Project 1: Projectile Motion Introduction: The human brain has evolved since the time of hunters and gatherers to very effectively predict the motion of objects falling through the air. That intuitive grasp of projectile motion is what allows us to catch a fly ball, toss a fish across a market, and launch ourselves off a high perch into cold water. Going beyond what our intuition can achieve requires making a mathematical model for projectile motion, allowing us to calculate the trajectory of an object falling through the air with some precision. Experiment: In this experiment you will use a spring loaded ball launcher to investigate the predictive power of the kinematic equations for projectile motion. You will begin by making a detailed description of the launcher, then design an experimental procedure for characterizing the velocity of the ball when leaving the ball launcher, then use that result to predict the range of the ball launcher when fired at a particular angle. Finally, you will test this prediction and characterize how close you were to your prediction. Below is a checklist of the items that you are required to submit at the end of this project. To get a sense of the goals of the project, look over the entire list before you get started. Then work through the list sequentially. Submit these items-numbered and in correct order-as your instructor advises. You are encouraged to ask questions and discuss with your instructor and classmates. In matters of uncertainty estimation and data analysis, you must apply the methods described in the appendix! Safety note: Never look directly down the barrel of a loaded ball launcher. Whenever firing the ball launcher, ensure that the way is clear and that everyone around you is aware of what you are doing. Items Due: (Week 1: Questions 1-7, Week 2: Questions 8-13) 1. Take a photograph of the ball launcher apparatus. Label and describe the function of the following parts: a. Power settings b. Plunger c. Plumb bob and protractor d. Initial position indicator 2. Draw a schematic representation of the projectile's expected trajectory labeling the important quantities such as initial angle, initial velocity vector, initial height, and range. 3. Fill out the following table of kinematic quantities for the general case drawn in the previous question. Use only symbols or question marks for unknown quantities where appropriate. 15y= X = V V x, 0 y, 0 V V x , f y . f a = a = t = 4. Describe an experimental procedure where you will determine the initial velocity of the projectile launcher at one of the three power settings (Instructor's choice). While doing this, consider the following: a. Projectile motion from a raised position at an arbitrary firing angle can be complicated. Can you choose a simplified case which makes obtaining the magnitude of the initial velocity easier to obtain? b. How will you quantify the uncertainty in your measurements? Which uncertainty do you expect to be larger: the measurement itself or the randomness of shots? 5. Derive an equation for the initial velocity of the projectile given your experimental procedure. 6. Prepare a spreadsheet for data collection and analysis in Google Sheets or Excel. Use the equation functionality in the sheet to prepare calculation cells for averages, standard deviations, and calculated values using the relevant measurements. (Check your spreadsheet with your instructor - organization matters) 7. Do a test run of your experiment to: a. Make sure your measurement method is practical. b. Consider how each member of your group can contribute to conducting the experiment efficiently and safely. c. Ensure that, to the best of your ability, you can control systematic factors that affect the projectile's range. 8. Carefully measure the initial height of your projectile. Describe the proper use of a plumb bob and the initial height indicator. 169. Measure the range of your launcher over many trials to determine the initial velocity of the launcher with the appropriate uncertainty. 10. The general range equation for a projectile fired from an elevated position at an arbitrary angle is given by _Vo cos R = - (vo sin 0 + vg sin2 0 + 2gh ) Ask your instructor for an angle to fire from and calculate the expected range. 11. Mark the position for that range on the ground. Set the launcher at the appropriate angle and test your prediction. Is your measurement consistent with your prediction? (Carefully consider what is required to make a "consistency" argument.) 12. What systematic effects may have affected your experiment here? (If you think "human error" was a factor, then you should retake your data until that is no longer the case.) 13. Challenge Problem: Use Excel to calculate the range for every angle between 0 and 90 degrees. Plot the resulting ranges as a function of angle then estimate the angle for which the range is maximum. Does the angle for maximum range depend on the initial velocity and/or the initial height? Discuss.A Projectile Launcher 1 Trial 1 2 3 4 5 6 4 8 9 10 Launch Angle 11 Plunger Setting 12 13 |Initial horizontal position 14 |Initial vertical position 15 |Final horizontal position 16 Final vertical position 17 |Initial horizontal velocity 18 |Initial vertical velocity 19 |Final horizontal velocity 20 Final vertical velocity 21 Acceleration horizontal 22 Acceleration vertical O bk WN = Ax (mm) Xo (M) Yo (M) X (m) (m) m/s m/s m/s m/s m/s*2 m/s*2 3050 3014 3055 3042 3035 3042 Ax (m) 3.050 3.014 3.055 3.042 3.035 3.042 Trial1 0 1.115 3.050 0 -9.81 D Trial 2 0 1.115 3.014 0 0 0 -9.81 Ax Mean (m) 3.040 Trial 3 0 1.115 3.055 0 -9.81 F Trial 4 0 1.115 3.042 0 0 0 -9.81 G Trial 5 0 1.115 3.035 0 0 0 -9.81 H Trial 6 0 1.115 3.042 0 0 0 -9.81 Mean STDEV