User:darkpulsaromega/Sandbox

:$f\text{'}(c)\; =\; \backslash lim\_\{h\; \backslash to\; 0\}\backslash frac\{f(c\; +\; h)\; -\; f(c)\}\{h\}$

:$\backslash int\_\{a\}^\{b\}f(x)dx\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\backslash sum\_\{i=1\}^n\; f(x\_i^*)\backslash Delta\; x$

:$v\; =\; \backslash sqrt\{\backslash frac\{F\}\{p\}\}$

:$v\; =\; f\backslash lambda\backslash !$

:$F\; =\; \backslash frac\{\; -G\; m\_E\; m\_O\; \}\{\; r\_E^2\; \}\; \backslash !$

:$F\; =\; m\_O\; g\; \backslash !$

:$G\; m\_E\; =\; r\_E^2\; g\; \backslash !$

:$T\_M\; =\; 2\; \backslash pi\; \backslash sqrt\{\; \backslash frac\{\; r\; \}\{\; a\_M\; \}\; \}\; \backslash !$

:$4\; \backslash pi^2\; r\; =\; T\_M^2\; a\_M\; \backslash !$

:$4\; \backslash pi^2\; r\; =\; T\_M^2\; \backslash frac\{\; G\; m\_E\; \}\{\; r^2\; \}\; \backslash !$

:$4\; \backslash pi^2\; r^3\; =\; T\_M^2\; R\_E^2\; g\; \backslash !$

:$r\; =\; \backslash left\; (\; \backslash frac\{\; T\_M\; R\_E\; \}\{\; 2\; \backslash pi\; \}\; \backslash right\; )^\{\; \backslash frac\{\; 2\; \}\{\; 3\; \}\; \}\; g^\{\; \backslash frac\{\; 1\; \}\{\; 3\; \}\; \}\; \backslash !$

:$F\; =\; m\_M\; a\_M\; \backslash !$

:$F\; =\; \backslash frac\{\; -G\; m\_E\; m\_M\; \}\{\; r^2\; \}\; \backslash !$

:$a\_M\; =\; \backslash frac\{\; G\; M\_E\; \}\{\; r^2\; \}\; \backslash !$

:$\backslash vec\; F\_\{s,max\}\; =\; -\backslash mu\_s\; \backslash cdot\; |\backslash vec\; F\_N|\; \backslash cdot\; \backslash hat\; F\_\{ap\}\; \backslash !$

:$\backslash mu\_s\; =\; \backslash frac\{\; m\_T\; \}\{\; m\_b\; \}\; \backslash !$

:$\backslash vec\; F\_k\; =\; -\backslash mu\_k\; \backslash cdot\; |\backslash vec\; F\_N|\; \backslash cdot\; \backslash hat\; v\; \backslash !$

:$\backslash mu\_k\; =\; \backslash frac\{\; m\_T\; \}\{\; m\_b\; \}\; -\; \backslash frac\{\; a\; \}\{\; g\; \}\; \backslash cdot\; (1\; +\; \backslash frac\{\; m\_T\; \}\{\; m\_b\; \})\; \backslash !$

:$\backslash mu\_s\; =\; \backslash frac\{\; 1\; \}\{\; \backslash sqrt\{\; (L\; /\; h\_\{max\})^2\; -\; 1\; \}\; \}\; \backslash !$

:$\backslash mu\_k\; =\; \backslash frac\{\; 1\; \}\{\; \backslash sqrt\{\; (L\; /\; h\_\{max\})^2\; -\; 1\; \}\; \}\; \backslash cdot\; (1\; -\; \backslash frac\{\; a\; \}\{\; g\; \}\; \backslash frac\{\; L\; \}\{\; h\; \})\; \backslash !$

:$\backslash mu$

:$\backslash frac\{\backslash tan\{x\}\}\{\backslash sec\{x\}-cos\{x\}\}$

:$\backslash frac\{\backslash sin\{x\}\}\{\backslash cos\{x\}(\backslash sec\{x\}\; -\; \backslash cos\{x\})\}$

:$\backslash frac\{\backslash sin\{x\}\}\{1\; -\; \backslash cos^2\{x\}\}$

:$\backslash frac\{\backslash sin\{x\}\}\{\backslash sin^2\{x\}\}$

:$\backslash frac\{1\}\{\backslash sin\{x\}\}\; =\; \backslash csc\{x\}$

:$V\; =\; l\; w\; h\backslash !$

:$\backslash Delta\; V\; =\; \backslash sqrt\{\; (w\; h\; \backslash Delta\; l)^2\; +\; (l\; h\; \backslash Delta\; w)^2\; +\; (l\; w\; \backslash Delta\; h)^2\; \}\; \backslash !$

