restart:trajectoire elliptiquer:=theta->p/(1-e*cos(theta-theta0));

NiM+SSJyRzYiZio2I0kmdGhldGFHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSomSSJwR0YlIiIiLCZGLkYuKiZJImVHRiVGLi1JJGNvc0dGJTYjLCY5JEYuSSd0aGV0YTBHRiUhIiJGLkY4RjhGJUYlRiU=

Exercice 1Calcul de e^2 et cos^2(theta0) en fonction de p,R et du demi grand axe ara:=r(theta0); apog\351e

NiM+SSNyYUc2IiomSSJwR0YlIiIiLCZGKEYoSSJlR0YlISIiRis=

rp:=r(theta0+Pi); p\351rig\351e

NiM+SSNycEc2IiomSSJwR0YlIiIiLCZGKEYoSSJlR0YlRighIiI=

eq_a:=2*a=rp+ra; Equation du demi-grand axe

NiM+SSVlcV9hRzYiLywkSSJhR0YlIiIjLCYqJkkicEdGJSIiIiwmRi1GLUkiZUdGJUYtISIiRi0qJkYsRi0sJkYtRi1GL0YwRjBGLQ==

eq_R:=R=r(0); Condition initiale du tir

NiM+SSVlcV9SRzYiL0kiUkdGJSomSSJwR0YlIiIiLCZGKkYqKiZJImVHRiVGKi1JJGNvc0c2JEkqcHJvdGVjdGVkR0YxSShfc3lzbGliR0YlNiNJJ3RoZXRhMEdGJUYqISIiRjU=

On resout le systeme compos\351e des deux \351galit\351s pr\351cendentes pour les inconnues e et cos(theta0)sol:=solve({eq_a,eq_R},{e,cos(theta0)});

NiM+SSRzb2xHNiI8JC9JImVHRiUtSSdSb290T2ZHNiRJKnByb3RlY3RlZEdGLEkoX3N5c2xpYkdGJTYkLChJImFHRiUhIiIqJkYwIiIiSSNfWkdGKyIiI0YzSSJwR0YlRjMvSSZsYWJlbEdGJUkkX0wxR0YlLy1JJGNvc0dGKzYjSSd0aGV0YTBHRiUsJCooLCZJIlJHRiVGMUY2RjNGM0ZCRjFGKUYxRjE=

sol:=allvalues(sol);on explicite l'expression de e \340 l'aide de la fonction allvalues

NiM+SSRzb2xHNiI2JDwkL0kiZUdGJSokLCQqJiwmSSJhR0YlISIiSSJwR0YlIiIiRjFGLkYvRi8jRjEiIiMvLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjhJKF9zeXNsaWJHRiU2I0kndGhldGEwR0YlLCQqKCwmSSJSR0YlRi9GMEYxRjFGP0YvRisjRi9GM0YvPCQvRiksJEYqRi8vRjVGPQ==

sol_part:=[op(sol[1])];on ne garde que que la solution v\351rifiant e>0

NiM+SSlzb2xfcGFydEc2IjckL0kiZUdGJSokLCQqJiwmSSJhR0YlISIiSSJwR0YlIiIiRjBGLUYuRi4jRjAiIiMvLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjdJKF9zeXNsaWJHRiU2I0kndGhldGEwR0YlLCQqKCwmSSJSR0YlRi5GL0YwRjBGPkYuRiojRi5GMkYu

edeux:=subs(sol_part,e)^2;on calcule e^2

NiM+SSZlZGV1eEc2IiwkKiYsJkkiYUdGJSEiIkkicEdGJSIiIkYsRilGKkYq

costheta0deux:=subs(sol_part,cos(theta0))^2;on calcule cos(theta0)^2

NiM+SS5jb3N0aGV0YTBkZXV4RzYiLCQqKiwmSSJSR0YlISIiSSJwR0YlIiIiIiIjRikhIiMsJkkiYUdGJUYqRitGLEYqRjBGLEYq

Exercice 2Constante de la loi des aires: conservation du moment cin\351tiqueC:=R*v0*sin(alpha);

