Background

the change in the

object's $PE$ is equal to its change in $KE$ :

$PE=KE.$

In terms of relevant physical variables, this relationship is written as

$mg(h_1-h_2)=\frac{1}{2}m(V_2^2-V_1^2)$

where $m$ is the object's mass, $g$ is the $($constant$)$ acceleration due to gravity, $h$ is the object's height above some

reference point, $V$ is the object's velocity, and the subscripts $1$ and $2$ represent the initial and final states of the

object, respectively.

This classic falling object problem is often used to answer questions like, "What is the object's velocity right before

it hits the ground?" Once we have a grasp of these simple falling object problems, it is interesting to try to answer

new questions like, "What happens after the object hits the ground?" This question naturally leads to the study of

collisions. For perfectly elastic collisions, no energy is lost and the upward velocity of the ball just after it hits the

ground is exactly equal to the downward velocity of the ball just before it hits the ground. In this idealized scenario the

ball will bounce all the way back up to its initial height $(h_1)$ and continue doing so forever, clearly, the elastic collision

case does not describe real collisions.

A more realistic mathematical model for a bouncing object leads us to the study of inelastic collisions. For inelastic

collisions, the upward velocity of the object after rebounding off the ground is less than its downward velocity just

before striking the ground. The ratio of the magnitudes of these velocities is called the coefficient of restitution $(e),e=\frac{|\text{v}\text{e}\text{l}\text{o}\text{c}\text{i}\text{t}\text{y}\text{a}\text{f}\text{t}\text{e}\text{r}\text{s}\text{t}\text{r}\text{i}\text{k}\text{i}\text{n}\text{g}\text{g}\text{r}\text{o}\text{u}\text{n}\text{d}|}{|\text{v}\text{e}\text{l}\text{o}\text{c}\text{i}\text{t}\text{y}\text{b}\text{e}\text{f}\text{o}\text{r}\text{e}\text{s}\text{t}\text{r}\text{i}\text{k}\text{i}\text{n}\text{g}\text{g}\text{r}\text{o}\text{u}\text{n}\text{d}|}.$

Elastic collisions are described by $e=1$ and inelastic collisions are described by $0e<1.$ Leaving the details for

your physics class, the total distance traveled $(z)$ by an object initially dropped from a height $h_1$ is

$z=h_1+2h_1(e^2+e^4+e^6+$dots$)$

Note that, interestingly, the mass of the object and the acceleration due to gravity have no effect on this result. We

recognize the term in parenthesis as $($almost$)$ a geometric series. We can turn this into a geometric series by adding

and subtracting $2h_1($i$.$e$.,$ effectively adding $0)$ to $z,$ giving

$z=-h_1+2h_1(1+e^2+e^4+e^6+$dots$)=-h_1+2h_1{\displaystyle \underset{n=0}{\overset{}{}}}(e^2{)}^n$

where $n$ represents the number of bounces the object makes off the ground. The $-h_1$ term might seem strange at

first$-$does this somehow mean the object can possibly travel a negative distance? Test this expression for $0$ bounces

to assure yourself that this negative term out front isn't an issue. The second term in the final expression for $z$ is

clearly a geometric series with $r=e^2.$ Since we know $|e|<1,$ we can use the general result for the infinite sum of a

geometric series to show that the total distance traveled by the object is

$z=h_1(\frac{1+e^2}{1-e^2})$

Note that the total distance traveled is infinite when $e=1,$ which is the expected result when the collision is perfectly

elastic, but converges to some finite value for any e$<1.$

Your Task: Help me make a matlab code for this homework using the functions

h$1=1$;

e $=0\mathrm{.}7$;

$[$TotalDist$,$BounceOut,z$]=$ BallBounce$($h$1,$e$)$

Write a function called BallBounce that takes in values for the initial height of a ball above the ground $($h$1)$ and the coefficient of restitution between the ball and the ground $($e$)$ that does the following:

Finds the total distance traveled by the object in the limit that the number of bounces goes to infinity. This value should be the first output of your function.

Finds the number of bounces required for the ball to travel at least $95\%$ of the total distance it will travel. This value should be the second output of your function.

Define a row vector that stores the distance traveled as a function of the number of bounces. The last element of this row vector should be the distance traveledonce the condition described above is met. This vector should should be the third output of your function. You can perform a preliminary check of this variable by making sure its first element corresponds to the distance traveled for $0$ bounces.