L'angle\; optimal

Sol

Dans\;le\;tableur\;ci-contre,\
en\;face\;de\;chaque\;valeur\;de\;\alpha\
s'affichent\;les\;valeurs\;de\;P_{graph}\
et\; P_{form}\;correspondantes. \;On\;en\
conjecture\;que\;l'angle\;optimal\
\;\;\;\;\;\;\;est\;\;\;\alpha=41°    \;\;\;\; (ligne\;8)

P_{graph}\; \;et\;\; P_{form}\;\;sont\; les \;valeurs \;de\; la\;portée\
 associée\; à \;\alpha\;définies\; respectivement:\
*\;comme\; abscisse\; du \;point\; d'intersection\; de\;
 (C_{F_{\alpha}})\;avec (xx')\
*\;par\;la\;formule:\;P_{form}=\
\left(\frac{11.2^2×cos(\alpha)}{9.8}   \right) \left( sin(\alpha)+\sqrt{sin^2(\alpha)+\frac{2×9.8×1.8}{11.2^2}  \right)

\;\;Zoom\; de \;la\; bande:\
\left [x=14.3\;,\;x=14.5  \right ] 

Sol

1.8

α = 0.7330382858376184 rad (slider from 35*° to 55*°, step 0.017453292519943295)
F_α(x) = ((-(9.8 / 2)) * (x / ((11.2 * cos(α))))^(2)) + (x * tan(α)) + 1.8
g_α = Function(F_α, 0, 20)
A, B = Root(g_α)
P_{graph}: x(B)
C = Point(a)
liste1 = Sequence(If[0 ≤ x,((-(9.8 / 2)) * (x / ((11.2 * cos(α))))^(2)) + (x * tan(α)) + 1.8], α, 35*°, 55*°, 1*°)
P_{form}: ((11.2^(2) * cos(α) / 9.8) * (sin(α) + sqrt(sin(α)^(2) + (((2 * 9.8) * 1.8)) / 11.2^(2))))
A1 = Name(α)
B1 = Name(P_{graph})
C1 = Name(P_{form})
A2 = 0.6108652381980153 rad
B2 = 14.204804061333503
C2 = 14.204804061333498
A3 = 0.6283185307179586 rad
B3 = 14.284835843694996
C3 = 14.284835843695
A4 = 0.6457718232379019 rad
B4 = 14.351996176024567
C4 = 14.351996176024569
A5 = 0.6632251157578452 rad
B5 = 14.40602876077255
C5 = 14.406028760772557
A6 = 0.6806784082777885 rad
B6 = 14.446700472940714
C6 = 14.446700472940709
A7 = 0.6981317007977318 rad
B7 = 14.473801309777286
C7 = 14.473801309777281
A8 = 0.7155849933176751 rad
B8 = 14.487144318015245
C8 = 14.487144318015245
A9 = 0.7330382858376183 rad
B9 = 14.486565500731713
C9 = 14.486565500731714
A10 = 0.7504915783575616 rad
B10 = 14.471923705446615
C10 = 14.471923705446615
A11 = 0.767944870877505 rad
B11 = 14.443100494695928
C11 = 14.443100494695924
A12 = 0.7853981633974483 rad
B12 = 14.4
C12 = 14.4
A13 = 0.8028514559173916 rad
B13 = 14.342548759891052
C13 = 14.342548759891057
A14 = 0.8203047484373349 rad
B14 = 14.270695542458173
C14 = 14.27069554245817
A15 = 0.8377580409572781 rad
B15 = 14.184411152705042
C15 = 14.184411152705046
A16 = 0.8552113334772214 rad
B16 = 14.083688224889583
C16 = 14.083688224889585
A17 = 0.8726646259971647 rad
B17 = 13.968540999918783
C17 = 13.968540999918783
A18 = 0.890117918517108 rad
B18 = 13.839005087802512
C18 = 13.839005087802512
A19 = 0.9075712110370513 rad
B19 = 13.695137215121719
C19 = 13.695137215121719
A20 = 0.9250245035569946 rad
B20 = 13.537014957435998
C20 = 13.537014957435998
A21 = 0.9424777960769379 rad
B21 = 13.364736456539786
C21 = 13.364736456539784
A22 = 0.9599310885968813 rad
B22 = 13.178420122472415
C22 = 13.178420122472417
texte2: "P_{graph}=" + (LaTeX[P_{graph}]) + "\;m"
D = Point(a)
S = (0, 1.8)