I only need help with part $($b$).$ In part $($a$),$ I already found and plot the graph of y$($t$)$ versus x$($t$)$ by using the following code

Consider now a spherical object launched with a velocity $v$ forming an angle $$ with the horizontal ground. In the absence of air resistance, the trajectory followed by this projectile is known to be a parabola. This follows from writing Newton's law separately for the horizontal and vertical coordinates. The former scales linearly with time whereas the latter varies quadratically. Therefore, when time is eliminated, we are left with a quadratic equation that gives rise to a parabolic trajectory. Let's see how the trajectory changes when air resistance is no longer neglected. In the case of a resistive force that grows linearly with velocity $(c=0),$ we can still separate the motion between horizontal and vertical coordinates. Second Newton's law for both the horizontal and vertical coordinates become

$\frac{d{v}_x}{dt}=-\frac{b}{m}{v}_x$

$\frac{d{v}_y}{dt}=-g-\frac{b}{m}{v}_y$

The code written earlier can be applied to both directions separately, the difference being that gravity acts on the vertical direction $(y-$axis$)$ but not on the horizontal one $(x-$axis$).$ Once again, you will have results relating the coordinates $x$ and $y$ with the time $t.$

$($a$)$ Plot $y(t)$ versus $x(t),$ which will give you the trajectory followed by the object under the action

of air resistance. Superimpose this trajectory with the one which you would obtain in vacuum

to see how different the two cases are.

$($b$)$ Another well$-$known fact, often derived in introductory physics courses, is that the launching angle of $45$ leads to the maximum horizontal displacement in a projectile motion. This is the case in the absence of air resistance. The question we now pose is whether this is also the case when air resistance is not neglected. You can now use your code to determine what the optimum launching angle is$.$ How does that depend on the mass $m?$ Plot $-$optimum as a function of $m.$