NiM+SSJDRzYiKihJIlJHRiUiIiJJI3YwR0YlRigtSSRzaW5HNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjSSZhbHBoYUdGJUYo

Expression de p en fonction des conditions initialesp:=C*C/(G*M);

NiM+SSJwRzYiKixJIlJHRiUiIiNJI3YwR0YlRigtSSRzaW5HNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjSSZhbHBoYUdGJUYoSSJHR0YlISIiSSJNR0YlRjI=

p:=subs(G=vc*vc*R/M,p); injectons dans p l'expression de G

NiM+SSJwRzYiKipJIlJHRiUiIiJJI3YwR0YlIiIjLUkkc2luRzYkSSpwcm90ZWN0ZWRHRi5JKF9zeXNsaWJHRiU2I0kmYWxwaGFHRiVGKkkjdmNHRiUhIiM=

p:=subs(v0=T*vc,p); injectons dans p l'expression de v0=T*vc

NiM+SSJwRzYiKihJIlJHRiUiIiJJIlRHRiUiIiMtSSRzaW5HNiRJKnByb3RlY3RlZEdGLkkoX3N5c2xpYkdGJTYjSSZhbHBoYUdGJUYq

Exercice 3Expression de la conservation de l'\351nergie m\351caniqueeq_E:=(1/2)*m*v0*v0-G*M*m/R=-G*M*m/(2*a);

NiM+SSVlcV9FRzYiLywmKiZJIm1HRiUiIiJJI3YwR0YlIiIjI0YqRiwqKkkiR0dGJUYqSSJNR0YlRipGKUYqSSJSR0YlISIiRjIsJCoqRi9GKkYwRipGKUYqSSJhR0YlRjIjRjJGLA==

a:=solve(eq_E,a); expression de a

NiM+SSJhRzYiLCQqKkkiR0dGJSIiIkkiTUdGJUYpSSJSR0YlRiksJiomSSN2MEdGJSIiI0YrRilGKSomRihGKUYqRikhIiMhIiJGMg==

a:=subs(G=vc*vc*R/M,a); injectons dans a l'expression de G

NiM+SSJhRzYiLCQqKEkjdmNHRiUiIiNJIlJHRiVGKSwmKiZJI3YwR0YlRilGKiIiIkYuKiZGKEYpRipGLiEiIyEiIkYx

a:=subs(v0=T*vc,a); injectons dans a l'expression de v0

NiM+SSJhRzYiLCQqKEkjdmNHRiUiIiNJIlJHRiVGKSwmKihJIlRHRiVGKUYoRilGKiIiIkYuKiZGKEYpRipGLiEiIyEiIkYx

a:=simplify(a); on simplifie l'expression

NiM+SSJhRzYiLCQqJkkiUkdGJSIiIiwmKiRJIlRHRiUiIiNGKSEiI0YpISIiRi8=

Exercice 4Etude de la port\351ecostheta0deux;costheta0deux:=subs(sin(alpha)^2=1-cos(alpha)^2,costheta0deux); expression de cos(theta0)^2 en fonction de cos(alpha)^2

NiMqKiwmSSJSRzYiISIiKihGJSIiIkkiVEdGJiIiIy1JJHNpbkc2JEkqcHJvdGVjdGVkR0YvSShfc3lzbGliR0YmNiNJJmFscGhhR0YmRitGKUYrRiVGJywmKiZGJUYpLCYqJEYqRitGKSEiI0YpRidGKUYoRilGJ0Y1Ric=NiM+SS5jb3N0aGV0YTBkZXV4RzYiKiosJkkiUkdGJSEiIiooRigiIiJJIlRHRiUiIiMsJkYrRisqJC1JJGNvc0c2JEkqcHJvdGVjdGVkR0YzSShfc3lzbGliR0YlNiNJJmFscGhhR0YlRi1GKUYrRitGLUYoRiksJiomRihGKywmKiRGLEYtRishIiNGK0YpRitGKkYrRilGOUYp

Calcul des extremact2:=subs(cos(alpha)^2=x,costheta0deux);expression de cos(theta0)^2 en fonction de x=cos(alpha)^2

