# Projectile Motion Formula - PDF

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```					                                                                                                   CP Physics
Londonderry High School
Mr. Levergood
Mr. Cariello
Name:________________                       Period:________              Date: ________

Using Numerical Methods To Map
And Analyze Projectile Motion (100 Points)

Background: Numerical analysis involves the study of methods of computing numerical data.
Applications of numerical analysis occur throughout the fields of applied mathematics, physics,
and engineering.
Purpose: For this assignment you will use Microsoft Excel to develop a spreadsheet that allows
you to numerically analyze the trajectory of projectiles launched from the top of a 31-meter high
cliff at various angles and velocities.
Procedure:
a.        Open Microsoft Excel and give your file an original name. Save this file to your
Z-drive.
b.        In Row 1 fill cells A through N with the following column headings.
A          B       C       D      E       F       G       H       I       J         K      L       M        N

1     Vi          θi      Yi     θi      Vix     Viy     t       Xf      Yf     Vfx       Vfy     VR      θf       θf
(m/s)    (deg)      (m)    (rad)   (m/s)   (m/s)    (s)    (m)     (m)    (m/s)     (m/s)   (m/s)   (rad)    (deg)
2
3

c.        Color cells A1, B1, C1, A2, B2 and C2 yellow (or any color of your choice).
These will be your “Input” cells.
d.        Color the “Time” column light blue (or any other color of your choice) to remind
you that you will choose the time interval over which to calculate the physical
quantities associated with the projectile’s motion.

a.        Start with the kinematics equations on page 79 of the textbook (also listed below)
and derive algebraic expressions for each of the column headings above. The
algebraic expressions can be in terms of the four input values (Vi, θi, Yi, and t), in
terms of any physical quantity that you have previously calculated in the
spreadsheet, calculated using a math/trig formula you know, or calculated using a
math/trig formula in MS Excel.
Numerical Analysis of Projectile Motion

Kinematics Equations
r     r r            r     r r          1r           r 2 r2      r r          r
v f = vi + aave ∆t   x f = xi + vi t f + aavet f     v f = vi + 2aave ( x f − xi )
2

2
b.     In the box below enter the final formula to convert the Initial Angle in degrees, θi

c.     In the box below enter the final formula to calculate the Initial Horizontal
Velocity, Vix.

d.     In the box below enter the final formula to calculate the Initial Vertical Velocity,
Viy.

e.     In the box below enter the final formula to calculate the Final Horizontal Position,
Xf.

f.     In the box below enter the final formula to calculate the Final Vertical Position,
Yf.

g.     In the box below enter the final formula to calculate the Final Horizontal
Velocity, Vfx.

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Numerical Analysis of Projectile Motion

h.     In the box below enter the final formula to calculate the Final Vertical Velocity,
Vfy.

i.     In the box below enter the final formula to calculate the Final Resultant Velocity,
VR.

j.     In the box below enter the final formula to calculate the Angle of the Final

k.     In the box below enter the final formula to convert the Angle of the Final
Resultant Velocity, θf (rad), to degrees, θf (deg).

a.     Program column A in a way that you only need to change the value in cell A2 and
the next 28 cells in column A will change to that value.

b.     Program column B in a way that you only need to change the value in cell B2 and
the next 28 cells in column B will change to that value.

c.     Program column C in a way that you only need to change the value in cell C2 and
the next 28 cells in column C will change to that value.

d.     Program cell D2 to convert the angle in cell B2 from degrees to radians using the
formula you developed above. Auto-fill the entire column with this formula.

e.     Program cell E2 to calculate the initial horizontal velocity using the formula you
developed above. Auto-fill the entire column with this formula.

f.     Program cell F2 to calculate the initial vertical velocity using the formula you
developed above. Auto-fill the entire column with this formula.

