Written Assignment $\backslash$# $5$: Terminal Velocity $($Spreadsheets$)$

PHYS $100-$ SFU

$>$ Export your spreadsheet as a pdf$.$ Spend at least $5$ minutes making sure the pdf is readable for the grading TA$.$

$>$ Add an extra pdf page with written responses to the questions.

$>$ Submit only one pdf document.

$>$ Written solutions must be submitted via Crowdmark before Friday November $\backslash (29^\{\backslash$text $\{$th $\}\},11$: $59\backslash$mathrm$\{$pm$\}\backslash ).$

$>$ Grading will be out of $10$ points and will reflect both the responses to the actual questions and the calculations in the spreadsheet. Remember the TA will not have access to your original spreadsheet, just the PDF$,$ so you want to make sure they can understand your calculations for proper grading and partial marks.

In this assignment you will investigate the concept of terminal velocity. So far we have studied objects in free$-$fall using constant acceleration $-$g$.$ In reality objects in air will feel a "drag" force that opposes their motion. Distinct from kinetic friction, air drag force can be modeled as proportional to the speed squared of the object: $\backslash ($ F$\_\{$d$\}=$a v$^\{2\}\backslash )$ with some constant a$.$

The coefficient $\backslash ($ a $\backslash )$ is further broken down into $\backslash ($ a$=$D $\backslash$rho A $/2\backslash )$ where $\backslash (\backslash$rho $\backslash )$ is the air density $\backslash (\backslash$left$(\backslash$mathrm$\{$kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash$right$),$ A $\backslash )$ is the "frontal" area of the object $\backslash (\backslash$left$(\backslash$mathrm$\{$m$\}^\{2\}\backslash$right$)\backslash ),$ and $\backslash ($ D $\backslash )$ is a dimensionless "drag coefficient".

Question $1-$ Determine the units of $\backslash ($ a $\backslash )$ in terms of basic SI quantities only $(\backslash (\backslash$mathrm$\{$kg$\},\backslash$mathrm$\{$m$\},\backslash$mathrm$\{$s$\}\backslash ))$ and show how $\backslash (\backslash$mathrm$\{$av$\}^\{2\}\backslash )$ gives Newtons.

Suppose we have an object with mass $\backslash (\backslash$mathrm$\{$m$\}=1\backslash$mathrm$\{$~kg$\}\backslash ),$ falling from an initial height $\backslash ($ y$\_\{0\}=1000\backslash$mathrm$\{$~m$\}\backslash )$ through air with a density $\backslash (\backslash$rho$=1\mathrm{.}20\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ The frontal area area is $\backslash (0\mathrm{.}01\backslash$mathrm$\{$~m$\}^\{2\}\backslash ),$ the drag coefficient is $0\mathrm{.}5$ and it is falling due to a constant gravitational force of $\backslash ($ F$\_\{$g$\}=\backslash )$ mg $.$ This is a one$-$dimensional problem.

Question $2-$ Calculate the value of a for this scenario.

Question $3-$ Neglecting air drag for now, what is the time $\backslash ($ t$\_\{1\}\backslash )$ needed for the object to fall to $\backslash ($ y$=0\backslash ),$ using the kinematics we learned earlier in this course?

Set up a spreadsheet to calculate the height of the object at a series of regular time intervals $\backslash (\backslash$Delta t$=1\backslash$mathrm$\{$~s$\}\backslash )$ apart. You should have columns for: time $($ s$\backslash ()\backslash ),$ height $($ m $),$ velocity $\backslash ((\backslash$mathrm$\{$m$\}/\backslash$mathrm$\{$s$\})\backslash ),$ drag force $\backslash ((\backslash$mathrm$\{$N$\})\backslash ),$ net acceleration $\backslash (\backslash$left$(\backslash$mathrm$\{$m$\}/\backslash$mathrm$\{$s$\}^\{2\}\backslash$right$.\backslash )).$ The net acceleration results from the net force $($gravity plus drag force$),$ which changes the velocity. The velocity changes the position.

Note: The "Euler$-$Cromer" method from the tutorial from this week is not necessary for this scenario.

Drag your formulas down until you include your time $\backslash ($ t$\_\{1\}\backslash )$ from question $3.$

Question $4-$ What are the approximate height and velocity at $\backslash ($ t$\_\{1\}\backslash )?$ Use the line with the closest time.

Keep dragging your formulas down until the object hits the ground $(\backslash ($ y $\backslash )$ goes from positive to negative$).$

Question $5-$ What do you observe about the velocity and acceleration columns near the end?

Make graphs of height vs time and velocity vs time.

Question $6-$ How do your height $\backslash$& velocity graphs differ from the ones for regular freefall without air drag?

Question $7-$ Terminal velocity is the maximum velocity when air drag balances the graviational force for a falling object. You know $\backslash ($ F$\_\{$g$\}=-$m g $\backslash )$ and $\backslash ($ F$\_\{$d$\}=$a v$^\{2\}\backslash ),$ so calculate $\backslash ($ v $\backslash )$ directly and compare it to the value at the bottom of your spreadsheet calculations. How many time steps to you need to calculate so that the velocity is correct to two decimal places?