Commit e6255e67 by Maximilian Vitz

### Updated Solution 3

parent c5ba8877
 %% Cell type:markdown id: tags: ### Error Propagation %% Cell type:markdown id: tags: 1) %% Cell type:markdown id: tags: \$a = 6.1 \pm 0.1\$ cm %% Cell type:markdown id: tags: \$b = 2.8 \pm 0.1\$ cm %% Cell type:markdown id: tags: With 100% correlation one ends up with \$\rho = 1\$. %% Cell type:markdown id: tags: \$\Rightarrow \text{cov}(a,b) = \rho \sigma_{a} \sigma_{b} = \sigma_{a} \sigma_{b}\$ %% Cell type:markdown id: tags: \$\sigma_{f}^{2} = \left( \frac{\delta f}{\delta a} \right)^{2} \sigma_{a}^{2} + \left( \frac{\delta f}{\delta b} \right)^{2} \sigma_{b}^{2} + 2 \left( \frac{\delta A}{\delta a} \right) \left( \frac{\delta f}{\delta b} \right) \text{cov}(a,b) = \left( \frac{\delta f}{\delta a} \right)^{2} \sigma_{a}^{2} + \left( \frac{\delta f}{\delta b} \right)^{2} \sigma_{b}^{2} + 2 \left( \frac{\delta f}{\delta a} \right) \left( \frac{\delta f}{\delta b} \right) \sigma_{a} \sigma_{b} = \left( \frac{\delta f}{\delta a} \sigma_{a} + \frac{\delta f}{\delta b} \sigma_{b} \right)^{2}\$ %% Cell type:markdown id: tags: \$f = A(a,b) = a \cdot b\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{A} = \frac{\delta A}{\delta a} \sigma_{a} + \frac{\delta A}{\delta b} \sigma_{b} = b \sigma_{a} + a \sigma_{b} = 0.89 \text{cm}^{2}\$ %% Cell type:markdown id: tags: \$f = C(a,b) = 2a + 2b\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{C} = \frac{\delta C}{\delta a} \sigma_{a} + \frac{\delta C}{\delta b} \sigma_{b} = 2 \sigma_{a} + 2 \sigma_{b} = 0.4 \text{cm}\$ %% Cell type:markdown id: tags: 2) %% Cell type:markdown id: tags: With 0% correlation one ends up with \$\rho = 0\$. %% Cell type:markdown id: tags: \$\Rightarrow \text{cov}(a,b) = \rho \sigma_{a} \sigma_{b} = 0\$ %% Cell type:markdown id: tags: \$\sigma_{f}^{2} = \left( \frac{\delta f}{\delta a} \right)^{2} \sigma_{a}^{2} + \left( \frac{\delta f}{\delta b} \right)^{2} \sigma_{b}^{2} + 2 \left( \frac{\delta A}{\delta a} \right) \left( \frac{\delta f}{\delta b} \right) \text{cov}(a,b) = \left( \frac{\delta f}{\delta a} \right)^{2} \sigma_{a}^{2} + \left( \frac{\delta f}{\delta b} \right)^{2} \sigma_{b}^{2} \$ %% Cell type:markdown id: tags: \$f = A(a,b) = a \cdot b\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{A} = \sqrt{\left( \frac{\delta A}{\delta a} \right)^{2} \sigma_{a}^{2} + \left( \frac{\delta A}{\delta b} \right)^{2} \sigma_{b}^{2}} = \sqrt{b^{2} \sigma_{a}^{2} + a^{2} \sigma_{b}^{2}} = 0.67 \text{cm}^{2}\$ %% Cell type:markdown id: tags: \$f = C(a,b) = 2a + 2b\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{C} = \sqrt{\left( \frac{\delta C}{\delta a} \right)^{2} \sigma_{a}^{2} + \left( \frac{\delta C}{\delta b} \right)^{2} \sigma_{b}^{2}} = \sqrt{4 \sigma_{a}^{2} + 4 \sigma_{b}^{2}} = 0.28 \text{cm}\$ %% Cell type:markdown id: tags: 3) %% Cell type:markdown id: tags: \$\left(\begin{array}{cc} \sigma_{A}^{2} & \text{cov}(A,C)\\ \text{cov}(A,C) & \sigma_{C}^{2} \end{array}\right) = \left(\begin{array}{cc} \frac{\delta