Numerical Modeling: Including Air Resistance
Counts the same as one lab grade.
Many physical problems cannot be solved by plugging numbers into
an
equation. This is the case when air resistance acts on a projectile;
there
is no longer an equation which gives you the location of the projectile
at a given time like there is for the case with no friction. Since
friction acts
along the entire path the effects must be factored in at each split
second of the trip. One way to do this is a computer program which can
iterate the effects of friction over many time steps. A simpler way is
to us a spreadsheet where each cell of the spreadsheet factors in the
friction effects and calculates where the projectile is for the next
time interval.
In this exercise you will use a spreadsheet to find out what happens to
a
projectile when air resistance is included. You may work in groups of
three
but: There will quiz questions on this material so be sure
everyone
in the group knows how to run the spreadsheet. Be sure to read
the section
on numerical modeling in your text book.
Directions for getting started
Download the Excel spreadsheet
(or
construct your own as shown in the three figures below) and answer the
questions
at the bottom of this page in your report. (NOTE: if you are on
a
computer on campus (in the micro-labs; CV-028, CV-112, or KV-207) and
the
computer asks you which application you want to use, choose 'Browse'
and
then choose Microsoft Excel.
Hints for using a spreadsheet:
- Each entry in a spreadsheet has an adress. Letters denote
columns,
numbers denote rows. So C4 refers (in most spreadsheets) to the entry
in
the third column, row 4.
- To enter a name or number you click on the cell and type in the
entry
space. For example A2 in figure one has the word 'time' typed in it. H2
in
figure 2 has the numerical value of the friction coefficient typed in
it.
- To enter a formula you start (in most spreadsheets) with an
equal
sign (=).
- Multiplication is given by a *, addition by +, powers by ^,
division
by /. You also have a few special functions like sine and cosine.
- Look at the entry in B4 in figure one. This entry says to take
the
value in location B3, add it to the product of the numbers in locations
D3
and H5. The H5 has dollar signs around it which indicates that the
location of the value will not change in the formula if you copy the
formula to a
new location.
- Once you have the formula as you want it you press the enter key
to
record it.
- Most spreadsheets have a 'fill down" option which alows you to
fill
a column with the same formula down to row 1000. Notice that when you
do
that the spreadsheet automatically changes the references that are
entered
except for entries with dollar signs. So for example entry B4 has B3
and
D3 in the formula but B5 has B4 and D4 in it. Both have a reference to
H5.
The 'fill down' option changed the 3s in B4 to be 4s in B5 but did not
change the $H$5 reference since it has dollar signs.
- There will be a 'show numbers' option in the spreadsheet that
hides
the formulas but shows the results of preforming the calulations you
have
entered. This is shown in figure 3.
- Once the spreadsheet is constructed you don't change the
formulas
but you can modify the intial speed, friction coeficient, angle, etc.
in
column H by clicking on the value, changing it and typing the 'enter'
key.
- Most spreadsheets have graphing capability. For Excel you click
on
the Chart button then drag the mouse over the columns you want to
graph.
A series of choices will then be presented to you. (In the first screen
you can change the rows and columns you are plotting if you did not
select the
right rows or column. In the example to plot x vs y you want to select
B$1:B$1000$,C$1$:C$1000$.)
Figure one; the formulas for the first few columns:
Figure two; the formulas for the last few columns:
Figure three; the numbers for all columns in the first few rows:
Your report should include a discussion of each of the following:
Do not print out the numbers in the spreadsheet! Turn in a verbal
description of your results and a few sample graphs to make your point.
- Test the spreadsheet for the case where the friction coefficient
is
0: Click on the box H2, change the number to 0.0 and type the
'enter'
key. Record the value of x, vy and vx when y is
zero
(or as close to y = 0 as you can get- this will be approximately cells
B1000
for x, D1000 for vx, and E1000 for vy). Show a
calculation
of what the answers should be using the usual kinematic equations from
you
book with the time value for y = 0 (approximatley box A1000) as the
time.
How do the spreadsheet values compare (rel. error) with your calculated
values
(Note- your x, vy and vx values from the
spreadsheet
may not be exactly the same as the calculated values if the spreadsheet
doesn't
have a y value of exactly 0- pick the values when y is closest to 0)?
- Now change the coefficient to 0.0055. How do the spreadsheet
values
for x, vy and vx comapare with a calculation of
those values when y = 0?
- Run the spreadsheet for several different initial speeds (box
H8)
and several different angles (box H11) with a constant friction
coefficient
(try 0.0055). Describe the graph of x vs y position for several of
these
cases (the graph is already constructed in the version you downloaded).
- What differences in range do you see as speed is increased with
the
same angle? (Range is the x value when y = 0.)
- What differences in range do you see as angle is increased with
the
same speed?
- How are your last two answers different from the case of no
friction?
- Run the spreadsheet for several different friction coefficients
with
a constant speed and angle. What differences do you see as the friction
coefficient is increased with constant initial speed and angle?
- An angle of 45 degrees gives the greatist range in the case
where
there is no friction. Is the angle which gives the greatest range in
the
case with friction greater, less than or equal to 45 degrees? Turn in
two
or three graphs of x vs y position to prove your answer.
Extra Credit:
- By changing friction parameter, can you find a case where the x
velocity becomes negative? What does it mean when the output of the x
velocity becomes negative? Is this possible for a real projectile? Why
does this happen in
the spreadsheet but not in real life?
- Why should you be worried about the x velocity becomming
negative
but not the y velocity becomming negative?
- Experiment with changing the time step; try values of 0.001,
0.01,
0.1 and 1.0. In particular test the spreedsheet as you did above for a
zero friction case. Which of these choices of time step causes the
spreadsheet to give the wrong answers (and how do you know they are
wrong)? What conclusions can you make from this?
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