Lab 3 - Rolo - RRCC

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Apr 3, 2024

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Lab 3: Projectile Motion Lab Owen Rolo Eric Sheldrake Partners: N/A 2/10/2024 PHY2111 SC1

Objective The objective of this lab experiment was to familiarize students with the principles of projectile motion and demonstrate the lack of effect of horizontal velocity on time spent in the air or “hang time”. It should be clear to a student by the end of performing this experiment or reading this lab report; the distance the object is able to travel horizontally stems from differing horizontal velocities, not time spent in the air. Data & Calculated Results Table 1: Timing Data of Projectile Distance between tape strips on table = 20.0 cm in meters = .200 Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Time, s 0.17, 0.16 0.17, 0.16 0.17, 0.16 0.16, 0.14 0.17, 0.17 Velocity, m/s 1.21 1.21 1.21 1.33 1.18 Uncertainty Range = 1.23 m/s +/- .06 m/s Table 1 Trial 1 Velocity Calculation: (.17s + .16s)/2 = .165 s, .200m/.165s = 1.21 m/s Uncertainty Range Calculation (done in excel with unrounded values): (1.21 + 1.21 + 1.21 + 1.33 + 1.18)/5 = 1.228  1.23 m/s,  (x i – 1.228) 2 = .0144992,  (.0145/4) = .06 m/s Table 1 Coefficient of Variation Calculation (done in excel with unrounded values): (.06/1.23)*100 = 4.9% Table 2: Projectile Ranges & Calculated Horizontal Velocities

Height of table surface = 77.2 cm in meters = .772 Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Range, m .490 .495 .495 .510 .485 Velocity, m/s 1.24 1.25 1.25 1.29 1.22 Uncertainty Range = 1.25 m/s +/- .02 m/s Table 2 Trial 1 Velocity Calculation: (.490m) *  (9.81m/s 2 /1.544m) = 1.235  1.24 m/s Table 2 Coefficient of Variation Calculation (done in excel with unrounded values): (.02/1.25)*100 = 1.9% Table 1 & 2 Percent Difference (done in excel with unrounded values): 200 * | 1.23 – 1.25 |/(1.23 +1.25) = 1.5% Theoretical Time Spent in Air:  (1.544m/9.81m/s 2 ) = .3967 s Table 3: Timing & Ranges for Different Initial Velocities Velocity 1 Velocity 2 Velocity 3 Velocity 4 Velocity 5 Time 1, s .15, .16 .17, .18 .19, .18 .23, .22 .26, .25 Time 2, s .15, .15 .18, .18 .19, .19 .22, .23 .25, .25 Time 3, s .15, .14 .18, .16 .20, .19 .23, .23 .25, .26 Avg Time, s .150 .176 .190 .226 .252 Range 1, m .565 .460 .405 .335 .310 Range 2, m .550 .435 .400 .360 .315 Range 3, m .510 .465 .425 .340 .320

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Avg Range, m .542 .453 .410 .345 .315 Velocity, m/s 1.33 1.14 1.05 .885 .794 Table 3 Scatter Plot: 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 f(x) = 0.42 x − 0.03 R² = 1 Range vs Velocity Velocity (m/s) Range (m) Error Analysis: The investigation into any and all errors that may have occurred during the execution of these experiments has concluded that all errors were random and there were no systematic errors while performing trials. Reviewing the averages and uncertainty ranges for tables 1 and 2 shows that both values fall within each other’s ranges. While the coefficient of variations showed that the data from table two had less variation within its sample than table one’s [sample] (calculated at 1.9% and 4.9% variance respectfully), both standard deviations were relatively small for what was being measured so it was safe to assume that they were both precise enough for our use case. By calculating the percent difference between the results from table one and two, a metric accuracy level was obtained where the lower the percentage, the more accurate the results were.

The percent difference was only 1.5% between the methods used for tables one and two and since that is less than 5% it was safe to assume the trials were accurate. Judging the times and ranges from table three it would be safe to assume that the trials in part two were not as precise as the trials in part 1 because the lesser amount of trials created a less precise average range. More calculations would be necessary to determine exactly how precise part 2 was but the change in times and ranges did provide a noticeable pattern/spread that is comparable and in accordance with data from part one and provided reasonable assumption that part 2 was precise and accurate within its parameters. Impression: This lab clearly showed me, visually and audibly, that horizontal and vertical motion are independent of each other and the faster an object is moving the further it can travel. Every trial I heared the sound of the marble leaving the table and hitting the ground, over and over again the time spent in the air was the same. The only discerning difference between trials was how fast the ball was moving and how far from my table it landed. I was disappointed to see that this lab was really designed for 3 people to work on it, a minimum of two, and the only way I could accurately and precisely do this lab by myself was by videotaping each trial and watching it back which was very tedious and time consuming. It would be nice if human reaction time and a more practical approach was taken into account when assigning this lab in the future. Questions

