BLAW1002
”The Parabola of Motion: Understanding Free Fall Through Manual Measurement”
Introduction
Welcome to an exciting exploration of physics and mathematics! In this project, you will study the motion of free-falling objects of varying sizes to analyze their trajectories and calculate the acceleration due to gravity. Additionally, you will explore the effects of an initial upwards velocity on the motion, introducing a layer of complexity to the equations of motion.
Objective
Using manual data collection through stop-motion capture, analyze the free fall of three objects of similar shape but varying size, both with and without an initial upwards velocity. Calculate the acceleration due to gravity, derive all the SUVAT equations, and determine the percentage error compared to a = −9.81m/s2 .
Materials
• Three rounded objects of similar shape but varying size (e.g., a marble, golf ball, and tennis ball)
• Smartphone or camera for stop-motion capture
• Free or accessible software for frame analysis (e.g., Tracker or similar apps)
• Rulers, Tape, String, Measuring sticks, Measuring tape
• Graph paper, Calculators, Notebooks
Group Formation
Work in pairs. Collaboration and communication are key.
Project Phases
Phase 1: Experiment Design
Design an experiment to capture the motion of three objects of varying size dropped from the same height. Choose a drop height between 5 meters and 20 meters, ensuring safety and accuracy in data collection. For the second set of trials, incorporate an initial upwards velocity to make the objects travel higher than their original drop point. Plan how you will measure the time of motion and distances, ensuring minimal errors. Consider environmental factors, such as wind, and control for them as much as possible.
Phase 2: Data Collection
Using your smartphone or camera, capture the motion of the three objects during both sets of trials:
• Free-fall trials: Drop the objects from rest at the chosen height and record the motion.
• Upwards velocity trials: Launch the objects upwards with an initial velocity so they travel higher than the original drop point before falling back down.
Analyze the stop-motion videos to determine precise time intervals for each object. Perform three trials for each object in both scenarios to ensure reliability. Record all data in an organized manner for ease of analysis.
Phase 3: Data Analysis and Acceleration Calculation
For each trial, analyze your data to calculate the acceleration due to gravity. Use the equations of motion, modifying them as needed to include the initial upwards velocity for the second set of trials.
Key equations:
• For free-fall trials:
s = ut + 2/1at2,
where u = 0 (initial velocity).
• For upwards velocity trials:
s = ut + 2/1at2,
where u > 0 (initial upwards velocity).
Compare your calculated values of a for each object to −9.81m/s2 , and calculate the percentage error using:
Percentage Error = |Theoretical Value/Experimental Value − Theoretical Value| × 100%.
Phase 4: Deriving the SUVAT Equations
Using the base equation s = ut+2/1at2, derive all the other SUVAT equations. Show all mathematical manipulations step by step and clearly define each variable:
• s: Displacement (m)
• u: Initial velocity (m/s)
• v: Final velocity (m/s)
• a: Acceleration (m/s2)
• t: Time (s)
Write a short explanation of the physical significance of each equation in relation to both free-fall and upwards velocity trials.
Phase 5: Presentation and Reflection
Present your findings, including your calculated values of a, percentage errors, and an analysis of the results. Include the derivations of the SUVAT equations as part of your presentation. Reflect on the learning experience, discussing challenges faced and how you addressed them. Compare the complexities of analyzing free-fall motion with and without an initial upwards velocity. Discuss the impact of ignoring factors like air resistance and drag, and hypothesize how these might have affected your results.
Reflection and Discussion Questions
In your final report, address the following reflection and discussion questions. Ensure your responses demonstrate the use of appropriate mathematical language, accurate representations, logical struc- ture, and connections to real-world contexts:
1. Accuracy and Degree of Confidence: How accurate were your calculated acceleration values compared to the theoretical value of −9.81m/s2 ? Justify the degree of accuracy and explain any factors that may have contributed to discrepancies in your results.
2. Impact of Initial Velocity: How did introducing an initial upwards velocity affect the motion and your calculations? Describe the mathematical strategies used to model this situation and discuss how the solution aligns with the context of the experiment.
