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{SECT 0 {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 8 "restart:" }
{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 203 22 "trajecto
ire elliptique" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "> " 0 "" 
{MPLTEXT 1 202 36 "r:=theta->p/(1-e*cos(theta-theta0));" }{MPLTEXT 1 
202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"rG6\"f*6#I&thetaGF%F%6
$I)operatorGF%I&arrowGF%F%*&I\"pGF%\"\"\",&F.F.*&I\"eGF%F.-I$cosGF%6#,
&9$F.I'theta0GF%!\"\"F.F8F8F%F%F%" }{TEXT 205 0 "" }}}{EXCHG {PARA 
200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 
206 10 "Exercice 1" }}{PARA 201 "" 0 "" {TEXT 204 0 "" }}}{EXCHG 
{PARA 203 "" 0 "" {TEXT 204 10 "Calcul de " }{TEXT 207 3 "e^2" }{TEXT 
204 4 " et " }{TEXT 207 13 "cos^2(theta0)" }{TEXT 204 16 " en fonction
 de " }{TEXT 207 3 "p,R" }{TEXT 204 22 " et du demi grand axe " }
{TEXT 207 1 "a" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "> " 0 "" 
{MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 15 "r
a:=r(theta0); " }{TEXT 208 6 "apog\351e" }{MPLTEXT 1 202 0 "" }}{PARA 
202 "" 1 "" {XPPMATH 20 "6#>I#raG6\"*&I\"pGF%\"\"\",&F(F(I\"eGF%!\"\"F
+" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 15 "rp
:=r(theta0+Pi" }{MPLTEXT 1 202 3 "); " }{TEXT 208 7 "p\351rig\351e" }
{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I#rpG6\"*&I\"p
GF%\"\"\",&F(F(I\"eGF%F(!\"\"" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> \+
" 0 "" {MPLTEXT 1 202 17 "eq_a:=2*a=rp+ra; " }{TEXT 208 26 "Equation d
u demi-grand axe" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 
20 "6#>I%eq_aG6\"/,$I\"aGF%\"\"#,&*&I\"pGF%\"\"\",&F-F-I\"eGF%F-!\"\"F
-*&F,F-,&F-F-F/F0F0F-" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" 
{MPLTEXT 1 202 14 "eq_R:=R=r(0); " }{TEXT 208 25 "Condition initiale d
u tir" }{MPLTEXT 1 202 0 "" }}{PARA 204 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#>I%eq_RG6\"/I\"RGF%*&I\"pGF%\"\"\",&F
*F**&I\"eGF%F*-I$cosG6$I*protectedGF1I(_syslibGF%6#I'theta0GF%F*!\"\"F
5" }{TEXT 205 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 204 13 "On resout
 le " }{TEXT 207 7 "systeme" }{TEXT 204 49 " compos\351e des deux \351
galit\351s pr\351cendentes pour les" }{TEXT 207 27 " inconnues e et co
s(theta0)" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 
202 40 "sol:=solve(\{eq_a,eq_R\},\{e,cos(theta0)\});" }{MPLTEXT 1 202 
0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I$solG6\"<$/I\"eGF%-I'RootOfG
6$I*protectedGF,I(_syslibGF%6$,(I\"aGF%!\"\"*&F0\"\"\"I#_ZGF+\"\"#F3I
\"pGF%F3/I&labelGF%I$_L1GF%/-I$cosGF+6#I'theta0GF%,$*(,&I\"RGF%F1F6F3F
3FBF1F)F1F1" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 
202 20 "sol:=allvalues(sol);" }{TEXT 209 29 "on explicite l'expression
 de " }{TEXT 208 1 "e" }{TEXT 209 25 " \340 l'aide de la fonction " }
{TEXT 210 9 "allvalues" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#>I$solG6\"6$<$/I\"eGF%*$,$*&,&I\"aGF%!\"\"I\"pGF%\"\"\"
F1F.F/F/#F1\"\"#/-I$cosG6$I*protectedGF8I(_syslibGF%6#I'theta0GF%,$*(,
&I\"RGF%F/F0F1F1F?F/F+#F/F3F/<$/F),$F*F//F5F=" }{TEXT 205 0 "" }}}
{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 23 "sol_part:=[op(sol[1])];"
 }{TEXT 209 42 "on ne garde que que la solution v\351rifiant " }{TEXT 
208 3 "e>0" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I
)sol_partG6\"7$/I\"eGF%*$,$*&,&I\"aGF%!\"\"I\"pGF%\"\"\"F0F-F.F.#F0\"
\"#/-I$cosG6$I*protectedGF7I(_syslibGF%6#I'theta0GF%,$*(,&I\"RGF%F.F/F
0F0F>F.F*#F.F2F." }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" 
