I start, or how do I make it explicit so I can solve the rest of it.

    This isn't a Real homework problem so much as a problem that I desperately need a solution to and couldn't find the right section.

    Pointers or solutions are soooo welcome. Thanx :)
     
  2. jcsd
  3. Nov 25, 2009 #2

    HallsofIvy

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    I am a bit confused by the fractions on the right side. Is that
    [tex]r_n- \frac{Mv^2}{r_n}\frac{GP}{r_n^2}[/tex]?

    Why not just
    [tex]r_n- \frac{MGP v^2}{r_n^3}[/tex]?

     
  4. Nov 25, 2009 #3
    I'm sorry there should actually be a - inbetween the radius and G not a *. Thanks for taking the time to reply though.

    [tex]r_n_+_1 = r_n- \frac{Mv^2}{r_n}-\frac{GP}{r_n^2}[/tex]

    (I stole your LaTex by the way:biggrin:)
     
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