How Can You Solve for x in Complex Elastic Collision Equations?

[luckis11]

(Tsinα)^2+(u+Tcosα)^2=((1+μ/m)((μu^2-μx^2-m((u-x)m/μ)^2)/(μ+μ^2/m))^0.5+Tsinα)^2+(Tcosα+(u-x)μ/m-x)^2

Can you isolate x? I.e. x=...
It's for elastic collision of unequal masses under angle of impact α.

 

[Ibix]

You haven't used LaTeX so your equation is very difficult to read, but it looks like a rather messy quartic.

You've flagged this as A level, meaning that you have postgraduate knowledge of the topic. Is that correct?

 

[A.T.]

Can you isolate x? I.e. x=...

https://www.symbolab.com/

 

[Arjan82]

Seems like a quartic indeed
https://en.wikipedia.org/wiki/Quartic_equation

 

[Arjan82]

Oh, wait, there is a square root in there somewhere... You make us work for it ;) But I would start to clean things up a bit by collecting all terms of x, then collect all terms not containing an 'x' and give it some other name. Then it is much easier to read.

 

[mfb]

$$(Tsinα)^2+(u+Tcosα)^2=\left((1+μ/m)\left(\frac{μu^2-μx^2-m((u-x)m/μ)^2}{μ+μ^2/m}\right)^{0.5}+Tsinα\right)^2+(Tcosα+(u-x)μ/m-x)^2$$
Like this?

Combine random constants:
$$a=\left(b\left(f+gx^2+hx\right)^{0.5}+c\right)^2+(d+ex)^2$$
You can expand the first square, then isolate the square root, square the whole equation, and you get a quartic equation, which has a closed solution but it's really awkward.

 

[luckis11]

I isolated the square root so got rid of it by squaring both sides of the equation, but I got (28 factors)^2=27+26+...+1 factors. So it's helpless.

 

[Vanadium 50]

I don't think you mean factors (or helpless). I think you mean terms.

Anyway, " (28 factors)^2=27+26+...+1 factors " doesn't have an x in it, so we don't know where you are. It does not look like you followed mfb's path.

 

[Arjan82]

Continuing with @mfb's equation. If I haven't made a mistake:

$$
\left[ \left(\sqrt{a - (d+ex)^2} - c\right)/b \right]^2 = f + gx^2 + hx
$$

then

$$
\left(a - (d+ex)^2 + c^2 - 2c \sqrt{ a - (d+ex)^2}\right)/b^2 = f + gx^2 + hx
$$

Expanding the terms in the square root will leave a $\sqrt{x}$ term. So it is not a quartic... I wouldn't know how to solve this.

 

[Vanadium 50]

Bring everything except the square root onto the right hand side, then square. You will hAve a quartic.

 

[Arjan82]

Oh, whoops, you are right :). I wasn't paying attention. I thought my second equation already had powers of four in x. (that happens when you are doing things simultaneously...)

 

[aheight]

(Tsinα)^2+(u+Tcosα)^2=((1+μ/m)((μu^2-μx^2-m((u-x)m/μ)^2)/(μ+μ^2/m))^0.5+Tsinα)^2+(Tcosα+(u-x)μ/m-x)^2

Can you isolate x? I.e. x=...
It's for elastic collision of unequal masses under angle of impact α.

In case you simply want an expresson to work with, you can solve for x in Mathematica easily:

[CODE title="Mathematica"]In[15]:= mySol[a_, d_, e_, c_, b_, f_, g_, h_] =
x /. Solve[(a - (d + e x)^2 + c^2 -
2 c Sqrt[a - (d + e x)^2])/(b^2) == f + g x^2 + h x, x];
mySol[1, 2, 3 + I, 4, 5, 6, 7, 8] // N

Out[16]= {-0.61495 - 0.545281 I, -0.559976 + 0.594679 I, -0.577998 +
0.710359 I, -0.56296 - 0.727542 I}[/CODE]

 

[luckis11]

I do not know this, what am I supposed to do with
{-0.61495 - 0.545281 I, -0.559976 + 0.594679 I, -0.577998 +
0.710359 I, -0.56296 - 0.727542 I}

 

[Arjan82]

I don't think that explicit solution is too relevant since it seems that for the chosen values of a to h all solutions have an imaginary part.

But if you want a solution then you have to do some of the work yourself. Start with the equation in post #9 and apply the hint in post #10. Then you'll have your quartic and can solve it using the equation in the wiki link in post #4.

 

[luckis11]

Ι told you I "solved" it but it has >27+26+...+1 terms. I don't see how imaginary numbers can help, can they? However, I do not have the equation of the op any more, I deleted it. I hope it's the same as the one I have now.

 

[vela]

You've flagged this as A level, meaning that you have postgraduate knowledge of the topic. Is that correct?

I moved the thread from the calculus forum to general math and reclassified it as level B as it's just a complicated algebra problem.

 

[luckis11]

The answer with imaginary numbers aheight mentioned is top level my friend.

 

[Mark44]

The answer with imaginary numbers aheight mentioned is top level my friend.

The fact that a solution you found involves complex numbers does not make this question a "top level" (i.e., A = post grad) type of problem.

 

[luckis11]

So what's the solution with complex numbers.

 

[Vanadium 50]

The same as it was in messages 12, 13, 15, 17 and 18.

 

[vela]

So what's the solution with complex numbers.

It's impossible to say since you didn't define what any of the quantities are. But if you were expecting a real root, a complex result usually means the equation you started with is wrong.

 

[cmb]

(Tsinα)^2+(u+Tcosα)^2=((1+μ/m)((μu^2-μx^2-m((u-x)m/μ)^2)/(μ+μ^2/m))^0.5+Tsinα)^2+(Tcosα+(u-x)μ/m-x)^2

Can you isolate x? I.e. x=...
It's for elastic collision of unequal masses under angle of impact α.

Yes.

 

[luckis11]

Don't bother, I solved it. Not exactly that equation, but a similar I have now.