Ganesh Sivararamakrishnan_Lab2

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Dec 6, 2023

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Lab Report #2: Kinematics Praneel, katie, Mary Introduction In this experiment we will continue to learn on how to use the capstone software and our aim is to perform two new experiments. The first experiment is to measure the acceleration of gravity using a picket fence while dropping it through our photogate to capture the acceleration of the picket fence. In our second experiment using a motion sensor, we start to measure the motion in one direction of the ball over the speaker. We continue to repeat each experiment 6 times to ensure accurate results. The reason for this lab is to understand better the effect gravitational pull has on objects by measuring acceleration, and to understand motion in one direction by measuring the rising and falling of the ball. Experiment 1: Measuring the Acceleration of Gravity Procedure: Starting this experiment, we first connect the Photogate to the PASCO interface. Upon opening the Experiment 1 tab on the Capstone file that is available in canvas. Following, we press the record button simultaneously dropping the picket fence through the Photogate. Make sure to not touch the sides of the Photogate with the picket fence. After it finishes falling, click the record

button again to accumulate the information. Finally, we use the data and record the slope in a data table. Repeat the measurement at least five more times to gather accurate data. Data Analysis Trial no: 1 2 3 4 5 6 Slope (m/s^2) 7.61 9.18 9.77 9.77 9.74 9.78 Table 1: Slope of each velocity-time graph for each run What is the shape of the position vs. time graph for the free fall? What is the shape of the velocity vs. time graph? How is this related to the shape of the position vs. time graph?

The position-time graph shape for the picket fence should be a parabolic curve since it is in free fall. Capstone recorded that the curve goes upward from left to right, since the starting position (above the Photogate) was considered to be 0m and the picket fence had gained displacement since it moved from 0m falling towards the table. Figure 1: Position-time graph for each run The shape of the velocity-time graph is a straight line with velocity increasing since acceleration due to gravity results in an increase in velocity as time passes. The straight line is due to velocity changing at a constant rate in free fall (acceleration remains at 9.8 m/s^2 throughout the free fall). This is related to the shape of the position-time graph since the slope of the position-time graph provides the velocity at that given time. The velocity-time graph is the derivative of the position-time graph. Since the position-time graph is an upwards parabolic curve, the velocity-time graph is an upwards increasing linear curve.

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Figure 2: Velocity-Time graph for each run From the six trials, we determine the minimum, maximum, and average values for the acceleration of the Picket Fence. One of the ways to state the precision is to take half of the difference between the minimum and maximum values and use the result as the uncertainty of the measurement. Trial 1 had the minimum value for acceleration at 7.61 m/s^2. Trial 6 had the maximum value for acceleration at 9.78 m/s^2. The average value for acceleration was 8.69 m/s^2. (9.78-7.61)/2=1.085 m/s^2 is the uncertainty of the measurement. Express your final experimental result as the average value, plus or minus the uncertainty value. 8.69 m/s^2+1.085 m/s^2= 9.775 m/s^2 8.69 m/s^2-1.085 m/s^2= 7.605 m/s^2

7.605 m/s^2 - 9.775 m/s^2 is the range of the average value (considering uncertainty value) Express the uncertainty as a percentage of the acceleration. This is the precision of your experiment. (1.085/8.69)(100%)=11.51% (1.085/7.605)(100%)=14.27% (1.085/9.775)(100%)=11.09% Compare your measurement to the generally accepted value of g from your textbook. Do the accepted values fall within the range of your values? The generally accepted value of g is 9.8 m/s^2. The accepted value does fall within the range of our values since 7.605 m/s^2 < 9.8 m/s^2 <= 9.775m/s^2. Additional Analysis In this experiment, the effect of air resistance is negligible because the picket fence is too thin, but how will a larger air resistance acting on the Picket Fence change the measurements? What effect would it have on the shape of your plots? Derive position, velocity, and acceleration kinematics equations for an object undergoing a changing acceleration. What would be the shape of graphs for these equations? Do you see any such effects on your data? A larger air resistance will slow the velocity since it counteracts the acceleration due to gravity. This means that it will take longer for the picket fence to hit the table. The position-time graph won’t be an exact parabolic curve (although it’ll still have the general parabolic curve shape). The velocity-time graph will also not be a straight line when the object gets closer to its terminal

velocity. The acceleration-time graph will obtain a value of 0 m/s^2 when there’s terminal velocity. Terminal velocity is when -acceleration due to gravity = drag force = 0 and the object reaches its maximum speed. Figure 3: Equations relating drag force to kinematic equations A=(F-D)/m - A=acceleration - F=constant - D=drag - m=mass

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Figure 4: Equation to calculate drag force ma=mg-Dv - m=mass - a=acceleration equation - g=gravity acceleration - D=drag force - v=velocity It might be possible to see these effects in this data, but it will be negligible since drag force isn’t that great because the picket fence does not have a very high mass and also was not dropped from a great height.

