A MATLAB function M$-$file for Euler$$s method $($as described in class and on pages $17-19$ of

the textbook$)$ for solving the differential equation

dv

dt $=$ g $$ c

mv

$($this is $(1\mathrm{.}9)$ on page $14)$ is given below. As discussed in class, this numerical method

is obtained by approximating dv

dt at time ti by v$($ti$+1)$v$($ti$)$

ti$+1$ti $,$ which results in the computed

approximation

v$($tt$+1)=$ v$($ti$)+$

$[$

g $$ c

mv$($ti$)$

$]$

$($ti$+1$ ti$).$

$1$

To create a MATLAB function M$-$file, either type edit in the Command Window or

select HOME $->$ New $->$ Script or select HOME $->$ New $->$ Function $($the latter gives

you a template for creating a function$).$

Each of these options will open a new window $($an Editor window$)$ in which to type in the

MATLAB statements for Euler$$s method. Enter the following, $($Note $-$ don$$t copy and paste

as the quotes might not work$).$ Statements starting with $\%$ are comments, documenting the

MATLAB code.

function Euler$($m$,$c$,$g$,$t$0,$v$0,$tn$,$n$)$

$\%$ print headings and initial conditions

fprintf$($values of t approximations v$($t$)$

$)$

fprintf$(\%8\mathrm{.}3$f$,$t$0),$fprintf$(\%19\mathrm{.}4$f

$,$v$0)$

$\%$ compute step size h

h$=($tn$-$t$0)/$n;

$\%$ set t$,$v to the initial values

t$=$t$0$;

v$=$v$0$;

$\%$ compute v$($t$)$ over n time steps using Euler$$s method

for i$=1$:n

v$=$v$+($g$-$c$/$m$*$v$)*$h;

t$=$t$+$h;

fprintf$(\%8\mathrm{.}3$f$,$t$),$fprintf$(\%19\mathrm{.}4$f

$,$v$)$

end

To save your M$-$file, select

EDITOR $->$ Save $->$ Save As$...$

At the top of this window, it should say

File name: Euler.m

Save as type: MATLAB files $(*.$m$)$

Select

Save

to save your file, and close the Editor window.

In order to use the above function, you must specify values for the $7$ local parameters in

the function Euler:

m is the mass of the falling object

c is the drag coefficient

g is the gravity constant

t$0$ is the initial time, v$0$ is the initial velocity

tn is the final time at which the velocity is to be computed

n is the number of time steps into which $[$t$0,$ tn$]$ is divided

Thus, in the function Euler, the step size h $=($tn $$ t$0)/$n is computed, and Euler$$s method

is used to compute an approximation to the solution v$($t$)$ of the differential equation at the

$2$

n points $($values of time$)$

t$0+$ h$,$ t$0+2$h$,$ t$0+3$h$,$ t$0+4$h$,...,$ t$0+$ nh $=$ tn$.$

For example, in order to use Euler to solve the problem given in Example $1\mathrm{.}2$ on page $17$

and to save your results in a file for printing, you could enter the following $($in the MATLAB

Command Window$)$:

diary filename

Euler $(68\mathrm{.}1,12\mathrm{.}5,9\mathrm{.}81,0,0,12,6)$

$\{$the desired results should appear here$\}$

diary off

$($a$)$ Create a working copy of the Euler function using MATLAB or Python MATLAB.

Upload a copy of your function to Crowdmark.

$($b$)$ Use Euler to solve the differential equation using m $=75\mathrm{.}3$ kg$,$ c $=13\mathrm{.}1$ kg$/$s and

initial conditions v$(0)=0$ on the time interval $[0,16]$ using $40$ time steps and for a

free$-$falling body on Venus where the gravitational constant is g $=8\mathrm{.}87$ m$/$s$2.$

$($c$)$ Using the same parameters from Q$1($b$)$ calculate the corresponding velocities using the

analytic solution. That is$,$ create a function for the formula,

v$($t$)=$ gm

c $(1$ e$$ ct

m $)(1)$

and run it for t $=[$t$0$ : h : tn$].$

$($d$)$ What is the terminal velocity of this free$-$falling body of mass $75\mathrm{.}3$ kg on Venus? $($You

may approximate this roughly with experimental results from your function or solve it

analytically.$)$