Comparing Trajectories
Objective: To compare the trajectory of an ideal projectile to one that experiences air resistance that is launched at an angle from ground level.
Discussion: In projectile (2 dimensional) motion air resistance is typically assumed to be disregarded. In reality, this is not a good assumption, but one that is made out of necessity because of the difficulty of quantitatively analyzing the motion of a projectile experiencing air resistance. Using Newton’s Second
Law, Kinematics and vector components modeling the motion of a projectile experiencing drag force can be done on a spreadsheet using only first year Physics equations.
Projectile motion without air resistance could be referred to as parabolic motion because the trajectory or path the object follows traces a parabola in this ideal case. First we will calculate the X and Y position as functions of time for the ideal case given velocity and launch angle. This is a simple application of kinematics equations of motion. To model the motion with air resistance the mass, cross sectional area, drag coefficient of the object in question must be defined in addition to the velocity and launch angle.
The resulting non-uniform acceleration of an object experiencing drag force would prevent use of the typical equations used in kinematics due to non-uniform acceleration. However, a spreadsheet can make this possible to do using the uniform acceleration equations by repeatedly recalculating acceleration using Newton's 2nd Law over very small time increments and modeling the motion as separate cases of uniformly accelerated motion for each tiny time increment. Then using the calculating power of a spreadsheet repeating over 1000 data points, the following first year Physics equations can be used to model projectile motion with drag force.
Governing Equations:
V
X

V cos
V
Y

V sin


V f

V
I
 at d

V t
V ave
 V f

V
I
2
F
D
 cAV
2
Eq. 1.1
Eq. 1.2
Eq. 1.3
Eq. 1.4
Eq. 1.5
Eq. 1.6 a
F

Net
F
Net
 m mg

F
D
Eq. 1.7
Eq. 1.8
Where V f
= Final Velocity, V
I
= Initial Velocity, a = acceleration, t = time interval, d=Displacement,

= launch angle, F
D
= Drag force, V x
= X component of velocity, V y
= Y component of velocity,
V ave
= average velocity, c = drag coefficient, A = cross sectional area, m = mass. SI units for all variables.
Setting Up:
1.
Open Microsoft Excel© www.spreadsheetlabmanual.com

Comparing Trajectories
5
6
5
6
2.
Set up your spreadsheet exactly like you see below.
A B C D E F
1
Initial Velocity (m/sec) Launch Angle (degrees) Drag Factor Cross Sectional Area (m 2 ) Mass (kg) Time Increment (sec)
2
3.
In Row 5 place the following column headings exactly as indicated below:
Time (s)
A
G
A x
(m/s 2 )
B
V x
Ideal (m/s)
F drag
H
X
(N)
C
V y
Ideal (m/s)
V y
I
Drag(m/s)
D
P x
Ideal (m)
F drag
J y
(N)
E F
P y
Ideal (m) V x
Drag (m/s)
F net
K
Y
(N)
L
A y
(m/s 2 )
5
6
M
Actual Px (m)
N
Actual Py (m)
4.
The time increments used in column A will depend on the object we are viewing. Objects that spend a shorter time in the air will require shorter time increments. For the first object we will use increments of 0.01 seconds. In cell A6, type “ =$F$2”. In cell A7 type “ =$F$2+A6” and then grab the box in the lower right hand corner of the cell, called the fill handle, and drag down to cell 1005.
5.
The ideal initial X and Y velocities are a function of time, the initial velocity and the launch angle.
Formulas can be written for them in cells B6 and C6 using Eq. 1.1 and 1.2. In cell B6 type the following “=$A$2*COS($B$2*3.14159/180)”. Since the ideal X velocity is constant, the formula in this cell can be copied down to cell 1005 by double clicking the fill handle in the lower right corner of the cell. Type the formula for the Y component of the initial velocity in cell C6 as follows: “=$A$2*SIN($B$2*3.14159/180)”. Note: “PI()” can also be entered into the formula instead of 3.14159.
6.
Since the Y component of velocity is changing uniformly in the ideal case, a continually changing formula for velocity is needed. In cell C7 type “=C6-9.81*$F$2”.
7.
The functions for position can be written using Eq. 1.5. The final and initial velocity for each interval is used to determine the average velocity and then the displacement using the definition of velocity (Eq. 1.4). In the X direction this is simple, since the velocity is not changing simply enter zero in cell D6. In cell D7 enter “=B7*$F$2+D6”. For the Y position in cell E6 type zero. Since the Y velocity is changing, Equations 1.5 and 1.6 can be combined to make the following formula that can be typed in cell E7 “=(C6+C7)/2*$F$2+E6”. Double click each fill handle and the formulas will be entered through to cell 1005. www.spreadsheetlabmanual.com

