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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:
       I. Setup your spreadsheet.
             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.


       II. Develop your equations.
             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
       (deg), to the Initial Angle in radians, θi (rad).




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.




                                        Page 2 of 7
                                                            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
             Resultant Velocity, θf (rad).




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




III. Program your spreadsheet.

      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
               your projectile.

       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.

       a.      Reset the initial height of your spreadsheet to 0.0 meters.

         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
velocity from IV.b. above. Enter these values into your new spreadsheet.

       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
heading neatly and properly completed.

II.    Pages 2 and 3 of this packet, showing the reduced form of each equation you
programmed into your Excel spreadsheet, should follow the coversheet.

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
should follow the question responses.

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
 THROUGH           thorough                                   adequate              minimal
                                     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
                                                                                     inadequate
                  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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