AP02 lab manual

Old Dominion University Physics 231N, Online Lab 1 OLD DOMINION UNIVERSITY PHYS 231 – Asynchronous Online AP02 – PROJECTILE MOTION Submitted By: 1. Brandon Davis 2. Raphael Afrim 3. Submitted on Date 10/01/2023

2 Experiment AP02: Projectile Motion Projectile Motion Experiment AP02 Objective  Investigate the x and y-components of a projectile’s path  Determine the acceleration due to gravity of a projectile Materials Spreadsheet Program (e.g., Excel) Theory A projectile is an object upon which the only force acting is gravity. There are a variety of examples of projectiles. An object dropped from rest is a projectile. An object that is thrown vertically upward is also a projectile. And an object which is thrown at an angle is also a projectile. A projectile is any object that once projected continues in motion by its own inertia and is influenced only by the downward force of gravity. The kinematic equations are a set of four equations that can be utilized to predict unknown information about an object's motion. The equations can be utilized for any motion that can be described as having a constant acceleration. They can never be used over any time period during which the acceleration is changing. Each of the kinematic equations include four variables. If the values of three of the four variables are known, then the value of the fourth variable can be calculated. In this manner, the kinematic equations provide a useful means of predicting information about an object's motion. The four kinematic equations that describe an object's motion are: x = x i + v i t + 1 2 at 2 (1) x = v i + v f 2 t (2) v f = v i + at (3) v f 2 = v i 2 + 2 ad (4) where x is the objects displacement, t is time, a is acceleration, v i is the initial velocity, and v f is the final velocity. Procedure Part A: Distance Our overall goal of this experiment is to help visualize that a projectile has motion is two separate directions: horizontal and vertical, or x and y . In each direction, the motion is either constant, or it is affected by gravity. Separating an object’s motion into its separate components can be difficult to get used to. We’re also going to again study our favorite constant in physics, the acceleration due to gravity, g.

4 Experiment AP02: Projectile Motion I. There are three sections of data. They are labeled as Initial Launch, 1 st Bounce, and 2 nd Bounce . II. In each section there are three columns: time, X, and Y. The time is measured in seconds and the time from one row to the next is the time between frames of the video. Again, these videos were recorded at 240 fps (1/240 = 0.00417 seconds between frames). X and Y are the position data recorded in meters. III. There is a blank column in each section where you will calculate the y-velocities. Use Excel to create graphs of x-position vs. time and y-position vs. time. (Time on the x-axis and position on the y-axis). Include in these graphs all three sections of the flight (initial, 1 st , and 2 nd bounce). Do not paste in your graphs to this report just yet. We’ll add some more information before pasting them in. Examples Note : When creating graphs, and we mention to create a graph of something vs. something else (e.g., displacement vs. time) this is always written as “y-axis data” vs. “x-axis data” (aka - vertical

Old Dominion University Physics 231N, Online Lab 5 axis vs. horizontal axis). It is standard nomenclature to write it this way and has been for a long time. Do not assume you can swap the axes! One common problem if you swap the axes (e.g., something else vs. something) is that we typically want to know the slope of the graphed data. Swapping the axes gives you the inverse slope (1/slope) of what we want! As you’re adding data, for each section, add it as a separate Series. Be sure to label the axes, add a title for the graph, and a legend. Examples are shown above from a launch at a 0° angle. Each Series of data is a different color, which Excel colors automatically when adding as a separate series. (There’s a reason we’re doing them as separate Series, which will become apparent later.) X-Direction In Excel, if you right-click on a data point within a graph, you can choose an option called Add Trendline. For the x-direction, add a trendline to the initial launch phase of the ball’s path. a) Once you right-click, choose Add Trendline b) Choose a Linear fit c) Check the box for D isplay Equation on Chart . An equation will appear on the graph. Compare the equation that Excel produces with the kinematic equation for an object in motion. What is the x-velocity of the ball in the initial launch? Record your value in Data Table 1. Excel: y = mx + b Kinematic Equation: x = v i t + x i (That kinematic equation is a variation of x = x i + v i t + 1 2 at 2 because a = 0 m/s 2 in the x-direction) Repeat your trendline steps and add a trendline (and display the equations) for each bounce of the ball. This can tell us if the ball slowed down or picked up speed as it bounced along. Y-Direction For the y-direction, add a trendline to your data. a) This time, choose a polynomial fit. b) Make sure the order is 2. (A second order polynomial is a parabola.) c) Check the box for Display Equation on Chart . Compare the equation that Excel produces with the kinematic equation for an object in motion. From the trendline equation, what is the acceleration of the ball? Record your value in Data Table 2. Do this for each bounce as well. Excel: y = a x 2 + bx + c Kinematic Equation: y = 1 2 gt 2 + v i t + y i