A}{\delta a} & \frac{\delta A}{\delta b}\\ \frac{\delta C}{\delta a} & \frac{\delta C}{\delta b} \end{array}\right) \cdot \left(\begin{array}{cc} \sigma_{a}^{2} & \text{cov}(a,b)\\ \text{cov}(a,b) & \sigma_{b}^{2} \end{array}\right) \cdot \left(\begin{array}{cc} \frac{\delta A}{\delta a} & \frac{\delta C}{\delta a}\\ \frac{\delta A}{\delta b} & \frac{\delta C}{\delta b} \end{array}\right) = \left(\begin{array}{cc} b & a\\ 2 & 2 \end{array}\right) \cdot \left(\begin{array}{cc} \sigma_{a}^{2} & \text{cov}(a,b)\\ \text{cov}(a,b) & \sigma_{b}^{2} \end{array}\right) \cdot \left(\begin{array}{cc} b & 2\\ a & 2 \end{array}\right)\$ %% Cell type:markdown id: tags: \$\Rightarrow \left(\begin{array}{cc} \sigma_{A}^{2} & \text{cov}(A,C)\\ \text{cov}(A,C) & \sigma_{C}^{2} \end{array}\right) = \left(\begin{array}{cc} \sigma_{a}^{2} b^{2} + \sigma_{b}^{2} a^{2} + 2 a b \text{cov} (a,b) & 2 (b \sigma_{2}^2 + a \sigma_{b}^2 + \text{cov}(a,b)(a+b))\\ 2 (b \sigma_{2}^2 + a \sigma_{b}^2 + \text{cov}(a,b)(a+b)) & 4 \sigma_{a}^2 + 4 \sigma_{b}^2 + 8 \text{cov}(a,b) \end{array}\right)\$ %% Cell type:markdown id: tags: \$\Rightarrow \text{cov}(A,C) = 2 (b \sigma_{2}^2 + a \sigma_{b}^2 + \text{cov}(a,b)(a+b))\$ %% Cell type:markdown id: tags: \$\rho_{A,C} = \frac{\text{cov}(A,C)}{\sigma_{A}\sigma_{C}}\$ %% Cell type:markdown id: tags: With 100% correlation one ends up with \$\rho_{a,b} = 1\$, \$\sigma_{a}= \sigma_{b}\$. %% Cell type:markdown id: tags: \$\rho_{A,C} = \frac{\text{cov}(A,C)}{\sigma_{A}\sigma_{C}} = \frac{4(a+b)}{4(a+b)} = 1\$ %% Cell type:markdown id: tags: With 0% correlation one ends up with \$\rho_{a,b} = 0\$, \$\sigma_{a}= \sigma_{b}\$. %% Cell type:markdown id: tags: \$\rho_{A,C} = \frac{\text{cov}(A,C)}{\sigma_{A}\sigma_{C}} = \frac{2(a+b)}{\sqrt{8(a^{2}+b^{2})}} = 0.938\$ %% Cell type:markdown id: tags: ### Triangulation %% Cell type:markdown id: tags: 1) %% Cell type:markdown id: tags: \$l = 101.2\$ m, \$\sigma_{l} = 0.1\$ m %% Cell type:markdown id: tags: \$\alpha = 87.4^{\circ}\$, \$\sigma_{\alpha} = 0.1^{\circ}\$ %% Cell type:markdown id: tags: \$\beta = 87.8^{\circ}\$, \$\sigma_{\beta} = 0.1^{\circ}\$ %% Cell type:markdown id: tags: \$d = \frac{l\sin{(\alpha)} \sin{(\beta)}}{\sin{(\alpha +\beta)}}\$ %% Cell type:markdown id: tags: \$\sigma_{d}^{2} = \sigma_{\alpha}^{2} \left( - \frac{l\sin{(\alpha)} \sin{(\beta)} \cos{(\alpha +\beta)}}{\sin^{2}{(\alpha +\beta)}} + \frac{l\cos{(\alpha)} \sin{(\beta)}}{\sin{(\alpha +\beta)}} \right)^{2} + \sigma_{\beta}^{2} \left( - \frac{l\sin{(\alpha)} \sin{(\beta)} \cos{(\alpha +\beta)}}{\sin^{2}{(\alpha +\beta)}} + \frac{l\sin{(\alpha)} \cos{(\beta)}}{\sin{(\alpha +\beta)}} \right)^{2} + \sigma_{l}^{2} \left( \frac{\sin{(\alpha)} \sin{(\beta)}}{\sin{(\alpha +\beta)}} \right)^{2}\$ %% Cell type:markdown id: tags: \$\Rightarrow d =1207.36\$ m, \$\sigma_{d} = 35.63\$ m %% Cell type:markdown id: tags: 2) %% Cell type:markdown id: tags: Assuming \$\sigma_{l} = 