:$V\; =\; \backslash pi\; (\backslash frac\{\; d\; \}\{\; 2\; \})^2\; h\; \backslash !$

:$V\; =\; \backslash pi\; r^2\; h\backslash !$

:$\backslash Delta\; V\; =\; \backslash sqrt\{\; (\backslash frac\{\; \backslash pi\; h\; \}\{\; 2\; \}\; d\; \backslash Delta\; d)^2\; +\; (\backslash frac\{\; d\; \}\{\; 2\; \}^2\; \backslash pi\; \backslash Delta\; h)^2\; \}\; \backslash !$

:$f\; =\; f(x\_1,\; x\_2,\; ...,\; x\_n)\; \backslash !$

:$\backslash Delta\; f\; =\; \backslash sqrt\{\; \backslash sum\_\{i=1\}^n\; \backslash left\; (\backslash frac\{\; \backslash partial\; f\; \}\{\; \backslash partial\; x\_i\; \}\; \backslash Delta\; x\_i\; \backslash right\; )^2\; \}\; \backslash !$

:$t\; =\; \backslash sqrt\{\; \backslash frac\{\; 2d\; \}\{\; g\; \}\; \}\; \backslash !$

:$P\; =\; 0.67\; \backslash frac\{\; \backslash sqrt\{\; \backslash frac\{\; 1\; \}\{\; n-1\; \}\; \backslash sum\_\{i=1\}^n\; (x\_i\; -\; \backslash overline\{\; x\; \})^2\; \}\; \}\{\; \backslash sqrt\{\; n\; \}\; \}\; \backslash !$

:$v(t)\; =\; \backslash frac\{\; x(t)\; -\; x(t\; -\; \backslash Delta\; t)\; \}\{\; \backslash Delta\; t\; \}\; \backslash !$

:$rv(t)\; =\; \backslash frac\{\; x(t\; +\; \backslash Delta\; t)\; -\; x(t)\; \}\{\; \backslash Delta\; t\; \}\; \backslash !$

:$a(t)\; =\; \backslash frac\{\; rv(t)\; -\; v(t)\; \}\{\; \backslash Delta\; t\; \}\; \backslash !$

:$a(t)\; =\; \backslash frac\{\; \backslash frac\{\; x(t\; +\; \backslash Delta\; t)\; -\; x(t)\; \}\{\; \backslash Delta\; t\; \}\; -\; \backslash frac\{\; x(t)\; -\; x(t\; -\; \backslash Delta\; t)\; \}\{\; \backslash Delta\; t\; \}\; \}\{\; \backslash Delta\; t\; \}\; \backslash !$

:$a(t)\; =\; \backslash frac\{\; x(t\; -\; \backslash Delta\; t)\; -\; 2x(t)\; +\; x(t\; +\; \backslash Delta\; t)\; \}\{\; \backslash Delta\; t^2\; \}\; \backslash !$

:$\backslash vec\; V\_x\; =\; \backslash left\; |\; \backslash vec\; V\; \backslash right\; |\; \backslash cos(\backslash vec\; V\_\{angle\})\; \backslash !$

:$\backslash vec\; V\_y\; =\; \backslash left\; |\; \backslash vec\; V\; \backslash right\; |\; \backslash sin(\backslash vec\; V\_\{angle\})\; \backslash !$

:$\backslash sqrt\{\; \backslash vec\; V\_x^2\; +\; \backslash vec\; V\_y^2\; \}\; \backslash !$ @ $\backslash arctan\{\; \backslash left(\; \backslash frac\{\; \backslash vec\; V\_y\; \}\{\; \backslash vec\; V\_x\; \}\; \backslash right)\; \}\; \backslash !$

:$\backslash left|\; \backslash vec\; C\; \backslash right|\; =\; \backslash sqrt\{\; \backslash left|\; \backslash vec\; A\; \backslash right|^2\; +\; \backslash left|\; \backslash vec\; B\; \backslash right|^2\; -\; 2\; \backslash left|\; \backslash vec\; A\; \backslash right|\; \backslash left|\; \backslash vec\; B\; \backslash right|\; \backslash cos(\; \backslash vec\; B\_\{angle\}\; )\; \}\; \backslash !$

