Root$-$Finding: fzero $($The MATLAB Built$-$in Function$)$

A simple formula to estimate the upward velocity of a rocket $($neglecting the aerodynamic drag$)$ is:

$v=$uln$(\frac{m_0}{m_0-qt})-gt$

where $t=$ time, $v=$ upward velocity, $u=$ the velocity at which fuel is expelled relative to the rocket, $m0=$ the initial mass of the rocket at time $t=0,q=t$ fuel

consumption rate, and $g=$ the downward acceleration of gravity.

Develop a MATLAB function which, for given parameters $u,m0,q,$ and $g,$ and a specified velocity $v^{**},$ computes the time $t^{**}$ that the rocket reaches this velocity.

Hint:

$(1)$ Formulate this problem into a root$-$finding problem, and solve it using the MATLAB built$-$in function "fzero".

$(2)$ The initial bracketing interval $($or initial guess$)$ for "fzero" must be within the domain of the root$-$finding function. Note that the above function $v(t)$ has a bounded

domain $($due to $"$ln$").$

Function $8$

function tstar $=$ fzero$\_$rocket$\_$example$($u$,$ g$,$ q$,$ m$0,$ vstar$)$

$\%$ Inputs $($all scalars$)$

$\%,$ mo: initial mass of rocket at time $t=0$

$\%$ g: gravitational constant

$\%q$ : fuel consumption rate

$\%,u$ : velocity at which fuel expelled relative to rocket

$\%$ vstar: upward velocity value for which tstar is to be determined

$\%$ output

$\%$ tstar: the time that the rocket reaches the specified velocity, vstar $($scalar$)$

$$8$ Write your code here.

tstar $=0$;

end

Code to call your function $?$

$m0=160e3$;$\%kg$

$u=1800$;$\%\frac{m}{s}$

vstar $=750$;$,9\frac{m}{s}$

$q=2600$; $\%k\frac{g}{s}$

$g=9\mathrm{.}81$;$\%\frac{m}{s^{{?}^{{?}^{?}}}}2$

tstar $=$ fzero$\_$rocket$\_$example$($u$,$ g$,$ q$,$ m$0,$ vstar$)$

Assessment:

Test $1$: $m0=160e3,u=1800,$vstar$=750($Pretest$)$

Test Code:

clear; clc

$m0=160e3$;$\%kg$

$u=1800$;$\%\frac{m}{s}$

vstar $=750$; $\%\frac{m}{s}$

$q=2600$;$\%k\frac{g}{s}$

$g=9\mathrm{.}81$;$\%\frac{m}{s^{{?}^{{?}^{?}}}}2$

tstar $=$ fzero$\_$rocket$\_$example$($u$,$ g$,$ q$,$ m$0,$ vstar$)$

tstarCorrect $=$ reference.fzero$\_$rocket$\_$example $,$ vstar