A small rocket with an initial weight of $3000lb($including $2400lb$ of fuel$),$ initially at rest, is launched vertically upward. The rocket burns fuel at a constant rate of $80l\frac{b}{s},$ which provides a continuous thrust, $T,$ of $8000lb.$ The instantaneous weight of the rocket is given by $w(t)=3000-80tlb.$ The drag force, $D,$ experienced by the rocket is given by $D=0\mathrm{.}005g(\frac{dy}{dt}{)}^{2}lb,$ where $y$ is the distance in $ft,$ and $g=32\mathrm{.}2f\frac{t}{s^2}.$ The equation of motion for the rocket is given by:

$\frac{w}{g}\frac{d^{2}y}{d{t}^2}=T-w-D.$

Determine and plot the rocket's position, velocity, and acceleration $($three separate panels of one figure$)$ as a function of time from $t=0$ to $t=3s.$ To do this, you must reduce the second$-$order ODE to a system of two first$-$order ODEs.

$($a$)$ Solve the problem using your own $4$th$-$order Runge$-$Kutta algorithm $($that accepts systems of linear equations$)$ using a step size of $0\mathrm{.}05s.$ You should be able to do this by modifying the function we defined in class to specify the system of linear equations for the simple harmonic oscillator.

$($b$)$ Use one of MATLAB's built$-$in functions $($like ode$45)$ to solve the problem. You should put this code in the same script as part $($a$).$ You do not need to make a new plot since the results should be the same as what you found in part $($a$).$

$($c$)$ Make sure to label the plots.