Physics I - Motion 2 Lab Report

Physics I - Motion 2 Lab Student: Havana Perez Partner: Ariella Pevzner Section 024 Lab Date: 09/28/23 Due date: 10/05/23

Objective: The objective of this experiment is to further explore the relationships between position ( x ), velocity ( v ), acceleration ( a ), and time ( t ) of an object moving in one dimension using the motion sensor. Note that in this experiment there is the assumption that acceleration ( a ) is a constant. Another aim is to successfully learn how to utilize air tracks to measure the change in position when there is an absence of friction and to practice reading the position, velocity, and acceleration vs. time graphs to observe the physics concepts learned in class in real life. Description: The motion sensor apparatus is set up on a metal rod that is clamped onto a table. This sensor detects the distance an object is from the motion sensor by using sound waves and the sound wave reflections and measures the time it takes for each pulse to be reflected. We utilize a tilted air track + 2 gliders of different mass and index cards; the air track has a blower pumping air through small holes, moving the gliders with low friction. Theory: An object’s motion can be described by its position relative to a reference point, the speed and direction with which it is traveling, and the changes to its rate of motion. In other words, the position, velocity, and acceleration of an object are necessary to accurately describe the object’s motion. Utilizing the given kinematic equations in the lab, the position of the gliders is determined while the velocity and acceleration of the gliders are calculated by Capstone. The position equation is derived by measuring the change in an object’s position over time resulting in a quadratic formula as the observed object is moving at constant acceleration, a , away from the starting point, X 0 , where the initial speed was V 0 ; hence, the position graph will be a parabola. The velocity equation can be derived by taking the derivative of the position equation as velocity is the rate of change of the object’s position concerning time; therefore, the velocity graph will be linear. Similarly, the acceleration equation can be derived by taking the derivative of the velocity equation since acceleration is the rate of change of an object’s velocity concerning time; as such, the acceleration graph will be a horizontal line. Note that the position equation has a quadratic form because the acceleration is constant; this is expected since the only acceleration in projectile motion is governed by gravity (9.8 m s 2 ) which is always constant. Procedure: Part I 1. Set up Capstone and connected the interface to the computer a. Plugged the motion sensor into two digital ports

Data And Calculations:  Length of air track = 1.14 m  Length of blocks = 0.254 m  Thickness of big block = 0.034 m  Thickness of small block = 0.020 m  θ of small block = tan (0.020 m / 0.254 m) = 4.502°  θ of big block = tan (0.034 m / 0.254 m) = 7.624° Table 1. Notes and formulas used in this experiment: A. small glider + small block B. big glider + small block C. big glider + big block D. small glider + big block Mean acceleration (a) = 0.01690 a = g × sin θ 0.0169 = g × sin (4.502°) g = 0.2153 m/s 2 % error: ( 9.8 − 0.2153 ) 9.8 × 100% = 97.8% error Run 1 = 0.0121 Run 2 = 0.0189 Run 3 = 0.0093 Run 4 = 0.0111 Run 5 = 0.0095 Mean acceleration (a) = 0.01218 a = g × sin θ 0.01218 = g × sin (4.502°) g = 0.1552 m/s 2 % error: ( 9.8 − 0.1552 ) 9.8 × 100% = 98.4% error Run 1 = 0.2439 Run 2 = 0.2466 Run 3 = 0.2428 Run 4 = 0.2521 Run 5 = 0.2535 Mean acceleration (a) = 0.24778 a = g × sin θ 0.24778 = g × sin (7.624°) g = 1.867 m/s 2 % error: ( 9.8 − 1.867 ) 9.8 × 100% = 80.9% error Run 1 = 0.2351 Run 2 = 0.2411 Run 3 = 0.2347 Run 4 = 0.2345 Run 5 = 0.2363 Mean acceleration (a) = 0.24638 a = g × sin θ 0.0169 = g × sin (7.624°) g = 1.857 m/s 2 % error: ( 9.8 − 1.86 ) 9.8 × 100% = 81.0% error  all acceleration values have units of m/s 2  % error = ( actual − measured ) actual × 100%  a = g × sin θ  Actual g from lab = 9.8 m/s 2

Error Analysis : We know that the gravitational constant is 9.8 m/s^2, providing us with a benchmark to compare our theoretical gravitational constants and determine the percentage error in the first part of this experiment. However, our data significantly deviated from the expected value of 9.8. Overall, the experiment presented more error for the smaller block, for both the small glider and the big glider. The percent errors for A (small glider, small block) and B (big glider, small block) were 97.8% and 98.4%, respectively, whereas for C (big glider, big block) and D (small glider, big block) were 80.9% and 81.0% respectively. The most likely explanation for this discrepancy is the presence of friction on the Air Track, which interfered with acceleration. The angles created by the blocks were not sufficient to render the effects of gravity entirely negligible, especially for the smaller block, where our percentage error was nearly 100%. For instance, we were responsible for initiating and stopping data collection, as well as releasing the glider at the beginning of the Air Track, potentially with variations in force between trials. We also faced challenges in measuring the sides of the block, as a meter stick was not the most precise tool for this purpose. However, this should not have significantly impacted the recorded value of zero acceleration. Furthermore, there's a possibility that the paper clips affixed to the gliders scraped against the Air Track, preventing us from achieving the expected results. Although we did notice this issue in a few trials, we promptly resolved it and excluded those runs from our analysis. In light of the above, we aimed to calculate 'g' from our measured acceleration and the angle at which we elevated the Air Track. As you may recall from class, 'g' is approximately 9.8 m/s^2, a well-established value obtained from previous experiments. Comparing our calculated 'g' with this accepted value, we found a significant disparity. To address this discrepancy, we need to revisit our calculations and provide a thorough explanation for the divergence between our measurements and the expected 'g' value. Lab Questions: a. How will acceleration change when we double the mass of the glider? Explain. - According to Newton’s 2 nd law of motion, the acceleration of the glider is inversely proportional to the mass of the glider. As such, doubling the glider mass means that the acceleration will be 0.5X of the original, assuming that the applied force remains the same.

