update uebungen

2020-05-05 16:16:41 +02:00
parent b33c550fbb
commit fb7f3aad6c
6 changed files with 13 additions and 14 deletions
@@ -107,11 +107,10 @@
                \intertext{Zusammen folgt}
                &\Delta t = \int_{x_0}^{x_E} \frac{\sqrt{1 + f'(x)^2}}{\sqrt{\frac{2}{m} (E - V(\vec{x}))}} \d x
            .\end{align*}
-        \item Mit $V(z) = mgz$, $E = 0$, $x_0 = 0$, $x_E = 1$ und $f(x) = x$ folgt
+        \item Mit $V(z) = mgz$, $E = \frac{m}{2} v_0^2$, $x_0 = 0$, $x_E = 1$ und $f(x) = x$ folgt
            \[
-            \Delta t = \int_{0}^{1} \frac{\sqrt{2} }{\sqrt{\frac{2}{m} mgx} } \d x
-            = \int_{0}^{1} \frac{\sqrt{2} }{\sqrt{2 g x} } \d x = \frac{2}{\sqrt{g} } \sqrt{x}
-            \Big|_{0}^{1} = \frac{2}{\sqrt{g} }
+                \Delta t = \int_{0}^{1} \frac{\sqrt{2} }{\sqrt{\frac{2}{m} (\frac{m}{2}v_0^2 + mgx)} } \d x
+            = \int_{0}^{1} \frac{\sqrt{2} }{\sqrt{v_0^2 + 2 g x} } \d x = 2 \cdot \frac{1}{2g} \cdot \sqrt{2} \sqrt{v_0^2 + 2gx} \Big|_{0}^{1}  \d x 
            .\] 
    \end{enumerate}
 \end{aufgabe}