PHYS221+Lab+2+Report+Template+-+One+Dimensional+Kinematics copy (1)

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2024S 1 of 14 PHYS221 Lab Report 2 – One Dimensional Motion and Acceleration on an Inclined Plane Section 21 – Gabriel Coffee/Yitong Xu Cara Silverman, Dylan Hirshorn February 19, 2024 Introduction This experiment is important because through using the apparatus ’ we were able to visually learn a car’s motion in relation to position, velocity, and acceleration. Once these values are collected, we created graphs and visual representations of the relationships between gravity and angles. After the experiment and data values were collected, statistical analysis was performed to determine the percentage error of all relevant data. Procedure Apparatuses used in the experiment The procedure followed was the manual “PHYS2 21 – Lab 2: One Dimensional Motion and Acceleration on an Inclined Plane, Spring 2024 ” .

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 2 of 14 Data and Analysis Part A: Smart Cart Motion Sensor Figure 1. Position vs. Time graph when cart is slowly pushed to the right The slope (velocity) of the position vs the time graph starts at 0 and decreases as it travels right and plateaus when the cart stops moving. This plateau occurs because the cart has a constant position. The velocity graph increases negatively at the beginning because we pushed the cart, which caused the speed to increase to the right then plateaus when the speed is constant. The acceleration is negative because the cart is pushed at the beginning to the right, this is reflected by the sharp dip downward on the graph, then plateaus as it is traveling at a constant velocity then stops. In both cases, there is no change in velocity.

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 3 of 14 Figure 2. Position vs. Time graph when cart is slowly pushed in the other direction (to the left) The slope (velocity) of the Position vs Time Graph increases at the beginning, because we pushed the cart. Its slope begins to level out afterward signifying a constant velocity. The velocity graph increased at the beginning when we pushed the car, and the slope slowly decreased as the car's speed decreased (somewhat of a plateau). The acceleration graph only increases at the beginning because the car was pushed, thus changing the velocity. When no more force was applied, the acceleration is 0. Figure 3. Position vs. Time with a horizontal slope (no slope) To create a horizontal line on the graph the cart will not be moved, the acceleration and velocity graphs will also show a horizontal slope because there is no movement (change in direction or speed) by the cart. Thus, all slopes remain 0.

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Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 4 of 14 Figure 4. Position vs time graph with straight upward slope To create a straight upward slope on the Position vs Time graph, the crat would have to move at a constant slope from left to right. The velocity graph shows an increase in velocity in the beginning, then decreases into a horizontal line. The acceleration graph also shows an increase at the beginning for a moment then remains at 0 as the velocity was fairly constant. Part B: Matching a movement Plot Set 1. The cart was moved to 60 cm, the middle of the track. We slowly moved the cart from the middle toward the left. This created a shallow, consistent, negative slope, and did not create much change in velocity or acceleration.

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 5 of 14 Plot Set 2. To create these 3 graphs, we set up blocks on the right side and pushed the cart up the inclination, then let it fall back to its starting position. The position graph is pushed up the ramp creating the shallow slope upward, then slows at the top (peak of graph) and rolls back down the ramp (end of slope decreasing steadily). The velocity increases sharply when we push it and decreases steadily when it rolls back to the left. Acceleration was steepest at the beginning when we pushed the cart forward to the right, then decreased when it rolled back down to the left. Plot Set 3. We pushed the cart slowly up the inclination to the right and pulled it faster back down to the left. The position slope increased steadily then decreased sharply. The velocity slope increased only slightly when the cart was pushed to the right. Both acceleration and velocity slopes decreased sharply when the cart was pulled back to the left.

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 6 of 14 Part C: Motion with constant acceleration Plots predicting how motion will look: Position, velocity, acceleration

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Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 7 of 14 Run 1 cart on one block Run 2 Cart on two blocks

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 8 of 14 Run 3 Cart on three blocks Run 4 Cart on four blocks

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 9 of 14 Run 5 Cart with one weight on one block Run 6 Cart with two weights on one bloc k The physical meaning of x,y, and m on these graphs are as follows: x represents the progression of time, y represents the position, velocity, or acceleration of the cart as it rolls to the left and bounces back toward the right, creating curves. The m of each graph represents the slope. The slope of the position is viewed as velocity of the cart and the slope of velocity is its acceleration.