Page 3 of 7
Numerical Analysis of Projectile Motion

g.      Choose a time interval between 0.01 and 0.25 seconds. Start the sequence in cell
G2 with 0.0 and continue it in cells G3 and G4. Highlight cells G2, G3, and G4
and auto-fill the entire column with this sequence.

h.      Program cell H2 to calculate the final horizontal position using the formula you
developed above. Auto-fill the entire column with this formula.

i.      Program cell I2 to calculate the final vertical position using the formula you
developed above. Auto-fill the entire column with this formula.

j.      Program cell J2 to calculate the final horizontal velocity using the formula you
developed above. Auto-fill the entire column with this formula.

k.      Program cell K2 to calculate the final vertical velocity using the formula you
developed above. Auto-fill the entire column with this formula.

l.      Program cell L2 to calculate the magnitude of the final resultant velocity using the
formula you developed above. Auto-fill the entire column with this formula.

m.      Program cell M2 to calculate the direction of the final resultant velocity in radians
using the formula you developed above. Auto-fill the entire column with this
formula.

n.      Program cell N2 to convert the angle in cell M2 from radians to degrees using the
formula you developed above. Auto-fill the entire column with this formula.

IV. Plot the trajectory of this projectile.

a.      If you have not already done so, choose an initial velocity and an initial angle for

b.      Enter these values in cells A2 and B2, respectively. Your spreadsheet should
auto-calculate all of the physical quantities listed in row 1 for each value of time
entered in column G.

c.      Highlight the final horizontal and vertical position values.

d.      Open the chart wizard and choose an X-Y scatter plot with data points and smooth
lines. Properly label and format all axes, series, and the graph itself.

e.      Place your trajectory plot on a separate sheet of the MS Excel workbook.

f.      Ensure that you properly label the chart, the horizontal and vertical axes, and the
data series.

Page 4 of 7
Numerical Analysis of Projectile Motion

Analysis:

I.      Launch with a Positive Angle. Choose an angle that is greater than 20o and an initial
velocity that is greater than 10 m/s. Enter these values into your spreadsheet and look at the
trajectory plot.

a.    How long did it take for the projectile to reach the ground 31 meters below the
cliff? How can you tell from your data?

b.      Print off your data sheet that shows the time at which the projectile reached the
ground and your corresponding trajectory plot. Ensure that your Trajectory Plot is properly
labeled and that the Heading Row appears at the top of each data page.

c.     Highlight the entire two rows that bracket the time at which the projectile reached
the ground and highlight the corresponding data points on your trajectory plot.

d.      Describe three ways that you can tell from your data when the projectile has
reached the top of its trajectory.

e.      How long did it take for the projectile to reach its maximum height?

f.       Highlight in a different color the entire two rows that bracket the time at which
the projectile reached its maximum height and highlight the corresponding data points on your
trajectory plot.

II.     Launch with a Zero Angle. Choose an angle of 0.0o and an initial velocity that is
greater than 10 m/s. Enter these values into your spreadsheet and look at the trajectory plot.

a.    How long did it take for the projectile to reach the ground 31 meters below the
cliff? How can you tell from your data?

b.      Print off your data sheet that shows the time at which the projectile reached the
ground and your corresponding trajectory plot. Ensure that your Trajectory Plot is properly
labeled and that the Heading Row appears at the top of each data page.

c.     Highlight the entire two rows that bracket the time at which the projectile reached
the ground and highlight the corresponding data points on your trajectory plot.

III.    Launch with a Negative Angle. Choose an angle that is less than 0.0o and an initial
velocity that is greater than 10 m/s. Enter these values into your spreadsheet and look at the
trajectory plot.

a.    How long did it take for the projectile to reach the ground 31 meters below the
cliff? How can you tell from your data?

b.      Print off your data sheet that shows the time at which the projectile reached the
ground and your corresponding trajectory plot. Ensure that your Trajectory Plot is properly
labeled and that the Heading Row appears at the top of each data page.

Page 5 of 7
Numerical Analysis of Projectile Motion

c.     Highlight the entire two rows that bracket the time at which the projectile reached
the ground and highlight the corresponding data points on your trajectory plot.

