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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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Projectile Motion Formula document sample

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