{MPLTEXT 1 202 26 "edeux:=subs(sol_part,e)^2;" }{TEXT 209 11 "on calcu
le " }{TEXT 208 3 "e^2" }{MPLTEXT 1 211 1 " " }{MPLTEXT 1 202 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#>I&edeuxG6\",$*&,&I\"aGF%!\"\"I\"pGF%
\"\"\"F,F)F*F*" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" 
{MPLTEXT 1 202 44 "costheta0deux:=subs(sol_part,cos(theta0))^2;" }
{TEXT 209 11 "on calcule " }{TEXT 208 13 "cos(theta0)^2" }{MPLTEXT 1 
202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I.costheta0deuxG6\",$**,&
I\"RGF%!\"\"I\"pGF%\"\"\"\"\"#F)!\"#,&I\"aGF%F*F+F,F*F0F,F*" }{TEXT 
205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG 
{PARA 201 "" 0 "" {TEXT 206 10 "Exercice 2" }{TEXT 212 0 "" }}}{EXCHG 
{PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 203 "" 0 "" 
{TEXT 204 31 "Constante de la loi des aires: " }{TEXT 207 32 "conserva
tion du moment cin\351tique" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "> " 
0 "" {MPLTEXT 1 202 19 "C:=R*v0*sin(alpha);" }{MPLTEXT 1 202 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"CG6\"*(I\"RGF%\"\"\"I#v0GF%F(-I$s
inG6$I*protectedGF-I(_syslibGF%6#I&alphaGF%F(" }{TEXT 205 0 "" }}}
{EXCHG {PARA 203 "" 0 "" {TEXT 204 14 "Expression de " }{TEXT 207 1 "p
" }{TEXT 204 1 " " }{TEXT 207 36 "en fonction des conditions initiales
" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 13 "p:=
C*C/(G*M);" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I
\"pG6\"*,I\"RGF%\"\"#I#v0GF%F(-I$sinG6$I*protectedGF-I(_syslibGF%6#I&a
lphaGF%F(I\"GGF%!\"\"I\"MGF%F2" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 ">
 " 0 "" {MPLTEXT 1 202 23 "p:=subs(G=vc*vc*R/M,p);" }{TEXT 209 1 " " }
{TEXT 208 10 "injectons " }{TEXT 209 5 "dans " }{TEXT 208 1 "p" }
{TEXT 209 17 " l'expression de " }{TEXT 208 1 "G" }{MPLTEXT 1 202 0 ""
 }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"pG6\"**I\"RGF%\"\"\"I#v0GF%\"
\"#-I$sinG6$I*protectedGF.I(_syslibGF%6#I&alphaGF%F*I#vcGF%!\"#" }
{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 20 "p:=sub
s(v0=T*vc,p); " }{TEXT 208 9 "injectons" }{TEXT 209 6 " dans " }{TEXT 
208 1 "p" }{TEXT 209 17 " l'expression de " }{TEXT 208 7 "v0=T*vc" }
{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"pG6\"*(I\"R
GF%\"\"\"I\"TGF%\"\"#-I$sinG6$I*protectedGF.I(_syslibGF%6#I&alphaGF%F*
" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}
}{EXCHG {PARA 205 "> " 0 "" {TEXT 213 10 "Exercice 3" }{MPLTEXT 1 202 
0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG 
{PARA 203 "" 0 "" {TEXT 204 11 "Expression " }{TEXT 204 6 "de la " }
{TEXT 207 35 "conservation de l'\351nergie m\351canique" }{TEXT 204 0 
"" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 41 "eq_E:=(1/2)*m*v0*v
0-G*M*m/R=-G*M*m/(2*a);" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#>I%eq_EG6\"/,&*&I\"mGF%\"\"\"I#v0GF%\"\"##F*F,**I\"GGF%
F*I\"MGF%F*F)F*I\"RGF%!\"\"F2,$**F/F*F0F*F)F*I\"aGF%F2#F2F," }{TEXT 
205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 18 "a:=solve(eq_
E,a); " }{TEXT 209 14 "expression de " }{TEXT 208 1 "a" }{MPLTEXT 1 
202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"aG6\",$**I\"GGF%\"\"\"
I\"MGF%F)I\"RGF%F),&*&I#v0GF%\"\"#F+F)F)*&F(F)F*F)!\"#!\"\"F2" }{TEXT 
205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 23 "a:=subs(G=vc
*vc*R/M,a);" }{MPLTEXT 1 202 1 " " }{TEXT 208 9 "injectons" }{TEXT 
209 6 " dans " }{TEXT 208 1 "a" }{TEXT 209 17 " l'expression de " }
{TEXT 208 1 "G" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "
6#>I\"aG6\",$*(I#vcGF%\"\"#I\"RGF%F),&*&I#v0GF%F)F*\"\"\"F.*&F(F)F*F.!