Comparing Trajectories
8.
The ideal case is complete. Enter 30.0 as the initial velocity, 30.0 as the launch angle and a time increment of 0.01. If you spreadsheet is correct thus far then the Y position should go from positive to negative at time 3.06 seconds. Scroll down to row 311 and see if column E goes
from positive to negative from 311 to 312.
9.
Now the trajectory for a real projectile has to be determined. The projectile experiencing drag will have the same initial velocity as the ideal so enter “=B6” in cell F6.
10.
In reality a projectile encounters air resistance. The force acts 2 dimensionally in both the X and
Y direction but its effects can be calculated separately in each dimension. Since the only force acting in the X direction is air resistance, it is equal to the net force on the projectile in the X direction. It can be determined using equation 1.6. In cell H6 type “=$C$2*$D$2*F6^2”.
11.
With the net force the acceleration can be calculated using Newton’s Second Law. Using Eq. 1.7 the acceleration can be calculated by entering “=H6/$E$2” into cell G6.
12.
The new velocity can be calculated at the end of the time increment in cell F7. The velocity in cell F7 will be slightly different in magnitude than F6 because of the drag force so using equation
1.3, the following formula should be entered into cell F7: “=F6-G6*$F$2”. Double click each fill handle from cell F7 then G6 and H6.
13.
Now skip to cell I6 to enter the initial velocity. It should be the same as the initial velocity for the ideal case “=C6”.
14.
The formula for the drag force in the Y direction is similar to that of the X direction except that it uses the Y component of velocity. Enter “=IF(
I
6<0,$D$2*$C$2*
I
6^2,-$D$2*$C$2*
I
6^2)” into cell J6. Note: the formula reads "Letter then Number," cell
I
6, not the number 16. The “if function” of the formula is present to ensure that the drag force is always opposite the direction of motion. (if the velocity is negative, the force is positive, if the velocity is positive, the force is negative)
15.
When the projectile is on the way up, the drag force and the force of gravity are pointed down.
When it is on the way down the drag force is pointed up. With the sign adjustment in the previous formula, the net force can be calculated by summing the drag force and the force of gravity. Enter “=$E$2*(-9.8)+J6” into cell K6.
16.
In cell L6 the acceleration can be calculated with the following formula: “=K6/$E$2”.
17.
Now the change in velocity is known for each time increment and the subsequent Y velocities can be determined. To do so enter “= I 6+L6*$F$2” in cell I 7. Note: the formula reads "Letter then Number," cells
I
6 and
I
7, not the numbers 16 and 17. .You can now double click any fill handle in order to carry all of the formulas down to cell 1005.
18.
The last formulas to enter are those for the X and Y positions. In cell M6 and N6 enter 0 and in cell M7 enter “=(F6+F7)/2*$F$2+M6”.
19.
In cell N7 enter the following formula: “=( I 6+ I 7)/2*$F$2+N6” Note: the formula reads "Letter then Number," cells I 6 and I 7, not the numbers 16 and 17. Double click the fill handles starting from the left and working to the right for each formula that is to be entered for the entire column. Your spreadsheet is complete!
20.
An IF function in column O can isolate the actual range when the Y position returns to zero and a
Max function applied to column O can place it where it can be easily found. This can be done www.spreadsheetlabmanual.com

