# Find exact value

I want to know exact x'[t] value where z[t]=0

I know approximate x'[t] value is 107

but I cannot find how to find exact x'[t] value

    (*1 mph = 0.44704 m/s
50000ft=15240m*)

g = 9.81;
{d, Subscript[c, 1], Subscript[c, 2]} = {0.2 2, 1.55 10^-4 d, 0.22 d^2};
mass = 100;

ls0 = NDSolve[{x''[t] + Subscript[c, 2]/mass ((x'[t] + 26.8)^2 + z'[t]^2)^(1/2) (x'[t] + 26.8) == 0, z''[t] + Subscript[c, 2]/mass ((x'[t] + 26.8)^2 + z'[t]^2)^(1/2) z'[t] + g == 0, z[0] == 15240, z'[0] == 0, x'[0] == 107, x[0] == 0}, {x, z}, {t, 0, 1000}]

ParametricPlot[
Evaluate[{x[t], z[t]} /. ls0], {t, 0, 110}, PlotRange -> All, AxesLabel -> {speed, flying altitude}, AspectRatio -> 1/GoldenRatio]


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You're using a numerical method.... What do you mean by "exact"? – Dr. belisarius Jul 8 '14 at 4:24
I know What you mean.. But I don't know how to code numerically What I want – user16336 Jul 8 '14 at 4:37

t0 = t /. First@N@Solve[{Evaluate[z[t] /. ls0] == 0}, t]


104.606

z[t0] /. ls0


{-7.25357*10^-13}

x'[t0] /. ls0


{-26.3732}

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