G8-Lab 2_

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Jan 9, 2024

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Lab #3: Free Fall – p. 1 Physics 250 - Free Fall OBJECTIVE Measure the acceleration due to gravity in the vicinity of Skyline College. EQUIPMENT Computer In lecture, it is claimed that the size of the acceleration given to objects by gravity has an approximate value of g= 9.8 m/s 2 near the surface of planet Earth (actual value varies with particular geographic locations). We will test this in lab today, by carefully observing the motion of a freely falling object. We will use data collected from a video of a ball falling in the Physics lab at Skyline College. If the above statement is true, we expect that a plot of the velocity of the object vs. time will produce a straight line graph, whose slope represents the acceleration of the object. Note that we will take the downward direction to be the positive direction for convenience— this makes the measured acceleration positive (though still downward!). We will actually be directly measuring the position (y) information of the falling object, not velocity. This means we’ll need to crunch that raw data to generate velocity vs. time data and the corresponding graph above. In order to do this, we remember that the slope of the y vs. t graph gives the value of the instantaneous velocity. We will use two methods to measure g: 1. Method 1: We will first use raw data directly from the raw video; 2. Method 2: We will then make a further analysis data generated from the same video using a tracking video analysis tool. THEORY v t

Lab #3: Free Fall – p. 2 METHOD 1: 1. Go to the following link to access the video: https://youtu.be/8i7pG1t14jI 2. You will now use the video, to find a rough estimate of g using two data points: the launch point and the point right before the ball hits the ground. 3. The ball is dropped next to a 2-meter stick. We will estimate this to be the total distance traveled by the ball. To evaluate the time of fall, you can move the cursor on the video to move forward and backward as needed. 4. The timestamps are not precise enough on the video, so we will need to find a way to make a better estimate of time. To do so, take the total time of the video to be 2.77s. You can verify this by looking at the 1 second and 2 seconds marks. Using the ½ meter stick or another measuring tool, measure the total length of the bar on your screen. For example, on the screen shown, 1.00 second corresponds to 9.0 cm and 2.77 seconds correspond to 24.9 cm. This will allow you to find a more precise measurement of the elapsed time. 5. Find the time when the ball starts moving and record it in the Results Sheet. Repeat for the position right before the ball hits the ground. 6. Use your data to calculate g assuming that the ball started from rest, and calculate the % error. Show your work below and report the values in the Results Sheet. 7. You may have find that this measurement may not be very accurate due to a number of sources of errors. Discuss them below. In the next part, we will try to overcome some of these errors.

Lab #3: Free Fall – p. 3 METHOD 2: 1. Go to the following link to access the next video: https://youtu.be/YMIzT16wrsg 2. The video from Method 1 has now been analyzed by a tracking tool. This tool records the position of the ball every 1/30 seconds (sampling frequency of 30 Hz). The data is shown briefly in graphs and given in a separate spreadsheet. Make sure to pause and take a look at the graphs to make sure they make sense. 3. The spreadsheet should have the following columns (not all of which will have data in them yet): Pt # Time [s] y [m] v [m/s] a [m/s 2 ] 0 0.467 0 [no value] [no value] 1 0.500 0.024 0.75 [no value] 2 0.433 0.056 0.98 6.8 Pt #: runs from 0,1,2,… through your last point. Time (t): starting at the time the ball starts falling, with each subsequent data point being 1/30 s later than the previous one. Position (y): as measured by the tracker. Velocity (v): We want to get velocity info from the raw position (y) info. Here’s one way: If we figure out the displacement over time between any two adjacent points, this formula gives the average velocity of the plummet between those two data points: ࠵? "# = ∆& ∆’ = & ( )& * ’ ( )’ * (1) If we assume that the data points are made often enough that the object doesn’t have a chance to change velocity much from the beginning to the end of the time interval, then we can assume v av is similar to v f (and v i ) without losing too much accuracy: v av ≈ v f ( ≈ v i ) (2) This gives us a way to generate velocity info for any given point: ࠵? ࠵?࠵? ࠵?࠵?࠵?࠵? ࠵? = [& "’ ’234 ’ )& "’ 564#2789 :"’" 572;’ ] ∆’ (3) The time between adjacent points ( ∆࠵? ) is fixed at 1/30 s, per the setting of the tracking software.

