35.sagews

fmin,fmax,s0,v0,phi,tc,tg = var( 'fmin,fmax,s0,v0,phi,tc,tg' )
f = [ fmax*tc + fmin*(tg - tc) == v0, 
      fmin*(tg - tc)^2/2 + fmax * tc * (tg - tc) + fmax*tc^2 / 2 == s0  + v0*tg - phi]
assume( [s0 > 0, v0 > 0, tg > tc, tc > 0, fmin < 0, fmax > 0] ) 
solutions = solve(f,[tc,tg])
print solutions
print latex(solutions)
print

# some checking ;)
print bool(f[0].substitute(tc = (-sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 -
2*fmax*phi) - (fmax - fmin)*v0)/(fmax*fmin - fmax^2), tg = (fmin*v0 -
sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 -
2*fmax*phi))/(fmax*fmin)))

print bool(f[1].substitute(tc = (-sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 -
2*fmax*phi) - (fmax - fmin)*v0)/(fmax*fmin - fmax^2), tg = (fmin*v0 -
sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 -
2*fmax*phi))/(fmax*fmin)))

print bool(f[0].substitute(tc = (sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi) + (fmin - fmax)*v0)/(fmax*fmin - fmax^2), tg = (sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi) + fmin*v0)/(fmax*fmin)))

print bool(f[1].substitute(tc = (sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi) + (fmin - fmax)*v0)/(fmax*fmin - fmax^2), tg = (sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi) + fmin*v0)/(fmax*fmin)))

[ [tc == (-sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi) - (fmax - fmin)*v0)/(fmax*fmin - fmax^2), tg == (fmin*v0 - sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi))/(fmax*fmin)], [tc == (sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi) + (fmin - fmax)*v0)/(fmax*fmin - fmax^2), tg == (sqrt(fmax - fmin)*sqrt(-fmin)*sqrt(v0^2 + 2*fmax*s0 - 2*fmax*phi) + fmin*v0)/(fmax*fmin)] ] \begin{array}{l}[\begin{array}{l}[\mbox{tc} = \frac{{{-\left( \sqrt{ \mbox{fmax} - \mbox{fmin} } \right) \sqrt{ -\mbox{fmin} }} \sqrt{ {v_{0}}^{2} + {{2 \mbox{fmax}} s_{0}} - {{2 \mbox{fmax}} \mbox{phi}} }} - {\left( \mbox{fmax} - \mbox{fmin} \right) v_{0

{\mbox{fmax} \mbox{fmin}} - {\mbox{fmax}}^{2} },\\ \mbox{tg} = \frac{{\mbox{fmin} v_{0}} - {{\sqrt{ \mbox{fmax} - \mbox{fmin} } \sqrt{ -\mbox{fmin} }} \sqrt{ {v_{0}}^{2} + {{2 \mbox{fmax}} s_{0}} - {{2 \mbox{fmax}} \mbox{phi}} }}}{{\mbox{fmax} \mbox{fmin}}}]\end{array},\\ \begin{array}{l}[\mbox{tc} = \frac

\mbox{tg}  =  \frac{{{\sqrt{ \mbox{fmax} - \mbox{fmin} } \sqrt{ -\mbox{fmin} }} \sqrt{ {v_{0}}^{2}  + {{2 \mbox{fmax}} s_{0}} - {{2 \mbox{fmax}} \mbox{phi}} }} + {\mbox{fmin} v_{0}}}{{\mbox{fmax} \mbox{fmin}}}]\end{array}]\end{array}

True
True
True
True
}}}

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