OMAR MOHAMED - L_Projectile Motion & Inclined Plane Sept 2023

c SPH4U Lab: Projectile Motion // Motion on an Inclined Plane (L1) Purpose : To analyze the motion of an air puck on an inclined plane using a graphical approach and to compare its motion to that of a projectile. Materials : air table apparatus clinometer timing device (50.0 Hz) air pucks (2) rulers / meter stick newsprint scissors adhesive tape Procedure : 1. One end of the air table apparatus should be propped up in order to make an inclined plane. Measure and record the angle of incline to the nearest tenth of a degree. 2. Cut a piece of newsprint so that it just fits onto the air table’s surface. Use adhesive tape if necessary to attach the paper to the edge of the table (not to the carbon paper itself). 3. Place one air puck in the bottom left corner of the air table. It will remain stationary for the remainder of the lab. 4. Turn on the compressor and timing device attached to the air table. Use the 50 Hz setting for timing. 5. Hold the second air puck near the top right corner of the air table. Release it from rest and quickly press and hold the timing pedal down to take data as it falls down the incline. Release the timing pedal before the puck reaches the bottom. DO NOT MOVE THE PAPER AFTER THIS STEP, as this represents the “vertical” direction for your data collection. The paper must be undisturbed before carrying out the next step. 6. Hold the air puck near the bottom right corner of the paper. Launch it on an angle (upward and slightly to the left) while quickly pressing and holding the timing pedal. The puck will make a curved trajectory and the timing pedal should be released before the puck hits any of the metal edges of the air table. 7. Take the newsprint off the air table. Identify the first point in the curved trajectory of the air puck. This is point #0. Label all points from there in sequence (0, 1, 2, 3, …, 39, 40). 8. Draw a set of axes through point #0. The y-axis needs to be EXACTLY PARALLEL to the straight line produced by the air puck in step 5. The x-axis needs to be EXACTLY PERPENDICULAR to the straight line produced by the air puck in step 5. 9. Measure and record the x and y positions of the air puck for each even-numbered point on the curved trajectory. When measuring x positions, ensure that your ruler is parallel to the x- axis. When measuring y positions, ensure that your ruler is parallel to the y-axis. Observations : Angle of incline, measured to the nearest tenth of a degree: 6.5°

Mass of air puck that was launched in parabolic trajectory: 187.2g Complete the d x and d y columns of the table provided as a handout, and type the data into the following page: Table 1: Position-Time and Velocity-Time Information Point # time, t (s) horizontal position (x-component) d x (cm) vertical position (y-component) d y (cm) horizontal velocity (x-component) v x (cm/s) vertical velocity (y-component) v y (cm/s) 0 0.000 0.00 0.00 1 0.020 17 42 2 0.040 0.69 1.68 3 0.060 16.3 37.8 4 0.080 1.34 3.19 5 0.100 16.3 33.5 6 0.120 1.99 4.53 7 0.140 17.3 28.8 8 0.160 2.68 5.68 9 0.180 16.3 24.8 10 0.200 3.33 6.67 11 0.220 15.5 20.8 12 0.240 3.95 7.50 13 0.260 16.3 16.3 14 0.280 4.60 8.15 15 0.300 14.5 7.00 16 0.320 5.18 8.43 17 0.340 14.8 3.75 18 0.360 5.77 8.58 19 0.380 16.3 0.5 20 0.400 6.42 8.56 21 0.420 8.5 -5.50 22 0.440 6.76 8.34 23 0.460 21.5 -9.00 24 0.480 7.62 7.98 25 0.500 11.8 -12.5 26 0.520 8.09 7.48 27 0.540 7.3 -22.8 28 0.560 8.38 6.57 29 0.580 18.8 -24.5 30 0.600 9.13 5.59 31 0.620 12.8 -29.3 32 0.640 9.64 4.42 33 0.660 9.50 -34.0 34 0.680 10.02 3.06 35 0.700 6.8 -37.8

Equation of horizontal position vs time: d x 14 t + 0.48 Equation of vertical position vs time: d y − 60.9 t 2 + 46.0 t 3. Input the velocity-time data into graphing software. Plot the x and y velocity data on the same set of axes. Use the software to produce a trend line (line or curve of best fit, as appropriate) for each set of data. The equation of each trend line and correlation coefficient ( r -value) should be included in the spaces below. Velocity vs Time Graph of Air Hocket Puck

black: Equation of horizontal velocity vs time: v x − 25.5 t + 37.4 Correlation coefficient: r = 0.8625 orange: Equation of vertical velocity vs time: v y − 243 t + 91.8 Correlation coefficient: r = 0.998 4. From the perspective of an observer looking side-on at the inclined air table, draw a free- body diagram of the puck as it is sliding down the incline . Assume the ramp is frictionless. NOTE: this observer will not see any x-direction forces, as they are looking in the positive x- direction. However, they will see a dimension that is perpendicular to the paper, which you can call the z-direction for your free-body diagram. The positive y-direction should remain unchanged from how it was defined on your air table sheet (which should be [up the ramp]).

