Instructions:

$1.$ In addition to this HW$3$ document, you will also need the two excel files posted with

HW$3$: ProjectileData.xlsx and HydrogenPeroxide.xlsx$.$ Put both of this files in your

current MATLAB directory.

$2.$ Show your work! Make sure you include MATLAB commands.

$3.$ It is fine to work with other students, but what you turn in must be your own work $-$ not

something copied from someone else.

Problem $1$: Curve Fitting $$ Trajectory of Projectile

The excel file, ProjectileData, has three columns of data: time, distance, and height. Import

each of these columns into MATLAB using either xlsread or the Import Data tool. The variable

distance represents measurements of the x$-$position $($horizontal position$)$ of the projectile over

time and the variable height represents measurements of the y$-$position $($vertical position$)$ of the

projectile over time.

The equations for the x and y position of a projectile launched at an angle of $\backslash$theta $($rad or degrees$)$

with an initial velocity of V$0($m$/$s$)$ are:

$($a$)$ Plot time on the x$-$axis and distance on the y$-$axis. Add axis labels $($with units$)$ and a title to

your plot. We know that the x$-$position of the projectile increases linearly with time. So use

the curve fitting tool to fit a $1$st order polynomial $($line$)$ to the distance data. Display the

equation for the fitted polynomial on your graph with $5$ significant digits. Then copy and

paste your plot in the space below.

MATLAB PLOT:

$($b$)$ Plot time on the x$-$axis and height on the y$-$axis. Add axis labels $($with units$)$ and a title to

your plot. We know the height of the projectile follows a parabolic $(2$nd order$)$ curve. So use

the curve fitting tool to fit a $2$nd order polynomial $($quadratic$)$ to the height data. Display the

equation for the fitted polynomial on your graph with $5$ significant digits. Then copy and

paste your plot in the space below.

MATLAB PLOT:

$($c$)$ Look at the numerical coefficient for the squared term in the fitted polynomial for the height

data. Theoretically, this coefficient should be equal to $-1/2*$g$.$ How close is it$?$ Calculate a

percent error using the following formula with $-1/2*$g as actual value:

Show Calculations:

$\%$ Error $=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$($d$)$ The numerical coefficient for the linear term in the fitted polynomial for height should be

approximately V$0$ sin$(\backslash$theta $)$ and the numerical coefficient for the linear term in the fitted

polynomial for distance should be approximately V$0$ cos$(\backslash$theta $).$ Enter the values below then

solve for the initial velocity, V$0,$ and the launch angle, $\backslash$theta $.$

V$0$ sin$(\backslash$theta $)=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

V$0$ cos$(\backslash$theta $)=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$\backslash$theta $=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_($include units$)$

V$0=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_($include units$)$

Show Calculations:

Problem $2$: Curve Fitting $$ Decomposition of Hydrogen Peroxide $$ Estimating Reaction

Rates from Experimental Data

First order chemical reactions can be modeled using exponential functions. Hydrogen Peroxide

$($H$2$O$2)$ decomposes as a $1$st order reaction into water and oxygen gas:

The concentration of hydrogen peroxide decreases exponentially according to the following

equation:

In lab, we investigated the decomposition of hydrogen peroxide in air which was a very slow

reaction. It decomposes much, much faster in the presence of a catalyst such as Iodide $($I$-$

$).$

C$($t$)=$ Concentration at time t $($M or mols$/$L$)$

C$0=$ Initial Concentration $($M$)$

k $=$ Reaction rate $($s

$-1$

$)$

The HydrogenPeroxide.xlsx file has the results of five experiments measuring the concentration

of hydrogen peroxide using an Iodide catalyst at $5$ different temperatures. We will use curve

fitting to estimate the reaction rate, k$,$ for each of these five temperatures.

If we take the natural log of the concentration equation, we have:

$$ For each of the five temperatures, plot time on the x$-$axis and ln$($C$($t$))$ on the y$-$axis.

Create five separate plots. Remember, in MATLAB ln$($C$)$ is log$($C$)!$

$$ Next use the curve fitting tool to fit a line to the data. Display the curve fit equation with

five significant digits. The slope of the line should be roughly equal to $$k as long as the

measurements are good. Enter your estimated reaction rate values in the table below.

$$ Include your plots in the space indicated $($make sure they are labeled appropriately$)$ and

the MATLAB commands used to generate one of the five plots.

Absolute Temperature $($K$)$ Estimated Reaction Rate, k $($s$-1$

$)$

$280$

$285$

$290$

$295$

$300$

MATLAB COMMANDS TO GENERATE ONE OF THE PLOTS:

FIVE MATLAB PLOTS SHOWING CURVE FIT:

Problem $3$: Curve Fitting $$ Decomposition of Hydrogen Peroxide $$ Estimating Activation

Energy from Experimental Data

The reaction rate, k$,$ depends on the temperature according to Arrhenius$$ Equation $($also

exponential$)$:

We will again apply the very nice trick of taking the natural log of this equation which gives:

Using your five values from the previous problem, plot $1/$T on the x$-$axis and ln$($k$)$ on the y$-$axis.

Again, do a linear curve fit. Using the fact that the slope of your line should be approximately

A $=$ Fre