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**1. Homework Statement**

This problem is related to momentum and energy. I have set the mass to unity to simplfy things and reduce the variable count. u,v,w and x are velocities. The first equation is obtained from energy considerations and the second and third equations from conservation of momentum. I am trying to find velocity u in terms of w (or vice versa) and eliminate v and x or alternatively find v in terms of x and eliminate u and w.

**2. Homework Equations**

[tex]{1 \over \sqrt{1-u^2}}+{3 \over \sqrt{1-v^2}} = {1 \over \sqrt{1-w^2}} + {2 \over \sqrt{1-x^2}}[/tex]

[tex]v = {u \over \sqrt{9-8u^2}[/tex]

[tex]x = {w \over \sqrt{4-3 w^2}[/tex]

**3. The Attempt at a Solution**

I substituted the given values for v and x in the first equation and simplified to get an expression in terms of u and w only:

[tex]{1 \over \sqrt{1-u^2}}+3{\sqrt{9-8u^2} \over \sqrt{9-9u^2}} = {1 \over \sqrt{1-w^2}} + 2 { \sqrt{4-3w^2}\over \sqrt{4-4w^2}}[/tex]

By substituting y for [itex]9/u^2[/itex] and z for [itex]4/w^2[/itex] (hoping to extract u or w later) and further simplifying I get:

[tex]{1 \over \sqrt{1-9/y}}+3{ \sqrt{y-{8}} \over \sqrt{y-9}} = {1 \over \sqrt{1-4/z}} + 2{ \sqrt{z-3}\over \sqrt{z-4}}[/tex]

Despite the relatively simple appearance of the simplified equation the online solver I use still can not solve for z in terms of y (or vice versa). Any ideas?

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