Conclusion While performing the experiment it is very important that one must drop the picket fence as smoothly and straight as possible through the photogate. We repeated our process 6 times to try to minimize as much human error as much as possible, and as shown in the graphs we can visualize how the velocity changes throughout the 6 runs and how the acceleration stayed relatively similar for the 2 of our tests. From this we can infer that the gravity causes the picket fence to grow in acceleration when it reaches the bottom point of the line. All the data is shown in the graphs above. Experiment 2: Motion in One Dimension Procedure For the experiment, connect the motion detector to the port 1 of the PASCO interface. Then face the motion detector to where the speaker is pointed upwards. After that, open the Experiment 2 tab inside of the Capstone file that is found in canvas. Once everything is set up, hold the ball above the motion sensor and click the record button in Capstone. Then throw the ball 1 meter above the motion detector and let it drop straight onto the detector. After the ball drops, stop the recording. Repeat these steps however many times until the graph of the data is smooth.

Data Analysis Figure 5: The quadratic curve compared to the graph of position vs time The quadratic curve of the position vs time graph is -0.47x^2+9.6x-4.45. The coefficient of the x^2 is -0.47. The coefficient of the graph is not close to the coefficient of the free fall equation. The coefficient of the x^2 term is -0.47, while the coefficient of the (1⁄2)g equals out to be 4.9. One of the main differences that sets these two values apart is that the coefficient of the curve is positive and the coefficient of the free fall equation is negative. From that, the two coefficients are quite far from each other. The initial velocity equation is y=-9.31x + 9.61. The graph of the collected data for the velocity doesn’t match the initial velocity equation. The velocity of the ball at the top of its motion is 1.76 m/s. The acceleration of the ball at the top of its motion is 9.24m/s^2.

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Figure 6: The Linear curve compared to the graph of velocity vs time The equation of the linear curve is y=-9.3x+9.6. The coefficient x term of the linear curve is -9.3. The coefficient of g is 9.8. The coefficient of the x term is far from the coefficient of g because the value is negative. In turn, these two coefficients are not close to each other. The mean value for acceleration during the free-fall section of flight for Run#2 is -9.27 m/s^2. The acceleration derived from the position-time graph was 9.24 m/s^2. The acceleration derived from the velocity-time graph was 4.65 m/s^2. The mean acceleration value is completely different from the values of g found from the position vs time curve and the velocity vs time curve. The mean value is a negative number, while the two values of g are positive numbers. An object with a change in velocity, either in its magnitude or the direction, will experience acceleration. When the velocity vector and acceleration vector are in the same direction, it’s speeding up (accelerating). When they’re in opposite directions, it’s slowing down (decelerating). A change in direction but not speed is centripetal acceleration, which points to the exact center of the circular path.

Figure 7: The point of acceleration of the ball Additional Analysis If initial velocity was greater, the position-time graph would be having a steeper slope since it’d go farther in less time. If the acceleration was greater, the position-time graph would also have a steeper slope since it’d accelerate faster and go farther less in time. If less air resistance was present, the position-time graph would also have a steeper slope since there would be less negative force thereby covering more distance in less time. Changes in velocity, acceleration, air resistance, and much more can affect the position-time plot and can impact the slopes of the graph as these factors impact the speed at which the object travels.

Conclusion While doing the experiment, it is imperative to try and minimize the amount of hand movement used to toss the ball up and catch it again to avoid messing with the motion sensor. To eliminate human error we had one person do all 6 sets. The position for most of the runs were relatively the same being at or around 1m above the motion sensor. We know that as the ball goes upward gravity is constantly accelerating and when the ball is downward till it reaches a peak and begins its descent as we see in the graphs above. All of this is shown in the data above.

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Ganesh Sivararamakrishnan_Lab2

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