3. Simplifications and Realism: This project assumed ideal conditions, such as ignoring air resistance. How might these assumptions have impacted your results? Propose appropriate mathematical strategies to incorporate these factors in future experiments.
4. Connections to Real-Life Contexts: Relate the equations and methods used in this project to practical real-life applications, such as in sports, engineering, or space exploration. How do these connections enhance the significance and understanding of your work?
5. Evaluation of Experimental Design: Identify the relevant elements of your experimental design. What were its strengths and limitations? Discuss how the accuracy and reliability of the results could be improved through changes in the design or data collection process.
6. Collaboration and Communication: Reflect on how you and your partner used appro- priate mathematical language and representations to communicate ideas effectively. What strategies worked well, and what could be improved in future collaborative efforts?
7. Organizing and Presenting Findings: How did you ensure your report was logically organized and included accurate mathematical representations (e.g., graphs, tables)? Justify how your choices improved the clarity and coherence of your conclusions.
Submission Guidelines
The final report must:
• Be typed and formatted neatly, including:
– All calculations, graphs, pictures, and tables needed for the entire project.
– Answers to all reflection and discussion questions.
• Be submitted as a single PDF file with both students’ names clearly labeled on the front page.
• Be uploaded to the ManageBac Dropbox for this project no later than November 28th , 2024.
Reports submitted after the deadline will be subject to penalties as outlined in the course syllabus.
Criterion C: Communicating
This criterion evaluates your ability to communicate your mathematical work clearly and effectively, using appropriate language, accurate representations, and a logical structure.
Achievement Levels: Level 1-2:
• Uses limited mathematical language.
• Uses limited forms of mathematical representation to present information.
• Communicates through lines of reasoning that are difficult to interpret.
Level 3-4:
• Uses some appropriate mathematical language.
• Uses appropriate forms of mathematical representation to present information adequately.
• Communicates through lines of reasoning that are complete.
• Adequately organizes information using a logical structure.
Level 5-6:
• Usually uses appropriate mathematical language.
• Usually uses appropriate forms of mathematical representation to present information cor- rectly.
• Usually moves between different forms of mathematical representation.
• Communicates through lines of reasoning that are complete and coherent.
• Presents work that is usually organized using a logical structure.
Level 7-8:
• Consistently uses appropriate mathematical language.
• Consistently uses appropriate forms of mathematical representation to present information correctly.
• Moves effectively between different forms of mathematical representation.
• Communicates through lines of reasoning that are complete, coherent, and concise.
• Presents work that is consistently organized using a logical structure.
Criterion D: Applying Mathematics in Real-World Contexts
This criterion assesses the ability to apply mathematics to real-world contexts by identifying rel- evant elements, selecting and applying mathematical strategies, and evaluating the accuracy and relevance of solutions.
Achievement Levels: Level 1-2:
• Identifies some elements of the authentic real-world situation.
• Applies mathematical strategies to find a solution to the authentic real-world situation, with limited success.
Level 3-4:
• Identifies the relevant elements of the authentic real-world situation.
• Selects, with some success, adequate mathematical strategies to model the authentic real- world situation.
• Applies mathematical strategies to reach a solution to the authentic real-world situation.
• Discusses whether the solution makes sense in the context of the authentic real-world situa- tion.
Level 5-6:
• Identifies the relevant elements of the authentic real-world situation.
• Selects adequate mathematical strategies to model the authentic real-world situation.
• Applies the selected mathematical strategies to reach a valid solution to the authentic real- world situation.
• Explains the degree of accuracy of the solution.
• Explains whether the solution makes sense in the context of the authentic real-world situation.
Level 7-8:
• Identifies the relevant elements of the authentic real-world situation.
• Selects appropriate mathematical strategies to model the authentic real-world situation.
• Applies the selected mathematical strategies to reach a correct solution to the authentic real-world situation.
• Justifies the degree of accuracy of the solution.
• Justifies whether the solution makes sense in the context of the authentic real-world situation.