Lab 2 - Graphical Analysis of Kinematics

Lab 2 - Graphical Analysis of Kinematics Musa and Mado PCS211 Section 231 Manmeet Pal Singh Amir Haji 2 October, 2023

Introduction With the help of motion detectors and the video analysis program to examine the motion of the cart and projectiles, the main goal of this lab is to comprehend the graphical and mathematical relationship between the motion of an object's acceleration using velocity- time graphs. The relationships between an object's position, velocity, and acceleration ought to be understood by the end of the lab. Additionally, two distinct methods can be used to capture the cart's travel, and one video can be used to assess the projectile's motion. PICK ONE The goal of this experiment is to investigate how velocity and location of an item in motion behave with respect to time. To foresee and predict an object's behavior, we look at how it moves. It is necessary to understand how motion changes over time for this experiment. This lab will conduct three different sorts of experiments to investigate the motion behavior of carts and projectiles. By performing calculations based on the provided graphs, values, and reasons, we can eventually understand the graphical and algebraic depiction of motion. Theory The principles you need to know are velocity, acceleration, and motion under constant acceleration. To begin with, one needs to understand the relationship of an object’s acceleration using a velocity-time graph. The main formulas used in this lab during the procedure were as follows: a = Δv/Δt Δx = vt vf = vi + aΔt Δx = ½(vf + vi)t vf^2 = vi^2 + 2aΔx

Procedure - The angle we used in this lab was 8.00° ± 0.5 - The initial position in this lab was 20 cm ± 0.5 Part 1: - Adjust the Jack by placing the angle finder on the ramp and adjusting the angle using the screw mechanism on the stem. - Connect the Motion detector to the LabQuest device. - Set the car on the ramp no closer than 15 cm from the motion detector and hold it there - Launch the Graphical Analysis (GA) app on your computer - When ready, click the “Collect” button on the GA app and remove your hand from the car as quick as possible - Once the car reaches the bottom, click the “collect” button again to stop collecting data. - 2 graphs should appear on the GA app. A position-time graph and a velocity-time graph - Highlight the graph section from the point you let go of the car up until it hits the bottom of the Jack for the first time. - Click the data options button and click the “best fit” button. For the Position graph, click quadratic and for the velocity graph, click linear. - Record the equations for position and velocity - Click “rename” on the run option and name it. - Then, click “Save As” and name it in order to save the data from your run. - Repeat this 3 more times for a total of 4 runs. Part 2: - Place the Cart at the bottom of the ramp on the Jack - Practice pushing the car gently up the ramp so that it goes up and down without hitting the motion sensor - When ready, press “Collect” and push the car up the ramp and let it come back down - Press “Collect” to stop the collection of data - Repeat the Steps from part 1 for obtaining best fit lines and data from the best fit lines for position-time and velocity-time graphs Part 3: - Download the following files: PCS211_KinematicsVideo.trk & ball_bouncing_across_stage.mov from D2L - Unzip the Ball bouncing video - In the Tracker app on the lab PC, open the .trk file you downloaded - Click file > Open > choose file and open the .mov file

- Familiarize yourself with the controls of the Tracker app - Play the video first in normal speed - Then, replay the video frame by frame up until the point where the ball hits the ground for the first time - Click the “Track” button and select “bouncing ball” - Shift click on the ball from the moment it hits the ground the first time to the moment it does so the second time - Use the following steps to plot the data curve * Right click the x(m) vs t(s) and select analyze * Select curve fitter > line * Record the values * Repeat the steps for your y-data but use parabolic fit instead of line * Record data values for both graphs Results and calculations: Analysis 1, 2 and 3: Data obtained from cart coming downwards: Trial Angle Initial Position Slope Equation 1 8.00° ± 0.5 20 cm ± 0.5 m =1.362 v(t) = 1.362t + 0.3488 2 8.00° ± 0.5 20 cm ± 0.5 m = 1.292 v(t) = 1.292t + 0.5358 3 8.00° ± 0.5 20 cm ± 0.5 m = 1.31 v(t) = 1.31t + 0.3275 4 8.00° ± 0.5 20 cm ± 0.5 m = 1.292 v(t) = 1.292t + 0.3734 Data obtained from cart pushed up then coming down: Trial Angle Initial Position Slope Equation 1 8.00° ± 0.5 0 cm ± 0.5 m = 1.344 v(t) = 1.344t - 0.6624 2 8.00° ± 0.5 0 cm ± 0.5 m = 1.286 v(t) = 1.286t - 0.601 3 8.00° ± 0.5 0 cm ± 0.5 m = 1.313 v(t) = 1.313t - 0.58

Discussions and conclusion: Discussion: It can be seen that the magnitudes of acceleration differ when comparing the accelerations of the ball bouncing and the cart travelling down the ramp. Even while the ball and the cart's acceleration may come from gravity, their differences may result from the fact that an object's acceleration also depends on its mass. Since gravity is a constant, the force acting on both objects is directly proportional to their respective masses. As a result, the force pushing on the object grows along with its mass. This causes the object's rate of acceleration to rise.

Conclusion: Our findings allow us to draw the conclusion that downward acceleration has visible variation when it is studied and/or estimated. This is demonstrated in part one when we let an automobile accelerate freely down a ramp. However, some runs were faster than others, and this could be due to a variety of outside variables. For instance, we did not pull our hand away from the car swiftly after clicking "Collect" each time, changing the average speed and consequently the acceleration. Additionally, we found that vehicles that roll down the ramp freely accelerate more quickly than those that are initially propelled up the ramp and then rolled back down. The ball's acceleration shouldn't be comparable to the ground exerting an external force called "g" with a value of -9.81 m/s2. Additionally, as the ball returns to the sky, it is defying gravity and accelerating faster. So it makes sense to say that the acceleration is accurate. It's crucial to pay attention to the outside influences that prohibit an object from falling freely. The ground in stage 3 acted as an external push on the ball, keeping it from accelerating in freefall throughout the procedure. Assuming that friction is neglected for the purposes of this lab experiment, the force preventing the cart from falling to the ground was the normal force applied to it by the inclined ramp. Wrap-up questions: Q1. A cart's acceleration during its descent down the ramp and its ascent and descent back up the slope were inconsistent with one another. Additionally, since the uncertainty in the first experiment was noticeably higher than in the second, the results were less accurate. Q2. Despite being very near to one another, the theoretical forecast and the experimental result did not agree. This is due to the experimental value not being inside the expected uncertainty's range. This difference could be the consequence of a variety of things, including human mistake in choosing the precise angle at which each experiment was conducted, machine error, unstable equipment, and the force and position at which the cart was released. Inconsistency between the two measurements can also be caused by the anticipated computation not taking the force of friction into account. Despite being very near to one another, the theoretical forecast and the experimental result did not agree. Q4. The acceleration coming down the ramp is altered for a factor of sinθ since it will be the component of gravity acting along the cart to accelerate it. As a result, the acceleration on the cart moving down the ramp and the acceleration on the ball are not the same. The ramp's angle