0\$. %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{d}^{2} = \sigma_{\alpha}^{2} \left( - \frac{l\sin{(\alpha)} \sin{(\beta)} \cos{(\alpha +\beta)}}{\sin^{2}{(\alpha +\beta)}} + \frac{l\cos{(\alpha)} \sin{(\beta)}}{\sin{(\alpha +\beta)}} \right)^{2} + \sigma_{\beta}^{2} \left( - \frac{l\sin{(\alpha)} \sin{(\beta)} \cos{(\alpha +\beta)}}{\sin^{2}{(\alpha +\beta)}} + \frac{l\sin{(\alpha)} \cos{(\beta)}}{\sin{(\alpha +\beta)}} \right)^{2}\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{d} = 36.61\$ m %% Cell type:markdown id: tags: The error on the distance is dominated by the error on the angles. %% Cell type:markdown id: tags: 3) %% Cell type:markdown id: tags: \$\alpha = \beta\$, \$\sigma_{\alpha} = \sigma_{\beta} = \sigma\$ %% Cell type:markdown id: tags: \$\sigma_{d}^{2} = \sigma_{\alpha}^{2} \left( - \frac{l\sin{(\alpha)} \sin{(\beta)} \cos{(\alpha +\beta)}}{\sin^{2}{(\alpha +\beta)}} + \frac{l\cos{(\alpha)} \sin{(\beta)}}{\sin{(\alpha +\beta)}} \right)^{2} + \sigma_{\beta}^{2} \left( - \frac{l\sin{(\alpha)} \sin{(\beta)} \cos{(\alpha +\beta)}}{\sin^{2}{(\alpha +\beta)}} + \frac{l\sin{(\alpha)} \cos{(\beta)}}{\sin{(\alpha +\beta)}} \right)^{2}\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{d}^{2} = 2 \sigma^{2} l^{2} \left( \frac{-\sin{(\alpha)} \sin{(\alpha)} \cos{(2\alpha)}}{\sin^{2}{(2\alpha)}} + \frac{\cos{(\alpha)} \sin{(\alpha)}}{\sin{(2 \alpha)}} \right)^{2} = 2 \sigma^{2} l^{2} \left( \frac{-\sin^{2}{(\alpha)} \cos{(2\alpha)} + \cos{(\alpha)} \sin{(\alpha)}\sin{(2\alpha)}}{\sin^{2}{(2\alpha)}} \right)^{2}\$ %% Cell type:markdown id: tags: Use: \$\hspace{0.5cm}\$ \$\cos(2 \alpha) = \cos^{2}{(\alpha)} - \sin^{2}{(\alpha)}\$ \$\hspace{0.5cm}\$ and \$\hspace{0.5cm}\$ \$\sin(2 \alpha) = 2\sin{(\alpha)}\cos{(\alpha)}\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{d}^{2} = 2 \sigma^{2} l^{2} \left( \frac{-\sin^{2}{(\alpha)} \cos^{2}{(\alpha)} + \sin^{2}{(\alpha)} \sin^{2}{(\alpha)} + 2 \cos^{2}{(\alpha)} \sin^{2}{(\alpha)}}{\sin^{2}{(2\alpha)}} \right)^{2} = 2 \sigma^{2} l^{2} \left( \frac{ \sin^{2}{(\alpha)} \cdot [ - \cos^{2}{(\alpha)} + \sin^{2}{(\alpha)} + 2 \cos^{2}{(\alpha)} ]}{\sin^{2}{(2\alpha)}} \right)^{2}= 2 \sigma^{2} l^{2} \left( \frac{ \sin^{2}{(\alpha)} \cdot [ \cos^{2}{(\alpha)} + \sin^{2}{(\alpha)]} }{\sin^{2}{(2\alpha)}} \right)^{2}\$ %% Cell type:markdown id: tags: Use: \$\hspace{0.5cm}\$ \$\sin(2 \alpha) = 2\sin{(\alpha)}\cos{(\alpha)}\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{d}^{2} = 2 \sigma^{2} l^{2} \left( \frac{ \sin^{2}{(\alpha)}}{\sin^{2}{(2\alpha)}} \right)^{2} = 2 \sigma^{2} l^{2} \left( \frac{ \sin^{2}{(\alpha)}}{4 \sin^{2}{(\alpha)}\cos^{2}{(\alpha)}} \right)^{2} = \frac{ \sigma^{2} l^{2}}{8 \cos^{4}{(\alpha)}}\$ %% Cell type:markdown id: tags: \$\Rightarrow \sigma_{d} \approx \frac{ \sigma l}{\sqrt{8} } \frac{(l/2)^{2} + d}{(l/2)^{2}} \approx \frac{ \sigma l}{\sqrt{8} } \frac{4 d^{2}}{l^{2}} = \sqrt{2} \sigma \frac{d^{2}}{l}\$ %% Cell type:code id: tags: ``` python ```
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