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Bahnen, da wir das Licht sehen können. Außerdem ist + $E > 0$, also bewegen sich die Lichtteilchen auf einer Hyperbelbahn. + \item $L = b m v_{\infty}$, mit $b = R = R_{\text{Sonne}}$ und $v_{\infty} = c$ folgt $L = Rmc$. + \item $r(\varphi) = \frac{p}{1 + \epsilon \cos(\varphi)}$. Für $r \to \infty$ folgt + $1 + \epsilon \cos \varphi \to 0$. Damit folgt + $\overline{\varphi} = \arccos\left( -\frac{1}{\epsilon} \right) $. + + Für die numerische Exzentrizität gilt + \[ + \epsilon + = \sqrt{1 + \frac{2EL^2}{m \alpha^2}} + = \sqrt{1 + \frac{c^4 R^2}{G^2M^2}} + .\] $\epsilon$ ist also masseunabhänig. + + $\varphi_a = \overline{\varphi} \implies \varphi_e = \pi - \overline{\varphi}$. + \item Mit (c) folgt $\vartheta = \varphi_a - \varphi_e = 2\overline{\varphi} - \pi$. Damit folgt + \begin{align*} + \vartheta &= 2\cdot \arccos\left( -\frac{1}{\sqrt{1 + \frac{c^4 R^2}{G^2M^2}}} \right) - \pi \\ + &= 2\cdot \arccos\left( -\frac{1}{\sqrt{1 + \frac{(\SI{3e8}{ms^{-1}})^4 (\SI{7e8}{m})^2} + {(\SI{6.7e-11}{m^{3}kg^{-1}s^{-2}})^2(\SI{2e30}{kg})^2}}} \right) - \pi \\ + &= \ang{;;0.88} + .\end{align*} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe}[Freier Fall auf zwei Wegen] + \begin{enumerate}[a)] + \item Hier bezeichnen die Integrationskonstanten $v_0$ die Anfangsgeschwindigkeit und + $h_0$ die Anfangshöhe. + \begin{align*} + \ddot{x}(t) &= -g \\ + \dot{x}(t) &= -gt + v_0 \\ + x(t) &= -\frac{1}{2} gt^2 + v_0t + h_0 + .\end{align*} + \item Zunächst: Lösung der homogenen DGL $\ddot{x}(t) = 0$. Zu erwarten ist ein Fundamentalsystem + von zwei linear unabhängigen Lösungen: + \begin{align*} + x_1(t) &= 1 \\ + x_2(t) &= t + .\end{align*} + Diese sind offensichtlich linear unabhängig. Eine partikuläre Lösung der inhomogenen Gleichung + $\ddot{x}(t) = -g$ ist: + \[ + x_p(t) = -\frac{1}{2} gt^2 + .\] + Damit erhalten wir die allgemeine Lösung: + \begin{align*} + x(t) &= A x_1(t) + B x_2(t) + x_p(t) \\ + &= A + B t - \frac{1}{2} gt^2 + .\end{align*} + Mit $A = h_0$ und $B = v_0$ folgt damit erneut: + \begin{align*} + x(t) = h_0 + v_0 t - \frac{1}{2} gt^2 + .\end{align*} + \item Die allgemeinen Lösungen in a) und b) sind wie zu erwarten die gleichen. Durch das schrittweise + Integrieren in a) müssen Integrationskonstanten hinzugefügt werden, um die Allgemeinheit der + Lösung zu erhalten. + + Bei b) wird die Allgemeinheit durch die Linearkombination der zwei linear unabhängigen Lösungen + $x_1$ und $x_2$ sicher gestellt. + \end{enumerate} + + \begin{aufgabe}[Gekoppelte Wasserbecken] + \begin{enumerate}[a)] + \item Becken A: Der Wasserabfluss ist proportional zum Wasservolumen. $f_A$ ist der + Proportionalitätsfaktor. Wegen $f_A < 0$ fließt das Wasser ab. Es handelt sich + um eine homogene DGL. + + Becken B: Hier gibt es einen Wasserabfluss $-f_B V_B$, der proportional zum Wasservolumen ist. + $f_B$ ist hier der Proportionalitätsfaktor. Zudem gibt es einen Zufluss $f_AV_A$, der dem + Abfluss aus Becken A entspricht. Es handelt sich wegen $f_AV_A$ um eine inhomogene DGL. + \item Im folgenden sei $f = f_A = f_B$. Dann folgt: + \begin{align*} + \frac{\d V_A}{\d t} &= - f V_A \\ + \frac{\d V_A}{V_A} &= -f \d t \\ + \ln(\V_A) &= -ft + C \\ + \intertext{Mit $V_{A,0} = e^{C}$ folgt} + V_A &= V_{A,0} e^{-ft} + \intertext{Für den homogenen Teil von $V_B$ folgt analog} + V_{B_{h}} &= V_{B,0} e^{-ft} + .\end{align*} + \item Durch Variation der Konstanten $V_{B,0} = B(t)$ folgt $V_B(t) = B(t)e^{-ft}$. Damit + \begin{align*} + \dot{V}_B &= \dot{B}e^{-ft} - f B e^{-ft} = \dot{B}e^{-ft} - f V_B + \intertext{Durch Einsetzen in die Ausgangs DGL für B ergibt sich} + \dot{B}e^{-ft} - fV_B &= - f V_B + f V_A + \intertext{Mit $V_A = V_{A,0} e^{-ft}$ folgt} + \dot{B} &= f V_{A,0}e^{-ft + ft} = f V_{A,0} + \intertext{Durch Integration erhalten wir} + B &= f V_{A,0} t + C + \intertext{Mit $V_B(t = 0) = C \stackrel{!}{=}$ folgt $C = 0$. Damit folgt} + V_B(t) &= f V_{A,0}t \cdot e^{-ft} + .\end{align*} + \item Das Volumen von B steigt zunächst durch den großen Abfluss von A an. Für sehr kleine + $t$ ist $e^{-ft} \approx 1$, das heißt der Anstieg kommt durch den linearen Teil $f V_{A,0} t$ + zustande. Je größer das Volumen von B, desto mehr fließt auch ab, deswegen wird dann ein + Maximum erreicht, nach dem das Volumen monoton fällt. + \end{enumerate} + \end{aufgabe} +\end{aufgabe} + +\end{document}