Assignment: Finding Zeros Graphically Write a program to estimate the proper launch

angle for a projectile launched with velocity magnitude $v_i\frac{m}{s},$ at a target $h$

m below, and $Rm$ away from the launch point by graphically determining the

zeros of Equation $3.$ You will provide the initial values when prompted.

As always,

from pylab import $*$

Define a function which will pass an array of launch angles and return an

array of values for the expression on the left side of the equal sign of Equation $3,$

each element of which corresponds to an element of the launch angle array.

Make the program input values $($all float types$)$ for $v_i,h,$ and $R.$

$[$Note: The input instruction in python$3$ returns only strings, whereas in

python$2$ it returned a "best guess" as to the type. In python$3,$ then, the in$-$

struction needs its return type changed $($the technical term is "cast"$).$ So$,$ for

example, if you want the input stored as an integer:

aninteger $=$ int$($input$("$type a number $"))$

or$,$ as a float $($real number$)$:

afloat $=$ float$($input$("$type another number $"))$

Create an array of $100$ launch angle values between $0$ and $3\frac{}{8}($the curve

blows up at $\frac{}{2}-$try it$)$ using the linspace function.

Plot each returned function value $($ordinate$)$ against launch angle $($abscissa$,$

the horizontal or $x-$axis$)$:

plot$($theta$,$ function$($theta$)).$

$[$labels$]$

show$()$

Read from the plot $($using the zoom and pan utility in the plot window, if

necessary$)$ the angle$($s$)$ where the curve crosses zero on the ordinate $(y-$axis$).$

$($Grid lines, which may help the determination, can be added to the plot with

the instruction grid$($True$)$ before issuing the show$()$ instruction.$)$ Report this

value $($which will be in radians$)$ to two decimal places for:

$h=1\mathrm{.}2m,R=2\mathrm{.}5m,$ and $v_i=4\mathrm{.}8\frac{m}{s}.$

Then play around and see which initial values near these never give zeros

and therefore can never reach the target. Report the values you come up with.

Consider an object released from rest $($i$.$e$,v(0)=0\frac{m}{s})$ high enough from the

surface of a planet to fall for $5$ seconds $($that is$,0t5$; recall from the first

assignment how to create an ordered list of numbers between two values; in this

case, make the list $100$ numbers long between $0$ and $5$ seconds$).$ Consider the

highest point $y(0)=0m.$ Note also that python variables do not distinguish a

lists from a single value $($which afterall is a list with one element$).$

Write a program which contains two functions, a$)$ velocity as a function

of time, $t,$ and value of acceleration due to gravity, $g,$ and b $)$ position as a

function of time, $t,$ and value of acceleration due to gravity, $g.$ These should be

very simple functions with two passed arguments and only a return expression

instruction in the body.

Also, include an input instruction for the value of the acceleration. Run

the program three times with three different values for the acceleration due to

gravity, $g$ :

i$.$ the nominal value of $g$ on earth;

ii$.$ the nominal value of $g$ on the moon; and

iii. the nominal value of $g$ on mars.

Then:

You should be able to determine with a single print instruction the min$-$

imum height above the surface the object must be released in order that

it drops for $5$ seconds. Set up your program to do this and submit the

answers $(3$ of them, one for each acceleration$).$

Plot and submit velocity$-$ and position$-$versus$-$time graphs for each accel$-$

eration, being sure to title each plot and label each axis properly, including

the correct units $($a total of $6$ plots$).$