Projectile on a Slope

\(\gamma=\theta +\phi\)

\(\ddot { x } =0\\ \dot { x } ={ v }_{ 0 }\cos { \gamma } \\ x={ (v }_{ 0 }\cos { \gamma } )t\\ x\overset { set }{ = } L\\ { v }_{ 0 }\cos { \gamma } t=s\cos { \theta } \\ \Rightarrow t=\frac { s\cos { \theta } }{ { v }_{ 0 }\cos { \gamma } } \\ \ddot { y } =g\\ \dot { y } =gt+{ v }_{ 0 }\sin { \gamma } \\ y=\frac { g }{ 2 } { t }^{ 2 }+({ v }_{ 0 }\sin { \gamma } )t\\ y\overset { set }{ = } H\\ \Rightarrow s\sin { \theta } =\frac { g }{ 2 } { t }^{ 2 }+{ (v }_{ 0 }\sin { \gamma } )t\)

\[\begin{eqnarray} \Rightarrow s\sin { \theta } & = & \frac { g }{ 2 } { (\frac { s\cos { \theta } }{ { v }_{ 0 }\cos { \gamma } } ) }^{ 2 }+({ v }_{ 0 }\sin { \gamma } )(\frac { s\cos { \theta } }{ { v }_{ 0 }\cos { \gamma } } ) \\ s\sin { \theta } & = & (\frac { g{ s }^{ 2 }\cos ^{ 2 }{ \theta } }{ 2{ { v }_{ 0 } }^{ 2 } } )\frac { \sin ^{ 2 }{ \gamma } +\cos ^{ 2 }{ \gamma } }{ \cos ^{ 2 }{ \gamma } } +(s\cos { \theta } )\tan { \gamma } \\ (\frac { g{ s }^{ 2 }\cos ^{ 2 }{ \theta } }{ 2{ { v }_{ 0 } }^{ 2 } } )\tan ^{ 2 }{ \gamma +(s\cos { \theta } )\tan { \gamma } +(\frac { g{ s }^{ 2 }\cos ^{ 2 }{ \theta } }{ 2{ { v }_{ 0 } }^{ 2 } } -s\sin { \theta ) } } & = & 0 \\ \tan { \gamma } & = & \frac { -(s\cos { \theta } )\pm \sqrt { { (s\cos { \theta } ) }^{ 2 }-4(\frac { g{ s }^{ 2 }\cos ^{ 2 }{ \theta } }{ 2{ { v }_{ 0 } }^{ 2 } } )(\frac { g{ s }^{ 2 }\cos ^{ 2 }{ \theta } }{ 2{ { v }_{ 0 } }^{ 2 } } -s\sin { \theta ) } } }{ 2(\frac { g{ s }^{ 2 }\cos ^{ 2 }{ \theta } }{ 2{ { v }_{ 0 } }^{ 2 } } ) } \\ \gamma \left( g,s,{ v }_{ 0 },\theta \right) & = & \boxed { \tan ^{ -1 }{ \frac { \pm \sqrt { { { { { { v }_{ 0 } }^{ 4 } }-g }^{ 2 }{ s }^{ 2 }\cos ^{ 2 }{ \theta } +{ { v }_{ 0 } }^{ 2 }2g{ s }\sin { \theta } } } -{ { v }_{ 0 } }^{ 2 } }{ g{ s }\cos { \theta } } } } \end{eqnarray}\]

Note by Gordon Chan
1 month ago

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  Easy Math Editor

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[example link](https://brilliant.org)example link
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# up as a code block.

print "hello world"
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Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
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2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

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