Lab 109: One-Dimensional Motion -- Velocity as a Function of Time and Distance at Constant Acceleration PHYS 111A-115 Raj Kyada September 21 2021 Partner:-Abhi Kyada , Stephen Kelmanovich Objectives 1. To investigate one-dimensional motion of an object with constant acceleration and to improve an understanding of mathematical equations related to the motion as well as graphical presentations of displacement and velocity changes over time. 2. To learn how to determine the values of velocity and acceleration from displacement and time measured in this experiment. Background 1. When an object moves with constant acceleration, its velocity as a function of time is given by: v = v0 + at 2. Where v0 is the initial velocity (at t=0), v is the velocity of the object at time t and a is the constant acceleration. The velocity of the object as a function of distance is given by: v2 = v02 + 2a(x − x0) , where v0 is the initial velocity (at x0) , v is the velocity of the object at x and a is the constant acceleration. 3. When an object moves down an inclined path, its acceleration can be calculated as a function of the angle by: a = g(sin θ) , where θ is the angle of the incline, g is the acceleration of gravity (which is 9.81m/s2 on Earth), and a is the constant acceleration. Equipment - Computer with Capstone software installed - 850 Universal Interface - Air track - Air supply with a hose - Wooden block to lift the air track from one side - Glider with a reflector and cylindrical metal object (flag) - Photogate(x2) - C-clamp(x2) to fix the photogates to the air track - Motion sensor - Tripod stand (for the motion sensor) - Protractor - Vernier Caliper Procedures 1. Gather the required equipment above. 2. Set up the computer, log in with UCID and password, and attach the 850 Universal Interface, and ensure proper connection. 3. Open the “Lab 109 One Directional Motion” file in “~\Desktop\Physics 111A Lab Experiments” and go to the “Part I” page. 4. Measure the diameter (in meters) of the “flag” using the caliper, then, in the “Calculator” tool on screen, input this value for “FlagDiameter” and click accept. This allows the photogates to measure instantaneous velocity. - When performing this lab, this step had already been found to be completed. 5. Use the wooden block to lift the air track, making sure to lift from the side not connected to the compressor 6. Use the protractor to measure the angle of inclination at three separate points, then average those three measurements to determine the proper angle of inclination (θ). Part I 7. Connect the photogates to Digital Input ports 1 and 2 on the Universal Interface 8. Place the first photogate 30cm from the top of the track using the C-clamp, this photogate will remain in a constant position throughout the experiment. Record the exact position of this photogate. 9. Place the second photogate 50cm from the top, or 20cm from the first photogate. This photogate will be moved throughout the experiment. Record the exact position of this photogate. 10. For each run: a. Starting from the initial .50m placement of the second photogate, increase x by approximately .25m up to 1.75m. Record x2 as well as (x-x0) in the data table b. Click the “RECORD” button on screen to begin recording data, then release the glider c. Have someone pick up the glider at the bottom, and do not let it bounce off the bottom of the air track d. Click the “STOP” button on screen e. Repeat until x2 = 1.75m 11. Record the times and velocities in the data table 12. Remove the photogates and disconnect them from the Universal Interface Part II 13. Connect the motion sensor to “PASPort 1” of the Universal Interface 14. Place the sensor at the end of the air track and mount the reflector on the glider 15. Choose the “Part II” page on screen 16. Click the “RECORD” button on screen, then again release the glider from the top of the track 17. When the measurement is done, click the “STOP” button. Two plots will appear on screen. 18. Do this twice. 19. Disconnect the motion sensor, remove the wooden block under the air track, shut down the computer, and return the equipment Results In Part I, a data table was made which included the instantaneous velocities in each photogate, as well as the times and distances between the two photogates. Using this data, we plotted two different charts, v2 vs t and (v2) 2 vs (x-x0) . Then, we drew a regression line, which gave us the slopes and intercepts of each chart. In the first chart, we used the equation v = v 0 + at , which told us that the slope of the regression line was the acceleration. We recorded this as 0.542m/s2 In the second chart, we used the equation v 2 = v 0 2 + 2a(x − x0 ) , which told us that the slope was two times the acceleration. We recorded the acceleration as 1.12m/s2 consequently. In Part II, we fitted the two plots we were given by using the “Highlight Range of Points” tool, then used the curve fitting tool to find the regression lines. Knowing that v = v 0 + at , we again could find the acceleration by taking the slope. We recorded these as 0.851m/s2 and 0.846m/s2 for the trials. To find the theoretical acceleration, we used the formula a = g (sin θ) , which, after filling in 9.81 for g, Earth’s gravity, and our angle for θ, we found acceleration to be .854m/s2. This value was unfortunately clamped to two significant figures due to the fact that the protractor only measures to two decimal places. We then performed error analysis, comparing each of the four “measured” accelerations to our theoretical value: |0.542−.854| = . = 37% (from v2 vs t) −.854 |1.12−.854| = .68 = 68% (from (v2) 2 vs (x-x0) ) −.854 |0.851−.854| = .15 = 15% (from motion sensor trial 1) −.854 |0.846−.854| = .14 = 14% (from motion sensor trial 2) −.854 Data Table x₀ [m] x₁ [m] Δx [m] t₁ [s] Δt [s] t₂ [s] v₂² v₁ [m/s] v₂ [m/s] a [m/s²] [m²/s²] 0.28 0.500 0.3985 0.4780 0.6835 0.28 0.750 0.7320 0.4793 0.8267 0.28 1.000 1.0111 0.4793 1.0134 0.28 1.250 1.2465 0.4801 1.1431 0.28 1.500 1.4590 0.4808 1.2575 0.28 1.750 1.6532 0.4804 1.3655 Discussion After doing the error analysis for the results, we found that none of our measured accelerations were accurate to a statistical significance. That said, we did find that the values gathered in Part II using the motion sensor were much more accurate than those using the repeating tests in Part I. However, we could not figure out exactly why, as our data doesn’t suggest any testing anomalies took place. There were no individual trials that did not fit the trendline either, so we could not make any useful guess as to what caused the error. Being that we did not measure the value of the diameter of the flag, I imagine there could be a possibility that an incorrect value would cause for incorrect velocities to be reported by the photogates, but I believe that would cancel out when you consider that both gates would report the incorrect velocity, and so the acceleration would be the same. If the angle was wrong, we also would have seen that in Part II, which we did not. This leaves only some testing anomaly or bad equipment to blame. I would be interested in retrying this experiment with a different set of photogates to see if our result repeats. Conclusion In summary, this lab demonstrated our ability to use the provided equations to solve for acceleration when given velocity and distance or time, as well as helped us understand the importance of properly measuring and double checking our data. Carrying over into future experiments, we can utilize this knowledge to help improve the accuracy of our data moving forward. We know the relationships between velocity, distance, and time, and we can see that using the lab data from Part II, which was close to our predicted acceleration, and, when this value is put back into the formula for the angle of incline θ = sin−1 ( ag ) , we found the angle to be within one degree of our measured angle, meaning that we were not far off at all.
