Matlab

Provide the complete code explianing how you implemented it for the following problem please!

Rocket Trajectory Problem

A model describing the vertical trajectory of a rocket is given by:

$\frac{dh(t)}{dt}=v(t)$

$(m_s+m_p(t))\frac{dv(t)}{dt}=-(m_s+m_p(t))g+u_p(t)v_e-\frac{1}{2}v(t)|v(t)|A{C}_D$

$\frac{d{m}_p(t)}{dt}=-u_p(t)$

Where:

$h(t)$ is the height reached by the rocket in meters.

$v(t)$ is the velocity of the rocket in meters per second.

$m_p(t)$ is the mass of the fuel in kilograms.

$u_p(t)$ in $k\frac{g}{s}$ is the rate at which the fuel is consumed $($this is considered as the input of the system$).$

$m_s=50kg$ is the weight of the "shell. of the rocket.

$g=9,81\frac{m}{s^2}$ is the acceleration due to gravity.

$=1,091k\frac{g}{m^3}$ is the average air density.

$A=r^2$ is the maximum cross$-$sectional area of the rocket, where $r=0,5m.$

$v_e=325\frac{m}{s}$ is the exhaust velocity of the fuel.

$C_D=0,15$ is the drag coefficient of the rocket.

$m_{p0}=100kg$ is the initial mass of the fuel at time $t=0.$

The fuel consumption is $u_p(t)=20k\frac{g}{s}$ for the first $5$ seconds. After the $5$ s the fuel consumption $u_p(t)=0k\frac{g}{s}.$

Tasks

$($a$)$ Plot the height reached by the rocket until the moment it hits the ground. To do so$,$ implement the $4$th order

Runge$-$Kutta algorithm to solve the system of differential equations.

$($b$)$ Do the same as part $($a$),$ but instead of implementing the algorithm directly, use the ode $45$ function in MATLAB.

$($c$)$ Plot the responses obtained from the previous two parts on the same coordinate axis to compare the results.

Use the legend function to identify each height.