\"#!\"\"F1" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 
202 20 "a:=subs(v0=T*vc,a); " }{TEXT 208 9 "injectons" }{TEXT 209 6 " \+
dans " }{TEXT 208 1 "a" }{TEXT 209 17 " l'expression de " }{TEXT 208 
2 "v0" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"aG6
\",$*(I#vcGF%\"\"#I\"RGF%F),&*(I\"TGF%F)F(F)F*\"\"\"F.*&F(F)F*F.!\"#!
\"\"F1" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 
16 "a:=simplify(a); " }{TEXT 209 25 "on simplifie l'expression" }
{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I\"aG6\",$*&I
\"RGF%\"\"\",&*$I\"TGF%\"\"#F)!\"#F)!\"\"F/" }{TEXT 205 0 "" }}}
{EXCHG {PARA 205 "> " 0 "" {TEXT 213 10 "Exercice 4" }{MPLTEXT 1 202 
0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG 
{PARA 203 "" 0 "" {TEXT 204 12 "Etude de la " }{TEXT 207 6 "port\351e"
 }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 78 "cost
heta0deux;costheta0deux:=subs(sin(alpha)^2=1-cos(alpha)^2,costheta0deu
x); " }{TEXT 209 14 "expression de " }{TEXT 208 14 "cos(theta0)^2 " }
{TEXT 209 15 "en fonction de " }{TEXT 208 12 "cos(alpha)^2" }{MPLTEXT 
1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#**,&I\"RG6\"!\"\"*(F%\"
\"\"I\"TGF&\"\"#-I$sinG6$I*protectedGF/I(_syslibGF&6#I&alphaGF&F+F)F+F
%F',&*&F%F),&*$F*F+F)!\"#F)F'F)F(F)F'F5F'" }{TEXT 205 0 "" }}{PARA 
202 "" 1 "" {XPPMATH 20 "6#>I.costheta0deuxG6\"**,&I\"RGF%!\"\"*(F(\"
\"\"I\"TGF%\"\"#,&F+F+*$-I$cosG6$I*protectedGF3I(_syslibGF%6#I&alphaGF
%F-F)F+F+F-F(F),&*&F(F+,&*$F,F-F+!\"#F+F)F+F*F+F)F9F)" }{TEXT 205 0 ""
 }}}{EXCHG {PARA 203 "" 0 "" {TEXT 204 11 "Calcul des " }{TEXT 207 7 "
extrema" }{TEXT 204 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 
40 "ct2:=subs(cos(alpha)^2=x,costheta0deux);" }{TEXT 209 14 "expressio
n de " }{TEXT 208 14 "cos(theta0)^2 " }{TEXT 209 15 "en fonction de " 
}{TEXT 208 2 "x=" }{TEXT 208 12 "cos(alpha)^2" }{MPLTEXT 1 202 0 "" }}
{PARA 202 "" 1 "" {XPPMATH 20 "6#>I$ct2G6\"**,&I\"RGF%!\"\"*(F(\"\"\"I
\"TGF%\"\"#,&F+F+I\"xGF%F)F+F+F-F(F),&*&F(F+,&*$F,F-F+!\"#F+F)F+F*F+F)
F2F)" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 20 
"deriv:=diff(ct2,x); " }{TEXT 208 23 "d\351riv\351e par rapport \340 x
" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I&derivG6\"
,&**,&I\"RGF%!\"\"*(F)\"\"\"I\"TGF%\"\"#,&F,F,I\"xGF%F*F,F,F,,&*&F)F,,
&*$F-F.F,!\"#F,F*F,F+F,F*F3F*F-F.F5**F(F.F1F5F3F*F-F.F," }{TEXT 205 0 
"" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 23 "sol:=solve(deriv=0
,x); " }{TEXT 209 14 "recherche des " }{TEXT 208 17 "valeurs critiques
" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I$solG6\"6$
*&,&!\"\"\"\"\"*$I\"TGF%\"\"#F*F*F,!\"#*&F(F*,&F+F*F.F*F)" }{TEXT 205 
0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 69 "cos_carre_alpha_
optimum_1:=sol[1];cos_carre_alpha_optimum_2:=sol[2]; " }{TEXT 209 15 "
expression des " }{TEXT 208 33 "valeurs optimales de cos(alpha)^2" }
