Math Help - Isolating a variable

1. 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:

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?

2. 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.

3. Re: Isolating a variable

Originally Posted by batboyx2
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:

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.

4. 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?

5. Re: Isolating a variable

Originally Posted by batboyx2
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.

6. Re: Isolating a variable

When written as a function, this can be represented as follows:

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?

7. Re: Isolating a variable

This question is identical to your original one (it's the same equation).

8. Re: Isolating a variable

Originally Posted by HallsofIvy
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?