o The acceleration graph presents as a positive horizontal line because the velocity increases at a constant rate, reflecting a uniform force's impact (in this case, a constant and positive velocity rate). These relationships are mathematically confirmed using the kinematic equations, and it's possible to derive each vector from the others through differentiation and integration. d. What are the correlations of all the graphs? - Essentially, the slope of the prior graph determines the data points of the next graph. The position graph provides displacement data, and taking its derivative yields the velocity graph, which in turn provides information on how velocity changes over time. Finally, taking the derivative of the velocity graph gives the acceleration graph, which represents the constant rate of change in velocity due to the constant force (gravity) acting on the object. e. The velocity curve is noisier than the position curve, and the acceleration curve is noisier than the velocity curve. Why? - The aforementioned observations stem from non-ideal testing conditions, where all variables except position are uncontrolled. In simpler terms, external factors come into play during the experiment. One reason for increased graph noise is the variability in start and stop times, resulting in minor discrepancies in recorded time intervals that affect velocity and acceleration calculations. Furthermore, the glider experiences minimal friction, even on the air track, causing variations in its motion that the Capstone software doesn't account for. There may also be unevenness in the air track and air resistance, both of which, like friction, disrupt the glider's motion. Another potential issue is the calibration of the equipment, which can impact the glider's motion. Since calibration is a human-driven process, errors and variations likely arise from human factors. Finally, these distortions are magnified as Capstone collects position data and extrapolates velocity and acceleration data. f. For the velocity plot, the acceleration is given by the slope (m). Why? - Acceleration is the change in velocity over time, and the slope of velocity represents the rate at which velocity changes at a given point in time. Since the velocity graph has a constant slope, producing a positive linear graph, indicates constant acceleration. The slope, represented by 'm' in the velocity graph, corresponds to the magnitude of acceleration because it quantifies how much the velocity changes per unit of time. This is the fundamental definition of acceleration. This can also be depicted mathematically by the following equation: a = dv dt

g. The average acceleration for the data in the box is given by the mean of (y). Compare this to the slope. - The average acceleration for the run was 0.016 m/s², while the slope of the velocity graph for the same run measured 0.015 ± 9.0 x 10 . These values are ⁻⁵ quite close to each other, although they ideally should be identical under ideal conditions. The proximity of these values suggests that the experiment was well-conducted. Any differences observed are likely attributable to minor sources of error and inherent variability. h. What is the experiential n(power)? What does the theory say it should be? Why is there a difference? - The exponent in the position graph's power function was 1.92 m/s², differing from the theoretical value of 2 m/s². The theoretical value suggests that the expected relationship between acceleration and time is quadratic, resulting in a parabolic curve when plotting position against time. The difference between the experimental and theoretical values can be attributed to several factors, including systematic measurement errors, variations in start and stop times, friction, air resistance, setup mistakes, and equipment limitations. When fitting a power function to experimental data, it produces the best mathematical model based on the observed relationship, which is influenced by the aforementioned factors. It's important to note that practical experiments rarely match theoretical concepts and predictions precisely because experiments aren't conducted in a vacuum where all other variables are controlled. i. When the same θ is used for the different gliders is there a difference in acceleration? Why? - When the same angle (θ) is applied to different gliders on an inclined plane, assuming all other factors remain constant, there should be no variation in acceleration. This is because the acceleration of an object moving down an inclined plane is primarily determined by two constant factors: gravity and the angle of incline

Conclusion: In this experiment, we investigated the relationships between position, velocity, acceleration, and time, specifically when acceleration remains constant. By analyzing these relationships through the slopes and curves of their respective graphs, we concluded that constant acceleration is directly proportional to the value of theta, with its dependency on mass being minimal, for the most part. In the second part of the experiment, we observed that the peaks on the position graph, corresponding to the moments when the glider collided with the end of the air track, coincided with the instances when the velocity graph crossed the x-axis. Furthermore, the subsequent sections of the graph, following each collision, exhibited decreasing magnitudes. This decrease occurred because the glider's position, velocity, and acceleration changed; they did not revert to their original conditions after each collision. Notably, we did not account for friction in this experiment, as we assumed its insignificance. This assumption may have contributed to the relatively high percentage of errors in our results. Nonetheless, it's worth noting that the experiment confirmed the theoretical concepts, as the graphs generated by Capstone software aligned with our expectations Human error can be present because it was required to click on the stop button once the glider hit the end of the track. That could have affected the way each of the graphs

would then be presented. Another possibility for error could be due to the clips placed on the index card and the glider that could have slowed down the glider which would also affect how the graphs would appear. Conclusion: In conclusion, the data collected does correspond with the results that should have been

position curve were when the glider hit the end of the track, while the valleys were when the glider went backward and hit the end of the track again. In the velocity graph when the line crossed the origin, was when the glider hit the end of the track had a velocity of zero in that instance, causing it to slow down as the glider traveled back and forth.

Each part of the graph after the collision has a smaller magnitude than the previous one since the velocity was constant, and the acceleration was along the origin of the graph. The two negative peaks in the graph were when the glider collided with the end of the track