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+ \begin{enumerate} + \item On observe un mobile en mouvement et on décrit sa position verticale en fonction du temps $t$ en secondes par la fonction $z(t) = -4,9t^2 + 12$. + \begin{enumerate} + \item Déterminer la fonction décrivant la vitesse du module $v(t) = z'(t)$ (ou en notation physique $\dfrac{dz}{dt}$). + \item Déterminer la fonction décrivant l'accélération du module $a(t) = v'(t)$ (ou en notation physique $\dfrac{dv}{dt}$). + \item À quelle hauteur le mobile a été lâché? Quel était alors sa vitesse? Son accélération? + \end{enumerate} + \item On étudie un mobile en chute libre. On le lance à une hauteur de 10m au dessus du sol avec une vitesse de 1m/s. Un bilan des forces permet de connaître son accélération au cours du mouvement: $a(t) = -10$. + \begin{enumerate} + \item On rappelle que l'accélération est la dérivée de la vitesse ($a(t) = v'(t)$). Déterminer la fonction vitesse du mobile. + \item On rappelle que la vitesse est la dérivée de la position ($v(t) = z(t)$). Déterminer la fonction position du mobile. + \item Est-ce que les deux fonctions déterminées aux questions précédentes sont conformes aux conditions initiales? + \end{enumerate} + \item On considère maintenant les frottements dûs à l'air. Ils exercent une force proportionnelle à la vitesse. Le bilan des forces mène à la fonction accélération suivante: $a(t) = v'(t) = kv(t)$. Pour simplifier on considèrera que $k = 1$ et on a donc $a(t) = v'(t) = v(t)$ + \begin{enumerate} + \item Est-ce que $v(t)$ peut-être une fonction constante? + \item Est-ce que $v(t)$ peut-être une fonction polynôme? + \item Est-ce que $v(t)$ peut-être une fonction exponentielle? + \end{enumerate} + \end{enumerate} \end{exercise} -\begin{solution} - <++> -\end{solution} - -\collectexercisesstop{banque} \ No newline at end of file +\begin{exercise}[subtitle={Vérifications}, step={2}, origin={Création}, topics={Équation differentielle}, tags={analyse, exponentiel, dérivation}] + Vérifier si les fonctions sont oui ou non solution des équations différentielles. +\end{exercise} +\collectexercisesstop{banque} diff --git a/TST_sti2d/07_Equation_differentielle/index.rst b/TST_sti2d/07_Equation_differentielle/index.rst index d1e78e4..a023549 100644 --- a/TST_sti2d/07_Equation_differentielle/index.rst +++ b/TST_sti2d/07_Equation_differentielle/index.rst @@ -2,7 +2,7 @@ ####################### :date: 2021-02-07 -:modified: 2021-02-07 +:modified: 2021-02-08 :authors: Benjamin Bertrand :tags: Analyse, Exponentiel, Dérivation :category: TST_sti2d @@ -13,14 +13,24 @@ À partir d'une position, on cherche à retrouver la vitesse puis l'accélération. Puis on renverse le problème, à partir d'une accélération, on va chercher à retrouver une fonction décrivant la position. +Le but est de passer assez vite sur la première partie pour se concentrer sur la 2e et introduire la 3e. + +.. image:: ./1E_pos_vitesse_acc.pdf + :height: 200px + :alt: Manipulation de position, vitesse et accélération. + Bilan: Définition d'une équation différentielle avec en particulier les différentes formes à connaître. +.. image:: ./1B_eq_diff.pdf + :height: 200px + :alt: Définition des éuqation différentielles + Étape 2: Solution d'une équation différentielle =============================================== Plusieurs fonctions candidates, on vérifie qu'elles sont solutions ou non d'équations différentielles. Puis on donne des équations différentielles uniquement avec la dérivée et on cherche des solutions. On expliquera que c'est une autre façon de retrouver la primitive. -Bilan: Trois famille d'équations différentielles $y'=a$, $y'=ay$ et $y'=ay+b$ avec les familles de solutions ou une méthode pour les résoudre. +Bilan: Trois famille d'équations différentielles *y'=a*, *y'=ay* et *y'=ay+b* avec les familles de solutions ou une méthode pour les résoudre. Étape 3: Famille de solutions ============================= @@ -32,6 +42,6 @@ Bilan: trouver une solution particulière Étape 4: Résolution d'une équation différentielle ================================================= -Exercices techniques de résolution. Puis problèmes de mise en situation. +Exercices techniques de résolution. Puis problèmes de mise en situation (bilan des forces - chute libre avec frottement ...). Bilan: vidéo de correction par un élève?