1644.sagews

Here's the working equations:

y2, y3, z = var('y2 y3 z')
energyEquation = y2 + 10000^2/(2*32.2*(40*y2)^2) + z == 200
momentumEquation = 40*y2^2/2 + 10000^2/(32.2*40*y2) == 40*y3^2/2+10000^2/(32.2*40*y3)
knownY3 = y3 == 100 - z
pretty_print( energyEquation )
pretty_print( momentumEquation )
pretty_print( knownY3 )
solution = solve( [ energyEquation, momentumEquation, knownY3 ], y2,y3,z )
pretty_print(Matrix( solution ))

\newcommand{\Bold}[1]{\mathbf{#1}}y_{2} + z + 970.496894409938 \, \frac{1}{y_{2}^{2}} = 200

\newcommand{\Bold}[1]{\mathbf{#1}}20 \, y_{2}^{2} + 77639.7515527950 \, \frac{1}{y_{2}} = 20 \, y_{3}^{2} + 77639.7515527950 \, \frac{1}{y_{3}}

\newcommand{\Bold}[1]{\mathbf{#1}}y_{3} = -z + 100

\newcommand{\Bold}[1]{\mathbf{#1}}\left($$\begin{array}{rrr} y_{2} = 49.0151187905 & y_{3} = \left(-50.5809248555\right) & z = 150.580912863 \\ y_{2} = 2.69155446756 & y_{3} = 36.6554809843 & z = 63.3445121951 \\ y_{2} = \left(-2.96556297758 - 0.497581433136i\right) & y_{3} = \left(-1.51297187707 - 35.528534053i\right) & z = \left(101.512971877 + 35.528534053i\right) \\ y_{2} = \left(-2.96556297758 + 0.497581433136i\right) & y_{3} = \left(-1.51297187707 + 35.528534053i\right) & z = \left(101.512971877 - 35.528534053i\right) \\ y_{2} = 3.9364461738 & y_{3} = \left(-33.4331140351\right) & z = 133.433121019 \\ y_{2} = 100.288 & y_{3} = 0.384498149358 & z = 99.615503876 \\ y_{2} = \frac{25}{322} \, \sqrt{5} \sqrt{322} & y_{3} = \frac{25}{322} \, \sqrt{2} \sqrt{5} \sqrt{7} \sqrt{23} & z = \frac{25}{322} \, {\left(4 \, \sqrt{2} \sqrt{7} \sqrt{23} - \sqrt{5}\right)} \sqrt{2} \sqrt{7} \sqrt{23} \\ y_{2} = -\frac{25}{322} \, \sqrt{5} \sqrt{322} & y_{3} = -\frac{25}{322} \, \sqrt{2} \sqrt{5} \sqrt{7} \sqrt{23} & z = \frac{25}{322} \, {\left(4 \, \sqrt{2} \sqrt{7} \sqrt{23} + \sqrt{5}\right)} \sqrt{2} \sqrt{7} \sqrt{23} \end{array}$$\right)

The only answer that makes sense:

y2 = 2.69155446756
y3 = 36.6554809843
z = 63.3445121951

Note: all solutions are in units of 'ft'