Comparing Trajectories because when it crosses the ground, this is the only point at which two consecutive cells in the actual Y position column have opposite signs. Enter the following into cell O6:
“=IF(N6*N7<0,M6,0)” and double click the fill handle. Cell O3 should be labeled "Range" and O4 should contain the following formula: “=MAX(O6:O1005)”. Note the cells refer to the letter O,
not zero.
21.
To check to make sure the spreadsheet works, test the following data points:
Initial Velocity: 65 m/sec, Launch angle: 20 degrees, drag factor: 0.15, CSA: 0.001034 m 2 , mass: 0.045 kg, time increment: 0.005 seconds. Your range for the ideal case should be roughly
277 meters and the real range accounting for wind resistance should be 188.7 meters.
22.
*See Appendix for guidance* The last thing to do is plot the X and Y position on a chart for both the ideal case and the case including air resistance. In order to do this insert a scatter plot using the curved lines. Remove any data series that are given and select the ideal X and Y values for the first data series. Add another data series and select the actual X and Y trajectories that account for wind resistance as the X and Y values in the series. Label each series and the graph axes appropriately and place the chart in a new sheet.
Development Questions:
1.
Why are there dollar signs any time a cell in Row 2 is referenced?
2.
What is the purpose of the IF function in column J?
3.
How can the IF and MAX functions in column O make your spreadsheet easier to use? Explain the logical test used with this IF function.
4.
How many calculating cells are used to predict the trajectory of one projectile with air resistance on this spreadsheet?
Questions:
1.
A 2700 pound (1225. Kg) armor piercing shell from the 16 inch (diameter) guns of the battleship
New Jersey has a muzzle velocity of 762. m/sec. What is its maximum range in miles if the drag factor is 0.12? 1mile = 1610 meters. Compare to the ideal case. Hint: Look at the graph to select a time interval. Convert the 16 inch diameter (d) to meters and calculate the CSA using
CSA=
 d 2 /4 and test for optimum angle. www.spreadsheetlabmanual.com

Comparing Trajectories
2.
The high concussion shells fired from the same gun have a higher muzzle velocity but contain more explosive charges and less steel. They have a muzzle velocity of 820. m/sec and a mass of
2240 pounds (1016. Kg). What is the range of a high concussion shell? Compare to the ideal case.
3.
What is the effect of (a) increasing and (b) decreasing the mass of the projectile? What is the effect of (c) increasing and (d) decreasing the cross sectional area of the projectile?
4.
A golf ball with an area of 0.001034 m 2 and a drag factor of 0.15 is hit at an angle of 15 degrees and has a mass of 0.045 kg at an initial velocity of 75.5 m/sec. Use a time increment of 0.01 seconds. Will it clear a hazard that ends 200 meters away? Will it clear the hazard ideally? At what initial speed would the actual trajectory clear the hazard by 10 meters? (Use Goal Seek*)
How much distance is lost due to air resistance in both cases? Skip to problem 9 before changing the data in row 2.
*Goal Seek is found under the "Data" tab in "What if Analysis" (or the tools menu for 2003 or earlier).
5.
A beach ball has a mass of 0.325 kg and a diameter of 0.85 meters. What distance will it travel if it is kicked at a speed of 25.0 m/sec at an angle of 30.0 degrees and its drag factor is 0.2? How does this compare to the ideal distance? Explain the large difference. Use a time increment of
0.003 seconds. Skip to problem 9 before continuing.
6.
An arrow is shot with an initial velocity of 85.0 m/sec. It is aimed 10.0 degrees above the horizontal. It has a mass of 0.300 kg. What is its ideal and actual range if the arrow is 1.75 cm in diameter and has a drag factor of 0.25? Skip to problem 9 before continuing.
7.
A baseball has a mass of 0.143 kg and an area of 0.00417 m 2 . A ball leaves the bat with a velocity of 55.0 m/sec at an angle of 40 degrees and just clears a 330 foot (100. m) wall that is
3.00 meters high. What is its drag coefficient? Use a time increment of 0.01 seconds. Use manual guess and check and the point (100,3) should be present on the actual trajectory.
8.
Balls hit at Coors Field in Colorado will have 20 percent less air resistance due to the high altitude (low air density). What distance can the same hit clear at Coors Field? Explain how this can give an advantage to the batter.
9.
One Dimensional Motion with air resistance: If a launch angle of 90 degrees is used, this spreadsheet can compare the actual maximum heights of objects thrown vertically. Without air resistance the height of the object can be determined using the equation d ideal
V
I
2