In the case of the graph showing time and vertical position, the curve was extremely close to the dots, with an r-value of 0.99, indicating a strong connection. However, both velocities were calculated using human-measured values, which might have led to slight inaccuracies. This means that using two measured values to calculate a third might not result in the most precise trend line. Moreover, during the measurement of horizontal displacement, there was less precision due to the lack of time, leading to potentially less accurate measurements. This inaccuracy affected the calculations which changed the graph and trend line. As a result, it was expected that the vertical measurements would depict a stronger trend line compared to the horizontal measurements, and this is indeed what was observed. 2. The theoretical acceleration of the air puck determined in Analysis #5 is likely not the same as the experimental acceleration obtained by the analysis of the v y -t graph. Provide three logical sources of error that were encountered during the experiment and/or analysis that resulted in this discrepancy. 1.There are some sources of error in the measurements that might have caused differences in the recorded values. One of them is that the air table may not have been set up very precisely, which could affect the values of the experiment. It's possible that the slope of the air table wasn't completely even, which could have changed the acceleration we calculated during the experiment. This is because a steeper slope would lead to a faster acceleration, and the opposite is true. 2. Another reason for potential errors in the experiment could be that the weight protractor might not have been completely accurate when measuring the angle of the air hockey table. There were a few things that could have caused this. First, if the string holding the weight was a bit torn, it could change where the weight sits, and that would affect the angle measurement. Also, if the weight was a little worn out, it might change where its center of gravity is, leading to an incorrect reading on the protractor. Finally, if the edges of the protractor were worn out from being used a lot, it could cause a balancing problem and, as a result, an incorrect angle measurement. Since the protractor was old and had been used many times, these three conditions were probably present to different extents. Still, they would have affected the final angle measurement, ultimately leading to a difference in the calculated accelerations. 3.The third source of error that contributed to the inconsistencies of the accelerations experimental and theoretical calculations is because some forces, like friction, were not considered in the theoretical calculations. For instance, when an object moves, there's kinetic friction that makes it slow down. In the theoretical calculations, we assumed there was no friction at all. When it comes to air resistance, it's a force that pushes against a moving object, making it slow down. Even though it didn't have a big impact in this case due to the object's light weight and short distance traveled, if we add up all the ignored forces, they could lead to a difference in the results.

3. Calculate the percent error of the experimental acceleration as compared to the theoretical acceleration of the air puck in the y-direction. Comment on the accuracy of the experimental result, given that less than 10% error would be expected in this laboratory. We found a 15% difference between the theoretical and experimental values. This happened because we didn't think about certain forces like air resistance and friction, which we thought wouldn't matter much. It turns out however that they did affect the accuracy of our predictions by making the hockey puck go faster than we thought in our experiment. I wasn't thinking our data would be super exact because some of our measurements and methods could introduce mistakes. For example, we had to rush some measurements due to time constraints, which made the graph and trend line a bit off. The acceleration we figured out relied on how accurate the graph and trend line were because it came from the slope. Since we didn't focus a lot on being super exact during the experiment, a small bit of inaccuracy was bound to happen. 4. What are the similarities between projectile motion and the motion of the puck on the air table? Use specific trends or values from your graphs to support your answer. Both projectile motion and the motion of the puck on the air table share similarities in terms of their vertical motion. This is evident from the position-time and velocity-time graphs. In both cases, the vertical position vs. time graph displayed a similar parabolic shape, and the trend lines for vertical position had high correlation coefficients (0.99 for the air table experiment). The vertical velocity vs. time graph also followed a clear trend in both scenarios, indicating consistent acceleration due to gravity. Additionally, the z-axis displacement in the free-body diagram was similar in both projectile motion and motion of the air puck, with the observer on the inclined table perceiving a dimension perpendicular to the paper, which aligns with the vertical displacement in both cases. However, it's important to note that while the trends are similar, there is a significant difference between the theoretical and experimental accelerations for the air puck due to the presence of factors like air resistance and friction, resulting in about a 15% error in this experiment, which shows the importance of considering real-world forces in these analyses.

In this lab, the goal was to analyze the motion of an air puck on an inclined plane and compare it to the motion of a projectile. An incline of 6.5 degrees was set up, and an air puck was used for the experiment. The motion was tracked and recorded carefully, with data points collected at specific intervals. The analysis revealed several key findings. The position-time and velocity-time graphs showed clear trends. Notably, the vertical position vs. time graph showed a remarkably strong correlation (r = 0.99), while the horizontal position vs. time graph displayed slightly less precision. The theoretical acceleration, considering a frictionless environment, was calculated based on the inclined plane's angle. However, when compared to the experimental acceleration obtained from the vertical velocity-time graph, a 15% error was observed. This error was caused due to unconsidered forces such as air resistance and friction. Overall, the lab successfully achieved its purpose of examining the motion of the air puck on an inclined plane. The numerical data emphasized how crucial it is to consider external forces in this experiment. Despite the variance between the theoretical and observed accelerations, the results offered valuable understanding of the complexities involved in projectile motion on incline planes.