User:Darkpulsaromega/Sandbox

Math Sandbox

Wikipedia:FORMULA

Calculus

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

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

Physics

$v={\sqrt {\frac {F}{p}}}$

$v=f\lambda \!$

Math Problem Solution

$F={\frac {-Gm_{E}m_{O}}{r_{E}^{2}}}\!$

$F=m_{O}g\!$

$Gm_{E}=r_{E}^{2}g\!$

$T_{M}=2\pi {\sqrt {\frac {r}{a_{M}}}}\!$

$4\pi ^{2}r=T_{M}^{2}a_{M}\!$

$4\pi ^{2}r=T_{M}^{2}{\frac {Gm_{E}}{r^{2}}}\!$

$4\pi ^{2}r^{3}=T_{M}^{2}R_{E}^{2}g\!$

$r=\left({\frac {T_{M}R_{E}}{2\pi }}\right)^{\frac {2}{3}}g^{\frac {1}{3}}\!$

$F=m_{M}a_{M}\!$

$F={\frac {-Gm_{E}m_{M}}{r^{2}}}\!$

$a_{M}={\frac {GM_{E}}{r^{2}}}\!$

${\vec {F}}_{s,max}=-\mu _{s}\cdot |{\vec {F}}_{N}|\cdot {\hat {F}}_{ap}\!$

$\mu _{s}={\frac {m_{T}}{m_{b}}}\!$

${\vec {F}}_{k}=-\mu _{k}\cdot |{\vec {F}}_{N}|\cdot {\hat {v}}\!$

$\mu _{k}={\frac {m_{T}}{m_{b}}}-{\frac {a}{g}}\cdot (1+{\frac {m_{T}}{m_{b}}})\!$

$\mu _{s}={\frac {1}{\sqrt {(L/h_{max})^{2}-1}}}\!$

$\mu _{k}={\frac {1}{\sqrt {(L/h_{max})^{2}-1}}}\cdot (1-{\frac {a}{g}}{\frac {L}{h}})\!$

$\mu$

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

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

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

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

${\frac {1}{\sin {x}}}=\csc {x}$

$V=lwh\!$

$\Delta V={\sqrt {(wh\Delta l)^{2}+(lh\Delta w)^{2}+(lw\Delta h)^{2}}}\!$

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

$V=\pi r^{2}h\!$

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

$f=f(x_{1},x_{2},...,x_{n})\!$

$\Delta f={\sqrt {\sum _{i=1}^{n}\left({\frac {\partial f}{\partial x_{i}}}\Delta x_{i}\right)^{2}}}\!$

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

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

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

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

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

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

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

${\vec {V}}_{x}=\left|{\vec {V}}\right|\cos({\vec {V}}_{angle})\!$

${\vec {V}}_{y}=\left|{\vec {V}}\right|\sin({\vec {V}}_{angle})\!$

${\sqrt {{\vec {V}}_{x}^{2}+{\vec {V}}_{y}^{2}}}\!$ @ $\arctan {\left({\frac {{\vec {V}}_{y}}{{\vec {V}}_{x}}}\right)}\!$

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

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

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

$\beta \gamma \theta \Phi$

Conservation of Momentum

$m_{A}v_{A0}+m_{B}v_{B0}=m_{A}v_{AF}+m_{B}v_{BF}\!$

Conservation of Kinetic Energy

${\frac {1}{2}}m_{A}v_{A0}^{2}+{\frac {1}{2}}m_{B}v_{B0}^{2}={\frac {1}{2}}m_{A}v_{AF}^{2}+{\frac {1}{2}}m_{B}v_{BF}^{2}\!$

Finding the Conservation of Relative Speed

Rearranging the Conservation of Momentum

$m_{A}v_{A0}-m_{A}v_{AF}=m_{B}v_{BF}-m_{B}v_{B0}\!$

$m_{A}(v_{A0}-v_{AF})=m_{B}(v_{BF}-v_{B0})\!$

Rearranging the Conservation of Kinetic Energy

${\frac {1}{2}}m_{A}v_{A0}^{2}-{\frac {1}{2}}m_{A}v_{AF}^{2}={\frac {1}{2}}m_{B}v_{BF}^{2}-{\frac {1}{2}}m_{B}v_{B0}^{2}\!$

$m_{A}(v_{A0}^{2}-v_{AF}^{2})=m_{B}(v_{BF}^{2}-v_{B0}^{2})\!$

Because:

$(v_{A0}^{2}-v_{AF}^{2})=(v_{A0}+v_{AF})(v_{A0}-v_{AF})\!$

And:

$(v_{BF}^{2}-v_{B0}^{2})=(v_{BF}+v_{B0})(v_{BF}-v_{B0})\!$

Therefore:

$m_{A}(v_{A0}+v_{AF})(v_{A0}-v_{AF})=m_{B}(v_{BF}+v_{B0})(v_{BF}-v_{B0})\!$

Dividing the two equations

${\frac {m_{A}(v_{A0}+v_{AF})(v_{A0}-v_{AF})}{m_{A}(v_{A0}-v_{AF})}}={\frac {m_{B}(v_{BF}+v_{B0})(v_{BF}-v_{B0})}{m_{B}(v_{BF}-v_{B0})}}\!$

$v_{A0}+v_{AF}=v_{BF}+v_{B0}\!$

$v_{A0}-v_{B0}=v_{BF}-v_{AF}\!$