lab_Projectile_Motion_S23(2)

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University of South Carolina *

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211

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Mechanical Engineering

Date

Oct 30, 2023

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In-Class Projectile Motion Demonstration/Activity Equipment: § projectile launcher, safety glasses, meter stick and tape measure § two photogates and one smart timer, strike plate

Introduction . All terrestrial projectile problems are analyzed with multi-dimensional kinematics. Examples include the motion of balls in virtually every sport, every ballistics and rocketry problem, and any other problem that involves position, velocity, and acceleration in more than one dimension. In this lab, we focus on the acceleration due to gravity alone (the effects of drag and non-inertial accelerations are neglected). Mini-lecture : Using the constant acceleration kinematics equations in two dimensions, derive the equations for range ( R ), time of flight ( T ), and maximum altitude ( H ) for a projectile launched with an initial speed v o from a height h at an arbitrary angle θ above the horizontal. Students should note these solutions in their lab books and may wish to add additional notes of the derivation as desired. Objectives of the In-Class Activity: This instructor-led lab explores the derivation of range and time-of-flight equations from two-dimensional kinematics and demonstrates measurements that verify the results: • how to find R and T theoretically • demonstrate an experiment to compare to theory • uncertainty calculations using Excel spreadsheet • comparison/discussion of predicted and measured R and T Activity: • Instructors will demonstrate the use of a spring-loaded projectile launcher and a timer, and measure parameters needed to determine a trajectory: initial speed v 0 , height h , and launch angle θ . • Students will predict values for R and T based on the derivation and measurements (using the mean values of v 0 , h , q , and assuming the acceleration due to gravity as 981 cm/s 2 ). All values should be recorded in the lab notebooks. • Instructors will use the spring-loaded projectile launcher and timer to measure the range R and time-of-flight T . • Instructors and students will compare the results to theory. o Uncertainty estimates are needed for making meaningful comparisons. • Instructors will demonstrate the use of a projectile spreadsheet. Changes can be made to the photo-gate separation, the photo-gate time, and the launch height. Values for initial speed, range, and time of flight will update for multiple angles. • Students will use the spreadsheet to explore the upper-lower bound uncertainty in the values of R and T , based on the known uncertainty of v 0 , height h , and launch angle θ . Together with the instructors, students will decide whether the experiment and the theory match one another.

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Challenge Questions: Not for credit but may be critical for understanding on exams or homework! 1. In general, how many solutions are there to the 2D kinematics equations for a given range and initial speed? 2. For a launch angle of ±90° (shooting a projectile straight up or down, respectively), what is the range of the projectile? 3. For a projectile launched at any angle that is not straight up or down, is the speed of the projectile when it lands larger than, less than, or the same as, the magnitude of either its horizontal or vertical velocities?

Studio 04 Deliverable – To Be Completed Outside of Studio This deliverable should be presented as the Analysis portion only of a lab report. Please refer to the Lab Reports document on Blackboard (in the General Resources for Studio folder) for information about how to write an Analysis. Due date: start of Studio 06 Important note: The experiment described here is NOT the experiment you did as the Studio 04 class activity! These experimental data below were acquired elsewhere – we are providing you with the data to analyze. Scenario: A projectile is launched at ground level and lands on the ground again downrange. The distance (range) from the launch point to the landing point is perfectly flat and horizontal. Identical projectiles are launched repeatedly at the same angle θ but with different initial speeds, and their ranges R and times-of-flight T are measured. A table of these initial speeds, ranges, and times of flight is given below. The goal is to use the data to empirically find the launch angle of the projectile and its uncertainty. Projectile V 0 (m/s) R (m) T (sec) 1 55 240 5.5 2 29 93 3.5 3 49 195 5.3 4 36 117 3.9 5 45 165 4.9 The uncertainties for these measurements are: • all speeds: ±2 m/s all ranges: ±5 m all times of flight: ±0.1 s a) Use your knowledge of 2-D kinematics to derive range ( R ) and time-of-flight ( T ) equations for this particular projectile problem, that depend on v 0 and θ . Do not show the derivation; instead, merely cite the appropriate equations as the first part of your analysis. b) In Excel, plot R vs v 0 T and determine the trendline slope and slope uncertainty. Make sure you understand why you are doing this (how does this particular plot help you to get the value of θ ?). From these data, calculate the value of θ and use the upper-lower bound method to find an uncertainty in θ . Your analysis must show the plot with error bars and trendline, and you must either show a calculation of your uncertainty, or your LINEST result. There is no need to explain the experiment in detail – just a sentence or two. Along with the instructions in a) and b) above, the analysis should: • Briefly explain the scenario. • Clearly mark the answers to the two tasks a) and b). • Be limited to 2 printed pages.