:$\backslash vec\; C\_\{angle\}\; =\; \backslash arccos\; \backslash left(\; \backslash frac\{\; \backslash left|\; \backslash vec\; C\; \backslash right|^2\; +\; \backslash left|\; \backslash vec\; A\; \backslash right|^2\; -\; \backslash left|\; \backslash vec\; B\; \backslash right|^2\; \}\{\; 2\; \backslash left|\; \backslash vec\; C\; \backslash right|\; \backslash left|\; \backslash vec\; A\; \backslash right|\; \}\; \backslash right)\; +\; \backslash vec\; A\_\{angle\}\; \backslash !$

:$\backslash vec\; B\_\{angle\}\; =\; 180\; -\; \backslash arccos\; \backslash left(\; \backslash frac\{\; \backslash left|\; \backslash vec\; B\; \backslash right|^2\; +\; \backslash left|\; \backslash vec\; A\; \backslash right|^2\; -\; \backslash left|\; \backslash vec\; C\; \backslash right|^2\; \}\{\; 2\; \backslash left|\; \backslash vec\; B\; \backslash right|\; \backslash left|\; \backslash vec\; A\; \backslash right|\; \}\; \backslash right)\; +\; \backslash vec\; A\_\{angle\}\; \backslash !$

:$\backslash beta\; \backslash gamma\; \backslash theta\; \backslash Phi$

:$m\_A\; v\_\{A0\}\; +\; m\_B\; v\_\{B0\}\; =\; m\_A\; v\_\{AF\}\; +\; m\_B\; v\_\{BF\}\; \backslash !$

:$\backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_A\; v\_\{A0\}^2\; +\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_B\; v\_\{B0\}^2\; =\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_A\; v\_\{AF\}^2\; +\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_B\; v\_\{BF\}^2\; \backslash !$

:$m\_A\; v\_\{A0\}\; -\; m\_A\; v\_\{AF\}\; =\; m\_B\; v\_\{BF\}\; -\; m\_B\; v\_\{B0\}\; \backslash !$

:$m\_A\; (v\_\{A0\}\; -\; v\_\{AF\})\; =\; m\_B\; (v\_\{BF\}\; -\; v\_\{B0\})\; \backslash !$

:$\backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_A\; v\_\{A0\}^2\; -\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_A\; v\_\{AF\}^2\; =\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_B\; v\_\{BF\}^2\; -\; \backslash frac\{\; 1\; \}\{\; 2\; \}\; m\_B\; v\_\{B0\}^2\; \backslash !$

:$m\_A\; (v\_\{A0\}^2\; -\; v\_\{AF\}^2)\; =\; m\_B\; (v\_\{BF\}^2\; -\; v\_\{B0\}^2)\; \backslash !$

Because:

:$(v\_\{A0\}^2\; -\; v\_\{AF\}^2)\; =\; (v\_\{A0\}\; +\; v\_\{AF\})\; (v\_\{A0\}\; -\; v\_\{AF\})\; \backslash !$

And:

:$(v\_\{BF\}^2\; -\; v\_\{B0\}^2)\; =\; (v\_\{BF\}\; +\; v\_\{B0\})\; (v\_\{BF\}\; -\; v\_\{B0\})\; \backslash !$

Therefore:

:$m\_A\; (v\_\{A0\}\; +\; v\_\{AF\})\; (v\_\{A0\}\; -\; v\_\{AF\})\; =\; m\_B\; (v\_\{BF\}\; +\; v\_\{B0\})\; (v\_\{BF\}\; -\; v\_\{B0\})\; \backslash !$

:$\backslash frac\{\; m\_A\; (v\_\{A0\}\; +\; v\_\{AF\})\; (v\_\{A0\}\; -\; v\_\{AF\})\; \}\{\; m\_A\; (v\_\{A0\}\; -\; v\_\{AF\})\; \}\; =\; \backslash frac\{\; m\_B\; (v\_\{BF\}\; +\; v\_\{B0\})\; (v\_\{BF\}\; -\; v\_\{B0\})\; \}\{$

m_B (v_{BF} - v_{B0}) } \!

:$v\_\{A0\}\; +\; v\_\{AF\}\; =\; v\_\{BF\}\; +\; v\_\{B0\}\; \backslash !$

:$v\_\{A0\}\; -\; v\_\{B0\}\; =\; v\_\{BF\}\; -\; v\_\{AF\}\; \backslash !$