Some equations

For clarification, [math]x[/math],[math]y[/math],[math]x_1[/math], and [math]y_1[/math] all depend on [math]t[/math], but [math]v_0[/math], [math]\vec{V}_0[/math], [math]\theta[/math], [math]\vec{n}[/math], and [math]a[/math] are all constants.

File: image.jpg (1MB, 3264x2448px) Image search: [Google]

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I'm not looking for an analytic solution, and I really doubt that a meaningful one exists at all, but what I'd like to do is seperate out all the symbolic and functional components so I can have something numerically solvable.

Also, here's a table of variable dependencies I scribbled.

just use euler lagrange doe

>>7700981
How is that even applicable? It's not an optimization problem.

Here's what I've gathered so far:

https://www.scribd.com/doc/292305422/Diff-Eqs

Wait, what if it's not [math]\vec{V}_0 = \hat{n}v_0[/math], but rather [math]v_0 = \sqrt{(x')^2+(y')^2}[/math]? Have I been reading this wrong the entire time?

>>7701863
Damn, that can't be it, because that would just result in [math]v_0 = \sqrt{v_0^2}[/math]

>>7701863
how could that be?
also isn't sqrt( (x')^2 + (y')^2 ) written in the first equation? so why wouldn't v_0 be written there instead?

>>7701901
My thought was perhaps the writer noticed that scalar term kept coming up in the equations, so he then defined [math]v_0[/math] to be that term.

>>7701907
I think I understand but just to clarify
you wrote vector V_0 but isn't is V_D? that's a D not a 0 right?

sorry I haven't figured out mathjax yet

>>7701922
I think it's [math]V_0[/math], but I can kinda see why it might look like a D. Nevertheless, they refer to the same thing.

Also, I renamed [math]a[/math] as [math]p_0[/math] since the author told me it was the initial position.

File: D.png (86KB, 373x183px) Image search: [Google]

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>>7701926
sorry to go off topic but the letter circled in yellow is what you're saying is a 0?
who writes their 0's like that

please tell me we're talking about the same letter here

>>7701928
We are talking about the same thing. It might be a D, I don't know, but I transcribed it as a 0, and I think I prefer it that way because it appears to be an initial velocity vector.

>>7701931
alright fair enough
carry on derp_commander

Ok, defining:
[eqn]
X := \left(x_1'(t)\right)_x\\
Y := \left(x_1'(t)\right)_y
[/eqn]
and
[eqn]
\chi := \left(y_1'(t)\right)_x\\
\Upsilon := \left(y_1'(t)\right)_y
[/eqn]
Eliminating the variable [math]t[/math] ultimately leads to these very interesting relations:
[eqn]
\left(v_0 X\right)^2 = \left(X^2 + Y^2\right)^2\\
\left(v_0 \Upsilon\right)^2 = \left(\chi^2 + \Upsilon^2\right)^2
[/eqn]
Any thoughts on what this means?

>>7700493
it has something to do with an intercept points of two projectiles ,i remember something about using something similar when trying to determine the point of collision between a projectile going on a balistic arc and another being dropped from a certain height

>>7702142
Any ideas then?

Also, changing the order of substitution removes a sign ambiguity:
[eqn]
v_0 X = X^2 + Y^2\\
v_0 \Upsilon = \chi^2 + \Upsilon^2
[/eqn]

>>7702142
Still, what are common techniques used to solve such problems?