Old Dominion University Physics 231N, Online Lab 7 (The jitter in Video Physics’ tracking is apparent here since the data don’t fall on nice, straight lines.) You’ll quickly notice there are three, discrete sections to this graph. Add a trendline to each set of data on this graph, choose a Linear fit , and display the equation on the chart. Compare the equation that Excel produces with the kinematic equation for an object in motion. Excel: y = mx + b Kinematic Equation: v = ¿ + v 0 What is the acceleration of the ball in set of data? Is it the same or similar? Record your values in Data Table 3. Error Analysis As we’ve seen in several experiments now, one key to science is understanding your confidence in your answer. What kind of error is associated with your calculations? How accurate are you? How precise? Next, we want to know how precise we were in our experiment. There are multiple, and statistically robust, ways to do this. We could launch an object over and over, calculating g many times, and then analyzing the distribution of that data. (This would be similar to we did in experiments 1 and 2.) We could run a Linear Regression on the velocity data using Excel, which is similar to, but more robust than, the basic trendline we added. Instead, we will again introduce a slightly different method from what we have done in previous experiments. You already know that repeated measurements allow you to not only obtain a better idea of the actual value, but also enable you to characterize the uncertainty of your measurement. In

8 Experiment AP02: Projectile Motion experiments 1 and 2 we had a large number of trials, N , but this time the number of trials is small (less than 10). In this case we use the formulae below: Mean ( g avg ) The average of all values of g (the “best” value of g ) g avg = g 1 + g 2 + … + g N N Range ( R ) The “spread” of the data set. This is the difference between the maximum and minimum values of g. R = g max − g min Uncertainty in a measurement ( ) σ Uncertainty in a single measurement of g . You determine this uncertainty by making multiple measurements. You know from your data that g lies somewhere between g max and g min . σ = R 2 = g max − g min 2 Uncertainty in the Mean ( σ avg ) Uncertainty in the mean value of g . The actual value of g will be somewhere in the neighborhood around g avg . This neighborhood of values is the uncertainty in the mean. σ avg = σ √ N = R 2 √ N Measured Value ( g ) The final reported value of a measurement of g contains both the average value and the uncertainty in the mean . g = g avg ±σ avg The average value becomes more and more precise as the number of measurements increases. Although the uncertainty of any single measurement is always , the uncertainty in the mean, σ σ avg , becomes smaller (by a factor of 1 / √ N ) as more measurements are made. Using the information in the table above, calculate your average value of g from all six values you have found during this lab and calculate the uncertainty. Finally, calculate the percent error of your overall average value and the accepted value of g . The most important part of this lab will be to report your findings. Make sure you report g = g avg ±σ avg and the percent error. This is your final value of g, your precision, and your accuracy all stated together.

10 Experiment AP02: Projectile Motion Data Table 4 Acceleration due to Gravity Accepted Acceleration Due to Gravity 9.81 m/s 2 45° Angle Average Acceleration due to Gravity g 10.00 m/s 2 Uncertainty in the Mean σ avg 0.04 m/s 2 % Error (Your Average vs. the accepted value) 1.93 % What to Turn In  Full Lab Report o Only grading the “ Data & Data Analysis” section. This section is worth 60 points.  Data Tables 1-4  Graph of x-position vs time with all three bounces graphed, trendlines added, and equations  Graph of y-position vs time with all three bounces graphed, trendlines added, and equations  Graph of y-velocity vs time with all three bounces graphed, trendlines added, and equations o Introduction and Conclusion sections are worth 20 points each. These are graded strictly as 0, 10, or 20 points with notes provided for improvements. o On Canvas, in the Content section, see the file “Laboratory Report Guideline” for information on considerations when writing a lab report. o Do not turn in the questions listed below. However, these questions can be used as a guideline for the types of things to consider when writing your report. However, do NOT directly rewrite and answer them. Questions to Consider For your x-position vs time graph, what shape does the data take? Is it linear? Exponential? Is this what you expect for the x-component of a projectile? Which data is the direction the ball travels forwards? X or Y? How can you tell? Sure, you saw the origin set in the Video Physics app, but how does the data support this? Does the ball slow down, speed up, or maintain its x-velocity throughout the video? Compare the three velocities you have and discuss.

Old Dominion University Physics 231N, Online Lab 11 Why is the y-velocity discontinuous and abruptly jump from negative to positive? What is going on here? What is the acceleration of the ball in set of data? Is it the same or similar? Record your values in Data Table 3.