After this lesson, why not add the effects of air? It is nearly impossible
to do with pencil and paper, but very easy on the computer. Assume that air presents a deceleration on the ball
opposite to $v$ given by $a_{air}=Cv^2$, where $C$ is some drag constant like $0.1$.

To use this, we need to know the direction in which the ball is flying at any instant. This is given by $\alpha=\tan^{-1}\frac{v_y}{v_x}$. If we know this flight angle, then the $x$ and $y$ deceleration components will be $a_x=a_{air}\cos(\alpha+\pi)$ and $a_y=a_{air}\sin(\alpha+\pi)$. Let's put this in, given that $\tan^{-1}\left(\frac{a}{b}\right)$ is

To use this, we need to know the direction in which the ball is flying at any instant. This is given by $\alpha=\tan^{-1}\frac{v_y}{v_x}$. If we know this flight angle, then the $x$ and $y$ deceleration components will be $a_x=a_{air}\cos(\alpha+\pi)$ and $a_y=a_{air}\sin(\alpha+\pi)$. Let's put this in, given that $\tan^{-1}\left(\frac{a}{b}\right)$ is

`math.atan2(a,b)`

.
Type your code here:

See your results here:

This code will not run. You need to fill in some launch angle $\theta$ in the

`theta=`

line and some launch speed
in the `v=`

line. You can try experimenting with the drag constant, making it larger or smaller. You might make
it $0$ and run it again at $0.01$ to compare the ranges of the two (i.e. it should go farther with no air drag).