NiM+SSRjdDJHNiIqKiwmSSJSR0YlISIiKihGKCIiIkkiVEdGJSIiIywmRitGK0kieEdGJUYpRitGK0YtRihGKSwmKiZGKEYrLCYqJEYsRi1GKyEiI0YrRilGK0YqRitGKUYyRik=

deriv:=diff(ct2,x); d\351riv\351e par rapport \340 x

NiM+SSZkZXJpdkc2IiwmKiosJkkiUkdGJSEiIiooRikiIiJJIlRHRiUiIiMsJkYsRixJInhHRiVGKkYsRixGLCwmKiZGKUYsLCYqJEYtRi5GLCEiI0YsRipGLEYrRixGKkYzRipGLUYuRjUqKkYoRi5GMUY1RjNGKkYtRi5GLA==

sol:=solve(deriv=0,x); recherche des valeurs critiques

NiM+SSRzb2xHNiI2JComLCYhIiIiIiIqJEkiVEdGJSIiI0YqRipGLCEiIyomRihGKiwmRitGKkYuRipGKQ==

cos_carre_alpha_optimum_1:=sol[1];cos_carre_alpha_optimum_2:=sol[2]; expression des valeurs optimales de cos(alpha)^2

NiM+STpjb3NfY2FycmVfYWxwaGFfb3B0aW11bV8xRzYiKiYsJiEiIiIiIiokSSJUR0YlIiIjRilGKUYrISIjNiM+STpjb3NfY2FycmVfYWxwaGFfb3B0aW11bV8yRzYiKiYsJiEiIiIiIiokSSJUR0YlIiIjRilGKSwmRipGKSEiI0YpRig=

Il s'agit bien d'un maximum de theta0 et donc un minimum de cos(theta0)^2 pour cos_carre_optimum_2 (seule valeur acceptable)delta:=factor(ct2-subs(x=cos_carre_alpha_optimum_2,ct2)); expression de la difference cos(theta0(x))^2-cos(theta0(xoptimum))^2

NiM+SSZkZWx0YUc2IiwkKipJIlRHRiUiIiUsKiokRigiIiMhIiIiIiJGLiomRihGLEkieEdGJUYuRi5GMCEiI0YsLCZGK0YuRjFGLkYxLCxGLUYuKiRGKEYpRi1GK0YsKiZGKEYpRjBGLkYuRi9GMUYtRi0=

assume(x>0 and x<1);assume(T>0 and T<1); is(delta>=0);cette diff\351rence est bien positive

NiNJJXRydWVHSSpwcm90ZWN0ZWRHRiQ=

Exercice 5Application Num\351riqueR:=6400000;T:=0.99;'cos_carre_alpha_optimum_2'=cos_carre_alpha_optimum_2;

NiM+SSJSRzYiIigrK1MnNiM+SSJURzYiJCIjKiohIiM=NiMvSTpjb3NfY2FycmVfYWxwaGFfb3B0aW11bV8yRzYiJCIkJz4hIiU=

Digits:=3:alpha:=arccos(sqrt(cos_carre_alpha_optimum_2));theta0:=arccos(sqrt(costheta0deux));

NiM+SSZhbHBoYUc2IiQiJFYiISIjNiM+SSd0aGV0YTBHNiIkIiRHIiEiIw==

'p'=p;e:=sqrt(edeux);'r(theta)'=r(theta);

NiMvSSJwRzYiJCIkOSciIiU=NiM+SSJlRzYiJCIkVyIhIiQ=NiMvLUkickc2IjYjSSZ0aGV0YUdGJiwkKiQsJiIiIkYsLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiY2IywmRihGLCQhJEciISIjRiwkISRXIiEiJCEiIiQiJDknIiIl

with(plots):

Warning, the name changecoords has been redefined

terre:=plot([R,theta,theta=0..2*Pi],coords=polar,scaling=constrained,color=blue,thickness=3):traj:=plot([r(theta),theta,theta=0..2*theta0],coords=polar,scaling=constrained,color=red,thickness=2):display(terre,traj);

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fleche=(ra-R)/1000; hauteur de la fl\350cehe de la trajectoire relativement \340 la surface terrestre en km

NiMvSSdmbGVjaGVHNiIkIiN4IiIi