{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I:cos_carre_al
pha_optimum_1G6\"*&,&!\"\"\"\"\"*$I\"TGF%\"\"#F)F)F+!\"#" }{TEXT 205 
0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I:cos_carre_alpha_optimum_2G6
\"*&,&!\"\"\"\"\"*$I\"TGF%\"\"#F)F),&F*F)!\"#F)F(" }{TEXT 205 0 "" }}}
{EXCHG {PARA 203 "" 0 "" {TEXT 204 20 "Il s'agit bien d'un " }{TEXT 
207 17 "maximum de theta0" }{TEXT 204 12 " et donc un " }{TEXT 207 11 
"minimum de " }{TEXT 207 14 "cos(theta0)^2 " }{TEXT 207 25 "pour cos_c
arre_optimum_2 " }{TEXT 204 25 "(seule valeur acceptable)" }{TEXT 204 
0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 58 "delta:=factor(ct
2-subs(x=cos_carre_alpha_optimum_2,ct2)); " }{TEXT 209 28 "expression \+
de la difference " }{TEXT 208 40 "cos(theta0(x))^2-cos(theta0(xoptimum
))^2" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#>I&delta
G6\",$**I\"TGF%\"\"%,**$F(\"\"#!\"\"\"\"\"F.*&F(F,I\"xGF%F.F.F0!\"#F,,
&F+F.F1F.F1,,F-F.*$F(F)F-F+F,*&F(F)F0F.F.F/F1F-F-" }{TEXT 205 0 "" }}}
{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 54 "assume(x>0 and x<1);assu
me(T>0 and T<1); is(delta>=0);" }{TEXT 209 6 "cette " }{TEXT 208 10 "d
iff\351rence" }{TEXT 209 10 " est bien " }{TEXT 208 8 "positive" }
{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" {XPPMATH 20 "6#I%trueGI*protec
tedGF$" }{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 
0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG 
{PARA 205 "> " 0 "" {TEXT 213 10 "Exercice 5" }{MPLTEXT 1 202 0 "" }}}
{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 203 "" 
0 "" {TEXT 207 21 "Application Num\351rique" }{TEXT 204 0 "" }}}
{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 47 "R:=6400000;T:=0.99;'cos_
carre_alpha_optimum_2'=" }{MPLTEXT 1 202 25 "cos_carre_alpha_optimum_2
" }{MPLTEXT 1 202 1 ";" }{MPLTEXT 1 202 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#>I\"RG6\"\"(++S'" }{TEXT 205 0 "" }}{PARA 202 "" 1 "" 
{XPPMATH 20 "6#>I\"TG6\"$\"#**!\"#" }{TEXT 205 0 "" }}{PARA 202 "" 1 "
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{XPPMATH 20 "6#/I\"pG6\"$\"$9'\"\"%" }{TEXT 205 0 "" }}{PARA 202 "" 1 
"" {XPPMATH 20 "6#>I\"eG6\"$\"$W\"!\"$" }{TEXT 205 0 "" }}{PARA 202 ""
 1 "" {XPPMATH 20 "6#/-I\"rG6\"6#I&thetaGF&,$*$,&\"\"\"F,-I$cosG6$I*pr
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}{TEXT 205 0 "" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 12 "with(
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{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 202 93 "terre:=plot([R,theta,the
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102 "traj:=plot([r(theta),theta,theta=0..2*theta0],coords=polar,scalin
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{PARA 200 "> " 0 "" {MPLTEXT 1 202 20 "display(terre,traj);" }
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