Program in Python please

The altitude of an object thrown upwards can be described by the following formula, which relates altitude

$($z$)$ to time $($t$)$:

z $=$ z$0+$ m

c

v$0+$ mg

c

$1$ e$($c$/$m$)$t

$$ mg

c t

In this equation, the variables are defined as follows: z$0$ represents the initial altitude from which the

object is launched, m stands for the mass of the object, c is the drag coefficient, and v$0$ denotes the initial

velocity with which the object is launched. It$$s important to note that positive velocity is assumed to be

in the upward direction.

$($a$)$ Given the following parameters:

g $=9\mathrm{.}81$ m$/$s$2$

z$0=10$ m

v$0=10$ m$/$s

m $=100$ kg

c $=5$ kg$/$s

Calculate the maximum altitude the object reaches, either through graphical $($include the figure$)$

or analytical methods $($show the analytical form$).[10$ pts$]$

$($b$)$ Implement a one$-$dimensional unconstrained optimization method to find the maximum height of

the object when $\backslash$epsi a You can choose from three methods: Golden$-$Section search, Parabolic

interpolation, or Newton$-$iteration. If you opt for Golden$-$Section search, you can refer to the

provided pseudo$-$code on page $367$ of the book. If you decide to implement Newton iteration, it is

recommended to use numerical differentiation for accuracy. Ensure that you select suitable initial

values for tlower and tupper based on your problem. $[50$ pts$]$

$($c$)$ If the kinetic energy with which the object can be thrown up is constant and is equal to ke $=$ mv$2$

$0$

$2=$

$2000$ kg $$ m$2$ s$2,$ what are the values of m and v$0$ for which the maximum altitude of the object

can be achieved, given the kinetic energy $($ke$),$ air resistance constant c $=8$ kg$/$s$,$ initial height

z$0=15$ m$,$ and acceleration due to gravity g $=9\mathrm{.}81$ m$/$s$2?$ Please use the random search method

described in Example $14\mathrm{.}1$ on page $371$ of the book. $[40$ pts$]$

$2$