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Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 10 of 14 Cart Rolling down an Inclined Plane Run Number Bounce Number Total Mass of Cart (kg) ± ?? kg Height of Blocks (m) ± ?? m Length Between Track Legs (m) ± ?? m sinθ Acceleration of the Cart (m/s2) Error in Acceleration of the Cart (m/s2) 1 1 0.271 kg 0.016 ± 0.001 1.02 ± 0.01 0.016 -0.160 ± 0.0011 2 1 0.271 kg 0.031 ± 0.001 1.02 ± 0.01 0.030 -0.298 ± 0.0013 3 1 0.271 kg 0.046 ± 0.001 1.02 ± 0.01 0.045 -0.453 ± 0.0015 4 1 0.271 kg 0.062 ± 0.001 1.02 ± 0.01 0.061 -0.626 ± 0.0017 5 1 0.524 kg 0.016 ± 0.001 1.02 ± 0.01 0.016 -0.159 ± 0.0011 6 1 0.776 kg 0.016±0.001 1.02 ± 0.01 0.016 -0.158 ± 0.0010

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 11 of 14 Calculated Acceleration Due to Gravity Run Number Bounce Number Acceleration due to Gravity (m/s 2 ) 1 1 -10.0 2 1 -9.93 3 1 -10.1 4 1 -10.3 5 1 -9.94 6 1 -9.88 The angle of the track was calculated by using the length of the track (hypotenuse) and the height of the inclination due to the bricks (opposite) to trigonometrically find the angle (θ) via the function: sin (θ) = h/l —> sin ⁻¹ (h/l) = θ Run 1: sin ⁻¹ ( 0.016) = 0.917° Run 2: sin ⁻¹ ( 0.030) = 1.72° Run 3: sin ⁻¹ ( 0.045) = 2.56° Run 4: sin ⁻¹ ( 0.061) = 3.50° Run 5: sin ⁻¹ ( 0.016) = 0.917° Run 6: sin ⁻¹ ( 0.016) = 0.917°

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 12 of 14 The chart below shows the components used to determine whether the experimental value for acceleration due to gravity is in agreement with the “accepted value” for gravity. The “accepted value” is referred to as. The actual value . The equation used to find the propagated error (uncertainties): Some of the values from the difference between the accepted/actual value and the experimental value were less than or equal to the propagated error, meaning the values were in agreement. Acceleration vs Mass graph As shown in the graph, the mass of the cart did not significantly affect the acceleration of the cart. There is effectively no slope (horizontal slope) to indicate that there is no change in acceleration when there is change in mass. However, there is a slight positive slope, but it is practically unnoticeable. The relationship in the graph should show that mass and acceleration are inversely proportional (a ∝ 1/m)

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Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 13 of 14 Acceleration vs sin (θ) Graph According to the trendline equation, if the cart were placed on a horizontal track, there would be no inclination and thus no angle. Because sin(0) = 0, there would be virtually no acceleration (y = - 0.00942). If the track were vertical, meaning the cart would fall straight down, the track would be at angle of 90 degrees, where sin(90) = 1. Using this value in the trendline equation produces an acceleration of -10.4 m/s ², close to the force of gravity and the acceleration due to gravity. The value of the slope of the acceleration vs sin θ graph is -10.4 . As the angle increases (θ) the acceleration increases. The graph below evaluates whether the measured accelerations agree with one another Determining if experimental and actual gravity are in agreement The difference between 2 measured accelerations in each case is less than the calculated propagated error. Thus, the measured accelerations are in agreement. The error of the slope represents how much the observed (experimental) values deviate from the trendline. This measure of uncertainty gives us a reasonable range to evaluate whether or not the data are consistent and reliable. Because of the relationship of a/sin( θ) = gravity, we know that sin(θ) and acceleration are directly (proportionally) related, as seen from the slope of the graph. Y-Intercept The y- intercept in both the acceleration vs mass and acceleration vs sin(θ) graphs are irrelevant. In the acceleration vs Mass graph, when there is no mass (x=0), acceleration should also be 0 (y=0). However due to the data being experimental, this is slightly offset to where the y- intercept is not non- existent. Similarly, in the Acceleration vs Sin(θ) graph, when there is no angle (x= sin(0)=0), there should be no acceleration (there was no other force acting on the cart, e.g. a push).

Student Names PHYS221 – Lab 2 - One Dimensional Motion Date Performed 2024S 14 of 14 Uncertainty, R ² value, and MSE value The uncertainty of variables gives us a reasonable range in which the actual value could be. The R ² value tells us how consistent the data are, in relation to the trendline. The MSE value tells us the average deviation of the experimental value from the actual value, i.e. how accurate the data are. In this case, we have been looking at the different trends between acceleration, mass, and angle. While the accuracy as shown by the MSE value is important, there is an allowed range of inaccuracy as it is limited experimental data. The consistency of the data is what makes the trend relevant, so the R ² may be more useful in this situation. Cart in Contact with Bumper On the acceleration vs time graph, the acceleration was in the +x direction when the cart and bumper connect with the wall. It was increasing down the ramp. It then slowed down as it bounced and proceeded back up the ramp, then fell again and gained speed (change in velocity). The magnitude of the acceleration was largest at the first bounce with the largest angle. Conclusion: The data shows that as the angle increases the acceleration of the car also increases. Next, it is found that as the mass of the car increases at the same angle the acceleration remains unchanged. The mean acceleration due to gravity was found to be – 10.03 m/s ², this agrees with the accepted value of – 9.81 m/s ² because the difference between 2 measured accelerations in each case is less than the calculated propagated error. Error and uncertainty are important in an experiment to ensure measurement values are accurate to a reasonable degree, as compared to expected values, and to determine an experiment's validity and replicability.

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