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Lab #3: Free Fall – p. 4 Acceleration (a): Using the fact that the acceleration is given by the slope of v vs. t, you can use a similar logic to eqn #1-3 (as well as similar Excel work) to add an acceleration column: ࠵? ࠵?࠵? ࠵?࠵?࠵?࠵? ࠵? = [# "’ ’234 ’ )# "’ 564#2789 :"’" 572;’ ] ∆’ (4) 4. Fill out the velocity column of your spreadsheet. Note that you need to leave the first velocity value blank , since there is no previous data point to refer to. Its value is NOT zero— we will explore this later. 5. Fill out the acceleration column of your spreadsheet. Note that you will now need to leave the first two acceleration values blank , until you have a previous velocity value to refer to. 6. Take a look at your acceleration column, and you should (hopefully) notice that, while the values fluctuate (sometimes wildly) it is nonetheless around a constant value, not increasing or decreasing as a general trend. If so, you are seeing that the acceleration is constant, to within experimental error. 7. Take the average of your acceleration column, using Excel. Then calculate the % error this value has with the accepted theoretical value. (Refer to the intro materials in this lab manual for help on these steps if needed.) Put your results on the result sheet. 8 . Graphing the data: Please make a plot of both y vs. t and v vs. t. [Note for plotting v vs. t graph: you will need to highlight equal length columns; a “t” without a “v” cannot be plotted!] We already mentioned that there were some approximations inherent in these calculations (see eqn #3-4), so now let’s use the more advanced linear least squares algorithm to get better results. 9. Perform a linear fit on your v vs. t data (using Excel’s linest). Record on result sheet. (Do NOT accidentally do this on your y vs. t data-- it is not expected to be linear!). 10. From this info, write down this method’s most probable value of the acceleration due to gravity (<g>) and then compute the 95% confidence interval for the value of g. Include units! Put on the result sheet. 11. Calculate your best estimate for the experimental value of the vertical-intercept ( b ) and its confidence interval, including units. Record your results on the result sheet.

Lab #3: Free Fall – p. 5 Questions: 1. For the y vs. t graph: a.) Briefly describe the rough shape of the graph. b.) Theoretically, what physical quantity does the instantaneous slope represent? What happens to the slope (and is this what you expect)? Explain. 2. For the v vs. t graph: a.) Briefly describe the rough shape of the graph. Slope (parts b-c): b.) Theoretically, what physical quantity does it represent? What happens to the slope (and is this what you expect)? What does the R 2 value tell you? Explain. c.) Does the accepted value of g fall within your confidence interval from method 2? What conclusions can you draw from this? Vertical axis intercept (parts d-e): d.) Theoretically, what physical quantity does it represent? Explain. e.) Is the experimental value statistically distinguishable from zero? Explain if this is consistent with your expectations.

Lab #3: Free Fall – p. 6 Application – Falcon 9 launch acceleration “On Saturday, May 30th, Falcon 9 successfully launched Crew Dragon’s second demonstration (Demo-2) mission from Launch Complex 39A (LC-39A) at NASA’s Kennedy Space Center in Florida. This test flight with NASA astronauts Bob Behnken and Doug Hurley on board the Dragon spacecraft initiated the return of human spaceflight to the United States.” SpaceX Team We will now apply the technique developed above to estimate the acceleration of Falcon 9 during the May 30 th , 2020 launch. To do so, we will use the following video: https://youtu.be/xY96v0OIcK4?t=15763 A few seconds after countdown, you will notice that 3 variables are being recorded: time (HH:MM:SS), altitude (km) and speed (km/h). We will estimate the motion to be one-dimensional and will be able to compare the speed to the actual data, and give an estimate of the acceleration. 1. Open a new tab in your spreadsheet or start a new one, as sown below. You will monitor the above variables every 5 seconds, starting at T +00:00:17 and until T +00:03:00. You may need to pause the video to be able to record the data. 2 . Fill out the y column using the measured altitude. Calculate v using y as you did in the Free Fall experiment. 3. Fill out the Measured v column using the measured speed, and convert it to m/s. Calculate the acceleration a using v as you did in the Free Fall experiment. Can we assume free fall in this scenario? Explain. Pt # t [s] y [km] Calculated v [m/s] Measured v [km/h] Measured v [m/s] a [m/s 2 ] 0 17 0.5 [no value] 226 62.8 [no value] 1 22 0.8 60 310 86.1 4.66 2 27 1.3 100 405 113 5.38 4. Compare the calculated and measured v values. Explain any difference you may notice. You may consider the precision given on the altitude in your explanation. 5. Make a graph of Measured v [m/s] vs. t and a vs. t. Listening to the commentary on the video, can you follow the trends and phases of the launch?

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Lab #3: Free Fall – p. 7 Results Sheet METHOD 1: Raw video analysis with Basic Error Analysis Include units! Initial Time (launch) Final Time (before hitting ground) Total elapsed time Include units! g (experimental, avg) g (accepted) % error METHOD 2: Direct point-by-point average with Basic Error Analysis Include units! g (experimental, avg) g (accepted) % error Linear Least Squares Fit (v vs. t) Parameter: Numerical value: Units: <m> s <m> s R 2 n/a

Lab #3: Free Fall – p. 8 Experimental estimate for g: (include units!) <g> = _________ s <g> = _________ 2 s <g> = _________ 95% confidence interval for g exp : Lower bound for g exp = _________ Upper bound for g exp = _________ Experimental estimate for vertical-intercept (b): (include units) What physical quantity does it represent? ________________ = _________ s = _________ 2 s = _________ 95% confidence interval for b: Lower bound = _________ Upper bound = _________

G8-Lab 2_

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