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Math Help - Isolating a variable

  1. #1
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    Isolating a variable

    I have an equation here, t=((arccos(sqrt(x/r))+sqrt((x/r)*(1-(x/r))))/sqrt(2m))*r^(3/2).
    or if you are a visual person:
    Isolating a variable-timetofall.png
    This equation is used to calculate the amount of time it takes to fall from height r, to height x, depending on what m (Gravitational Parameter, m is substituted for mu) is. I need to rearrange this equation in such a way that when I have everything except for x, I can find it in terms of all the other variables. So basically, I want to turn the above equation into x = ?. I have been working on this problem for 3+ hours now, and used up 3-4 sheets of graphing paper trying to find x in terms of r, t, and m. I have come to the conclusion that there is some process or operation I have to do which I haven't yet learned in school (This is what I do with my free time, this isn't schoolwork as I am only a highschool freshman). I have tried squaring both sides, subtracting and adding variables, multiplying and dividing by variables, and nothing seems to separate the equation how I want to. Can somebody please show me what process one may use to isolate x?
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  2. #2
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    Re: Isolating a variable

    That involves x both "inside" and "outside" a transcendental function, the arccosine. In such a case, there typically is no "algebraic" way to solve the equation.
    Thanks from Gusbob
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  3. #3
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    Re: Isolating a variable

    Quote Originally Posted by batboyx2 View Post
    I have an equation here, t=((arccos(sqrt(x/r))+sqrt((x/r)*(1-(x/r))))/sqrt(2m))*r^(3/2).
    or if you are a visual person:
    Click image for larger version. 

Name:	TimeToFall.png 
Views:	4 
Size:	1.6 KB 
ID:	27897
    This equation is used to calculate the amount of time it takes to fall from height r, to height x, depending on what m (Gravitational Parameter, m is substituted for mu) is. I need to rearrange this equation in such a way that when I have everything except for x, I can find it in terms of all the other variables. So basically, I want to turn the above equation into x = ?. I have been working on this problem for 3+ hours now, and used up 3-4 sheets of graphing paper trying to find x in terms of r, t, and m. I have come to the conclusion that there is some process or operation I have to do which I haven't yet learned in school (This is what I do with my free time, this isn't schoolwork as I am only a highschool freshman). I have tried squaring both sides, subtracting and adding variables, multiplying and dividing by variables, and nothing seems to separate the equation how I want to. Can somebody please show me what process one may use to isolate x?
    If a closed form solution for x exists, it may be much easier to derive the equation from the original setup, rather than trying to manipulate this one.
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  4. #4
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    Re: Isolating a variable

    So, what can I do if I have everything but X? Does it involve higher level math or is it simply impossible?
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    Re: Isolating a variable

    Quote Originally Posted by batboyx2 View Post
    So, what can I do if I have everything but X? Does it involve higher level math or is it simply impossible?
    Numerical solutions and/or approximations using inequalities.
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  6. #6
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    Re: Isolating a variable

    When written as a function, this can be represented as follows:
    Isolating a variable-free-fall-time.png
    where t is time after the start of the fall, y is the distance between the two bodies, and y0 is that starting value for y. μ is of course the gravitational parameter. Can you guys do anything more with this than you could with the other equation?
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  7. #7
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    Re: Isolating a variable

    This question is identical to your original one (it's the same equation).
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  8. #8
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    Re: Isolating a variable

    Quote Originally Posted by HallsofIvy View Post
    That involves x both "inside" and "outside" a transcendental function, the arccosine. In such a case, there typically is no "algebraic" way to solve the equation.
    Thus implying that there is a "nonalgebraic" way to solve the equation?
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