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\begin{document}

\frame{\frametitle{\b Vectors and 2D Motion} \bigskip

{\bf Cartesian and polar coordinates (1).}

Length of a 2D vector; angle $\phi$:
$$ r = \sqrt{x^2+y^2}\,,~~~~\phi = \arctan(y/x)\,. $$
Components of a velocity:
$$ v_x = v\,\cos(\phi)\,,~~~~v_y = v\,\sin(\phi)\,.$$
{\bf Not that CAPA wants asin, acos, atan and gives
wrong when useing arc instead.} \bigskip

{\bf Adding vectors (2):}
$$ \vec{v}_3 = \vec{v} = \vec{v}_1+\vec{v}_2 $$
with $\vec{v}_1$ along the positive $x$-axis. 
Maximum and minimum magnitudes of $\vec{v}$:
$$ v_{\max} = v_1+v_2\,, v_{min}=v_1-v_2\,. $$
\vskip 20pt }


\frame{\frametitle{\b Vectors and 2D Motion} \bigskip

{\bf Inexperienced pilot (3).}

Speed over ground $v$; speed in still air $v_y$:
$$ v = \frac{r}{t}\,,~~~~v_y=\frac{y}{t}\,. $$

Let the speed in still air now $v$ (our previous $v_y$).
Direction angle $\theta$:
$$ v_x = v\,\sin(\theta)\ \Rightarrow\ \theta=arcsin(v_x/v)\,.$$

{\bf Ferry boat (4).} \bigskip

{\bf Acceleration, Velocity and Displacement Vector (5).}

Velocity from acceleration $\vec{a}$ and initial velocity $\vec{v}_0$ 
at time $t=0$:
$$ \vec{v} = \vec{v}_0 + \vec{a}\,t\,,~~v=\sqrt{v_x^2+v_y^2}\,. $$
Angle with the  convention: 0 to $\pi$ for $v_y>0$, 0 to $-\pi$ for 
$v_y<0$. So,
$$ \alpha = sign(v_y)\,\arccos(v_x/v)\,,~~~sign(v_y)=\pm 1\,. $$
\vskip 20pt }


\frame{\frametitle{\b Vectors and 2D Motion} \bigskip

Position vector at time $t$ with initial position $\vec{r}_0$:
$$ \vec{r} = \vec{r}_0+\vec{v}_0\,t+\frac{1}{2}\,\vec{a}\,t^2\,,
   ~~~r=\sqrt{r_x^2+r_y^2}\,. $$

Angle as before:
$$ \alpha = sign(r_y)\,\arccos(r_x/r)\,,~~~sign(r_y)=\pm 1\,. $$

\bigskip {\bf Velocity in the $xy$ plane (6).} \bigskip

When particle is moving in $x$ direction:
$$ v_y=\frac{dr_y}{d\,t}=0~~\Rightarrow~~t~~\Rightarrow~~
   v_x=\frac{dr_x}{d\,t}\,. $$
When particle is moving in $y$ direction:
$$ v_x=\frac{dr_x}{d\,t}=0~~\Rightarrow~~t~~\Rightarrow~~
   v_y=\frac{dr_y}{d\,t}\,. $$
\vskip 20pt }



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