%&latex
%\documentclass[a4paper]{article}

\section{Formulario}
\subsection{.Cinematica}
\textbf{Moto uniforme}
\eal{&\v(t)=\v\;\;\forall t\in I \= \a(t)=0\;\;\forall t\in I\\
&\v=\v(t)\\
&\r(t)=\r(t_0)+(t-t_0)\v
}
\textbf{Moto accelerato uniforme}
\eal{&\a(t)=\a\;\;\forall t\in I \= \da(t)=0\;\;\forall t\in I.\\
&\a=\a(t)\\
&\v(t)= \v(t_0)+(t-t_0)\a\\
&\r(t)=\r(t_0)+\v(t_0)(t-t_0)+\frac{(t-t_0)^2}{2}\a
}
\textbf{Moto dei proiettili}
\eal{&\a(t)=(0,-g)\\
&\v(t_0)=(|\v(t_0)|\cos\t,|\v(t_0)|\sin
\t)\\
&\sist{\v_1(t) = |\v(t_0)|\cos\t \\
\v_2(t)=|\v(t_0)|\sin \t-(t-t_0)g
}\\
&\sist{
\r_1(t)=\r_1(t_0) + |\v(t_0)|\cos\t(t-t_0)\\
\r_2(t)=\r_2(t_0) + |\v(t_0)|\sin\t(t-t_0)-\frac{(t-t_0)^2}{2}g\\
}\\
&\sist{t(x)=  \frac{x-x_0}{v_0\cos\t} + t_0\\
y=y_0 + (x-x_0)\tan \t
- \frac{(x-x_0)^2}{v_0^2\cos^2\t}\frac{g}{2}
	}\\
&
git(\r)=\frac{v_0^2}{g}\sin 2\t\qq\T{[per $\r(t_0)=(0,0)$]}\\
&y_{\T{max}}=\frac{v_0^2\sin^2\t}{2g}
}
\textbf{Moto circolare}
\def\motocircvel{\v(t)=R\w(t)(-\sin\t(t), \cos\t(t))}
\def\motocircvelmod{|\v(t)|= R\w(t)}
\eal{&\r(t)=(R\cos \t(t) , R\sin \t(t))\\
&\w(t) = \t'(t)\\
&\ag(t) = \w'(t)\\
&\motocircvel\\
&\motocircvelmod\\
&\a(t)=
R\Big(-\ag(t)\sin\t(t)-\w^2(t)\cos\t(t),\;\;
\ag(t)\cos\t(t)-\w^2(t)\sin\t(t)\Big)
}
\textbf{Moto circolare uniforme}
\eal{
&\w(t)=\w\;\;\forall t\in I \= \ag(t)=0\;\;\forall t\in I\\
&\t(t)=\t(t_0)+(t-t_0)\w\\
&\a(t)=-R\w^2\.(\cos\t(t),\;\;\sin\t(t))\\
&|\a(t)|=R\w^2\\
&\v(t)=\w R\Big(-\sin\pton{\t(t_0)+(t-t_0)\w }, \;\;\cos\pton{\t(t_0)+(t-t_0)\w}\Big)\\
&\r(t)=\Big(R\cos(\t(t_0)+(t-t_0)\w) , R\sin (\t(t_0)+(t-t_0)\w)\Big)
}
\textbf{Moto armonico}
	\eal{
	&r(t)=R\cos(\t(t_0)+(t-t_0)\w)\\
	&v(t)=-\w R\sin\pton{\t(t_0)+(t-t_0)\w }\\
	&a(t)=-R\w^2\cos\pton{ \t(t_0)+(t-t_0)\w }\\
	}
\textbf{Moto relativo}
\eal{&\vR(t)=D\pqua{\vvec{OO_1}(t)}=\vR\;\;\forall t\\
	&\r(t)=\vvec{OO_1}(t)+\r_1(t)\\
	&\v_1(t)=\v(t)-\vR
}
\subsection{.Dinamica}
\eal{&\qmoto(t)=m(t)\v(t)\\
&\vec{F}(t){=}m\a(t)\;\;\T{ [con massa costante]}
}
\textbf{Blocco sul piano non inclinato}
\eal{&\R=\F+\p+\N+\Fatt\\
&\T{Caso statico:}\\
&\qq	\R=0\= R_y=R_x=0\= \sist{-mg+N+F\sin\t=0\\
	F\cos\t-\mu_sN=0
	}\=F=\frac{mg\mu_s}{\cos \t+\mu_s\sin \t}\\
	&\qq F\le \frac{mg\mu_s}{\cos \t+\mu_s\sin \t} \=\T{ il corpo non si
	muove}\\
	&\T{Caso dinamico: }\\
	&\qq 
	\R=0\= R_y=0,\;R_x=ma\= \sist{-mg+N+F\sin\t=0\\F\cos\t-\mu_dN=ma }
}
\textbf{Blocco sul piano inclinato}
\case{ Caso dinamico senza attrito }
	\eal{&\R=m\a\=\N+\p=m\a\=
	\sist{N=P\cos \t\\
		a=g\sin \t}
	}
	\case{Caso statico con attrito}
		\eal{&\R=0\= \tg \t = \mu_s\\
		&\tg \t \le \mu_s \= \T{ il corpo rimane fermo}
		}
	\case{Caso dinamico con attrito}
		\eal{&\R=m\a\=\sist{a=g(\sin\t-\mu_d\cos \t}\\
		&\tg \t \le \mu_d \= \T{ il corpo si ferma}
		}