where d=
2 a maximum height, a = 9.8 m/sec 2 and V
I
is given below. Predict the maximum height of the objects at the velocities given in each problem. To make your work easier, write the following formula in cell N1: “=MAX(N6:N1005)”. This is the maximum height. A. The golf ball from problem 4. B. The beach ball from problem 5. C. The arrow from problem 6. D. The baseball from problem 7. www.spreadsheetlabmanual.com

Comparing Trajectories
Making an X – Y Scatter Plot on Excel 2007 or later:
1.
Click on the Insert tab at the top of the screen.
2.
Select the Scatter icon and choose the graph without the data points on it (unless actual experimental data points are being plotted).
3.
The data must then be selected: Do not trust that the program will select the correct data,
even if the graph looks correct. To do this right click the chart area and choose “Select Data” from the drop down menu. Remove any data series that is automatically selected.
4.
To add the data, first name the series and then click the icon (shown below)
to the right of the data field for the X values. Once you do this the spreadsheet will be active and you can highlight any column of cells you wish to make the X values for the graph. Do not highlight column headings, just data points. Highlight your actual X position values (column M row 6-2005) and then click the same icon on the small window on your spreadsheet. Do the same for your Y position values (column N row 6-2005).
5.
Add another series and do the same for your Ideal X and Y position (columns D and E).
6.
You must highlight an equal number of X and Y values for a scatter plot.
7.
If you wish to move the graph after the data has been selected, right click the chart area and select “Move Chart” which will allow you to put it into a new sheet. This is recommended because it will allow you to place it out of the way of your active sheet and has a convenient chart tab at the bottom next to your sheet tabs. www.spreadsheetlabmanual.com

Comparing Trajectories
Making an X – Y Scatter Plot on Excel 2003 or earlier:
1.
Select the chart wizard option on the tool bar shown below:
2.
Choose X-Y Scatter as the type of chart. Do not choose line graph.
3.
Select the desired type of X-Y Scatter plot and click next. It is recommended that you choose the continuous line without data points if you are plotting a function
4.
There are two tabs up top, select the series tab.
5.
Remove any series that is present.
6.
Then click “Add” and you will see 3 fields for a new data series. The icon to the right of each field can be clicked to select the specific X and Y cells you would like to plot. Click the icon next to the X-values field. Highlight the desired numeric cells
(not column headings) for your actual X position values (column M) and then click the same icon on the small window on your spreadsheet. Do the same for your Y position values (column N).
7.
Add another series and do the same for your Ideal X and Y position (columns D and E)
8.
Assign names and units to your axes and your chart then click next.
9.
You can place your chart in your active sheet or as a separate sheet. I prefer a separate sheet so it is out of the way. The tabs beneath the chart or the worksheet shown below can switch you from one to the other.
10.
Adding an additional data series to a graph: Go to your chart and right click any open space on the chart area and select “source data”.
11.
Select the series tab and add a new series as you just did with new x and Y values. It will plot the graph of the same set of axes with a different color. This is good for comparing functions over a given domain in math or comparing graphs of a control to a dependent variable over a given independent variable in science. www.spreadsheetlabmanual.com