Problem $3$: A model rocket $($with mass m $=10$ kg$)$ initially at rest on the ground $($Zo

$=0)$ is launched upward. The motion of the rocket is described by the following differential

equations:

dw

dt $=$g $+$ T h

m $,(2)$

dz

dt $=$ w$,$

where t is time, z is altitude, w is velocity, and T h is the upward thrust due to the propulsion

of the rocket and g is gravity. In this exercise, the upward thrust Th varies with time as

follows:

T h$($t$)=670$N f or $0<=$ t $<2($s$),$

T h$($t$)=1366\mathrm{.}5$N f or $2<=$ t $<4($s$),$

T h$($t$)=0$N f or t $>=4($s$),$

and the gravity g varies with altitude z as follows:

g$($z$)=9\mathrm{.}81$

$1$

z

$10000$

$3$

$($m$/$s$2)$ f or $0<=$ z $<10,000($m$),$

g $=0($m$/$s$2)$ f or z $>=10,000($m$),$

Using Euler$-$Cromer method, the differential equations can be approximated by

Wn$+1=$ Wn $+$

$$gn $+$ T hn

m

$$t$,$

Zn$+1=$ Zn $+$ Wn$+1$t$,$

where $$t is the time step and capital letters are used to denote approximated quantities.

In this exercise, you will create a file rocket.m to include $3$ functions. The primary

function should be named rocket and two subfunctions should be named gravity and

thrust.

The rocket function solves the governing equations for the projectile motion of the rocket.

It should have the following header: function $[$T$,$ Z$,$ W$]=$ rocket$($ Tf$,$ dt$)$ where the

input Tf is the duration of the flight of the rocket and dt is the time step. The output

vectors T$,$ Z and W are the time, altitude and velocity of the rocket, respectively.

The thrust subfunction computes the thrust at a given time during the flight. It should

have the following header: function $[$Th$]=$ thrust$($t$)$ where the input t is the time and

the output Th is the upward thrust given above. Both input and output are single numbers.

The gravity subfunction computes the value of gravity at a given time during the flight.

It should have the following header: function $[$g$]=$ gravity$($z$)$ where the input z is the

altitude of the rocket and the output g is the altitude$-$dependent gravity given above. Both

input and output are single numbers.

Give all functions a description.

$($a$)$ Set p$3$a $=$ evalc$(\text{'}$help rocket'$).$

$($b$)$ Set p$3$b $=$ evalc$(\text{'}$help gravity'$).$

$($c$)$ Set p$3$c $=$ evalc$(\text{'}$help thrust'$).$

$($d$)$ Set p$3$d $=$ evalc$(\text{'}$help rocket$>$gravity'$).$

$($e$)$ Set p$3$e $=$ evalc$(\text{'}$help rocket$>$thrust'$).$

$($f$,$g$)$ Compute the altitude and velocity of the rocket at $120$ s using $$t $=0\mathrm{.}1$s$.$ Put the

answers in p$3$f and p$3$g$,$ respectively. The answers should be single numbers, not vectors.

$($h$)$ Create figure $2.$ Use function subplot to include $2$ panels with one on top of the

other. The top panel shows how the altitude of the rocket changes with time during the

$120-$second flight. The bottom panel shows velocity versus time. Set p$3$h $=$ 'See figure $2\text{'}.$