1. In Part 1, you computed the horizontal velocity of the projectile using two different methods. Use the uncertainty ranges to answer whether they appear to be the same value. Are they the same within random error, or is there a systematic error involved? Do your results seem to support the theory that the horizontal velocity remains constant? If not, what are some possible systematic errors that may explain the discrepancy? a. As touched upon in the error analysis, the averages of table 1 and table 2 are not the same, however, including either uncertainty range they do fall into each other ranges. Since the uncertainty ranges overlap it is safe to assume that the values would be the same if it weren’t for random error. If the uncertainty ranges hadn’t overlapped than that would have meant there was a systematic error. 2. Describe the shape and meaning behind the graph generated for Part 2. a. The graph generated for part 2 was shaped linearly and was constantly increasing. The trend line of the graph represents how far an object can travel horizontally at any given velocity; as velocity increases, the distance the object can travel also increases, velocity and distance have a positive linear relationship that is shown in the graph. The value for the slope should be roughly the same as the time spent in the air for each trial. 3. Given the range equation R = v x t the slope of the graph represents the hang time of the projectile. What does this imply about the relationship between the initial horizontal velocity and the hang time of the projectile? a. Since the relationship between horizontal range and horizontal velocity is linear, the quotient of R and v x will always be the same which implies that the initial horizontal velocity and the time spent in the air are completely independent of each other. 4. Compute the theoretical hang time using the measured height of the table and equation (6). How well does it compare with the slope of the graph? Use a percent error to help answer this question. a. The time spent in the air should have been theoretically, 0.3967 s, in actuality the average time spent in the air was 0.4232 s. The percent difference between .3967 and .4232 is roughly 6.5% which is higher than planned/expected but still under the 15% difference needed to state the values as comparable. 5. Suppose a rifle fires a bullet horizontally 200 meters from a target and the bullet drops into the target 10.2 cm below the center cross hair. Use the system of equations presented to determine the muzzle velocity of the rifle and show your work. a. Results and Interpretation:

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The purpose of this lab was to collect accurate data pertaining to the distance traveled and the horizontal velocities of the object. Since the statistical analysis proved the values found in this experiment were both accurate and precise, the results are relevant and can be easily interpreted. In both parts 1 and 2, the velocity is the independent variable and the distance the object can travel is dependent on the initial velocity. The relationship between velocity and distance is linear when there is no horizontal acceleration and velocity is constant. The values in table one are the horizontal velocities of each trial. They were calculated in two steps. First, the time it took between exiting the ramp to passing over the first piece of tape 20 cm away, and the time between the piece of tape and the edge of the table another 20 cm away, were averaged. Then by taking the distance between the ramp and tape or tape and edge of the table, .2 m and dividing it by the average time it took to cross them, obtained the initial velocities recorded. The horizontal velocity values in table two were obtained from the exact same trials they were obtained from in table one but were calculated with different variables so that precision and accuracy levels could be obtained and the values compared. The variables used in table two were much less susceptible to random/human error, partially due to the fact a system of equations was used, so the standard deviation and coefficient of variance was smaller than for the values found in table 1. In table two, to find velocity, R = vt was solved for v to create v= R/t. Then, since time is the one variable that can unify the motion in the x and y direction,  (2h/g) was substituted for t after solving for t in R y = .5 * g * t 2 . In table two, only one recorded variable, distance, is being used in calculation whereas in table one both time and distance were recorded values being used. This explains why table two was more precise and had a smaller coefficient of variation. Tables one and two showed that the horizontal velocity did not change between the time the ball was on the desk and the time it hit the floor. The time the ball took to cross .2 m and the time it took before hitting the ground a distance away previewed the same velocity with random error for both tables. Part 2 or table three, showed not only that the horizontal velocity did not change from start to finish, but that the horizontal distance traveled is completely independent of time and only changes due to change in velocity. The independence of horizontal distance from time is clearly shown by the graph of table three’s range vs velocity because the linearly increasing slope represents the time it took each trial to fall to the ground. This experiment confirms the lack of relationship between horizontal distance and time.

Error Sources and Concluding Remarks As stated before, the error analysis concluded that only random errors were involved in this experiment and no systematic errors were found. The largest source for random error that comes to mind first was the uneven, a bit bumpy, wood plank used as a ramp. The slight unevenness could have produced small bounces unidentifiable by eye that altered its vertical velocity or it could have artificially changed the horizontal velocity by slightly moving direction or creating a slip that decelerated the ball. There are also many shedding pets in the laboratory setting this experiment was done in so small pieces of hair on the desk could have affected the friction coefficient rolling over certain areas. To minimize random error in this experiment next time the plank of wood should be bought new or sanded down to make even, to reduce the error from friction the desk should be clean and made of the same material. If accessible, a space vacuum without air resistance would minimize any error caused by air resistance and provide a sterile work environment free from pet hair minimizing those sources of error as well. All in all, the values calculated for velocity were acceptably precise and accurate as were the values for time spent in the air. The calculations deemed horizontal velocity and distance dependent on one another and horizontal velocity and time completely independent of each other. The results ring true with accepted assumptions and physics phenomena. References Knight, R. D. (2023). Physics for scientists and engineers: A strategic approach with modern physics . Pearson Education, Inc.

Lab 3 - Rolo - RRCC

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