IV.    Bonus Challenge Question.

b.      Choose an angle that is less than 45o but greater than 0.0o and an initial velocity
that is greater than 10 m/s. Enter these values into your spreadsheet and look at the trajectory
plot.

c.      Label the data series that is displayed “Less than 45 Degrees”.

d.     Find the maximum horizontal displacement of the projectile when it hits the
ground. This displacement is called the “range” of the projectile.

e.      Make a copy of your calculation spreadsheet and give it a unique name.

f.     Choose an angle that is greater than 45o but less than 90o and the same initial

g.      Plot this data series on the same chart as IV.b. above. Label this data series
“Greater than 45 Degrees”.

h.       Find the maximum horizontal displacement of the projectile with this new angle.
Try several angles between 45o and 90o until you find one that yields the same range as the angle
that was less than 45o.

i.      Print your trajectory plots and the corresponding data sheets. Highlight the rows
that bracket the range on each data sheet.

j.     What general relationship can you develop that relates two different angles that
yield the same range with the same initial velocity?

Report Submission Requirements.

I.     The coversheet of your report should be the Grading Rubric on the next page with the

II.    Pages 2 and 3 of this packet, showing the reduced form of each equation you

III.   The responses to all questions asked during the analysis portion of this exercise should
follow pages 2 and 3. The question responses should be in complete sentences.

IV.    The trajectory plot and corresponding data sheets for each section of the analysis portion

V. Your report may be typed or neatly written.

Page 6 of 7
CP Physics
Londonderry High School
Mr. Levergood
Mr. Cariello
Name:________________                          Period:________                   Date: ________
Numerical Analysis of Projectile Motion

Grading Rubric (Based on LHS School-Wide Rubrics)
3-4 Points          0-2 Points
9-10 Points            7-8 Points          5 –6 Points                                                Points
DOES NOT            DOES NOT
EXCEEDS                 MEETS               MEETS                                                    Awarded
MEET                MEET
Does not
understand or
LITERACY       Demonstrates a                            Demonstrates an        Demonstrates a
Demonstrates                                                      makes no
understanding of                                                   effort to
READING       understanding of                           understanding of      understanding of
written material.                                                 understand
written material.                          written material.     written material.
written
material.
Effectively uses                                                Written work
LITERACY       Demonstrates                              Written expression
Written work needs     needs a great
THROUGH       mastery of written        written             needs some
improvement.            deal of
WRITING         expression.           expression.         improvement.
improvement.
Understands both
Shows lack of
oral and written                              Generally           Shows poor
LITERACY                            Understands most                                                comprehension
language and                            understands basic    comprehension of
THROUGH                              language and                                                        of the
concepts                               language and       the language and
THINKING                               concepts.                                                     language and
appropriate to the                             concepts.            concepts.
concepts.
course.
Independently
Connects new                                                  Does not make
connects new                                                  Occasionally makes
concepts with       Recognizes that                            connections
concepts with                                                    connections
previously learned   connections exist.                            between
previously learned                                              between concepts.
concepts.                                                     concepts.
concepts.
Effectively
CRITICAL                              Organizes and        Demonstrates        Demonstrates no
organizes and
THINKING and                             synthesizes      limited synthesis       synthesis of
synthesizes
PROBLEM                                information.       of information.       information.
information.
SOLVING
Poorly evaluates or
Thoroughly             Correctly                                                     Does not
misinterprets
evaluates           evaluates most         Evaluates                                 evaluate
evidence; draws
evidence; justifies   evidence; presents   evidence; presents                           evidence;
inaccurate or
a logical              a logical         a conclusion.                              draws no
conclusion.           conclusion.                                                   conclusion.
conclusion.
Ineffectively
Consistently
SELF-                             Manages time and     Manages time and      manages time and      Wastes time
manages time and
DIRECTED                             resources in an      resources with         resources; may        and/or
resources in an
LEARNER                             efficient manner.    some prompting.          require more       resources.
efficient manner.
frequent prompting.
Independently         Connects new        Completed the          Completed the      Completed the
connects new          concepts with        exercise and           exercise and       exercise but
concepts with       previously learned    recognizes that     occasionally makes    does not make
Challenge     previously learned        concepts.        connections exist.        connections       connections
concepts.                                                    between concepts.       between
Problem                                                                                               concepts.

5 Points              4 Points             3 Points              2 Points            1 Point

Total Points
> or = 63           58 /53 /48 /44       40 / 36 / 32         28 < pts > 14          14 > pts
Awarded
Letter
Grade                A             A-/ B+/ B/ B-        C+/     C / C-               D                   F
Awarded

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