Lab - Introduction to Kinematics Using the Motion Sensor

Lab 2: Introduction to Kinematics Using the Motion Sensor I. Objective The objective of this lab was to examine kinematics, the motion of objects, under various conditions. The Logger Pro Software and a motion sensor device were used to graph position, velocity, and acceleration and also examine the relationship between the three variables. II. Theory Motion detectors use high frequency sound waves to detect objects that are near them. When the detector is turned on, the sound waves are constantly emitted from the motion detector. When the sound waves hit an object, they reflect off of it, and a sensor on the detector receives the sound waves and registers it as a motion. To measure the velocity and acceleration, the Logger Pro software measures the change in position that it detects over time and it also graphs this change instantaneously. The motion sensor works because velocity is, by definition, the change in position over time (Equation 1) and acceleration is the change in velocity over time (Equation 2). v = Δx Δt Equation 1 In Equation 1, Δx is the change in position, distance, and Δt is the change in time, and the units for velocity, ,vare meters per second, ms . a = Δv Δt Equation 2 In Equation 2, Δv is the change in velocity and Δt is the change in time. The units for acceleration, a,are meters per second squared, m s2 . Based on the two equations, it can be understood that velocity is equal the slope of the position curve, and that acceleration is equal to the slope of the velocity curve. The motion detector reads velocity that goes towards the detector as negative and away from the detector as positive. When tested under various conditions, these formulas and knowing how the motion detector reads velocities, can be used to predict the general shape of the position, acceleration, and velocity graphs. An example would be when a cart is moving on a level track towards the sensor, the position graph will have a negative slope over time, the velocity will have a negative slope when the cart starts to move, and the acceleration should be relatively constant. When measuring motion from an incline, acceleration can be calculated using Equation 3, assuming friction does not affect it. a = g sin (θ) Equation 3 θ Where is g gravity, or 9.81 m/s 2 , and is the angle of the incline. The sine of the angle of the incline, however, is not known, but can be calculated by the height of the block ( )H divided by the length of the track ( ),Land by substituting that into Equation 3, the acceleration can be calculated using all known variables using the new Equation 4. a = (9.8 III. m H ) s2 L Equation 4 Procedure The lab was separated into four separate procedures, referred to as Procedures A, B, C, and D. For these procedures, the data collection of Logger Pro was set to sixty samples per second at a time of ten seconds. The small Δt and rapid sample rate were necessary for the velocity to nearly equal the slope of the position curve and the acceleration to nearly equal the slope of the velocity curve. The purpose of Procedure A was to test the motion sensor and confirm that it was working correctly with the Vernier interface and Logger Pro software. It also illustrated what trends would be seen in the graphs with regards to position and velocity. In this procedure, the detector was set up on the end of the table and one of the group members stood 2 meters away from the detector. When the data collection was started, the group member stood still for the first couple of seconds, then moved towards the motion sensor, and stopped before reaching it, allowing the software to graph the person standing still again. In Procedure B, the motion detector was placed at the end of a track and was used to record the motion of a cart that was placed on the track and pushed towards the detector. The cart and track were used again for Procedure C, but this time a 3.5cm block was placed under one end of the track, forming an incline. The motion detector was used to record the motion of the cart as it rolled down the incline after being released from rest. In Procedure D, the cart began at the bottom of the incline, and the motion detector recorded the cart as it was pushed up the incline and as it rolled back down. Once all the procedures had been performed, the slopes of the position curves and velocity curves were collected, as well as the averages of the velocity curves and acceleration curves, and they were placed in a table to determine the experimental percent differences in the factors that should have been the same. Equation 5 was used for these calculations. % dif f erence = Where the IV. | m − average| (100) m m is the slope and Equation 5 averageis the mean recorded from Logger Pro. Data Procedure A Procedure A was divided into two parts, Part A and Part B. The graphs of position, velocity, and acceleration over time from these parts are shown in Figures 1 and 2. Figure 1: Procedure A, Part A. The student walked towards the detector. The slope of the position graph is negative, because the distance between the student and the detector decreased. The velocity graph also shows a downward trend, because the student was moving towards the detector. Figure 2: Procedure A, Part B. The student walked away from the detector. The slope of the position graph is positive, because the distance between the student and the detector increased. The velocity starts at zero and it is positive, because the student is moving away from the detector. Procedure B The graph of the data collected during Procedure B are shown in Figure 3. Figure 3: The position graph is linear with a negative slope, because the cart rolled towards the detector. The position graph is scaled between 0 and 1.5 seconds to better illustrate the change in position over time. The velocity graph starts at 0, before the cart is pushed, and then becomes negative as the cart moves towards the sensor, then returns to 0 when the cart is stopped. Procedure C Figure 4 shows a screenshot of the position, velocity, and acceleration graphs from Logger Pro. It also shows the selected regions and the data from each graph for Procedure C. Figure 4: The position graph is quadratic in time; the velocity graph is linear, because the acceleration is near constant. The statistics and linear fit tools in Logger Pro were used for each of the selected region on the graphs. Procedure D The same Logger Pro tools used for Procedure C were also used for Procedure D, as shown in Figure 5. However, three regions were selected instead of two. Figure 5: The Statistics and Linear Fit tools in Logger Pro were used for each selected regions on the graphs. Since the statistics and linear fit tools hide parts of the graphs, Figure 6 shows just the graphs. Figure 6: The position graph is quadratic in time, and the velocity is linear, because the acceleration is essentially constant. The wild fluctuation at the start of the position and velocity graphs indicate the movement of the hand as it pushed the cart up the inclined track. V. Analysis A sample of the data taken during Procedure B is shown in Table 1. Procedure B Time (s) Slope ( )m 1.8315 0.932411 -0.18998 0.018139 1.8648 0.9261 0.010692 1.8981 0.919789 -0.18815 -0.02616 1.9314 0.913752 -0.19181 -0.04544 1.9647 0.906892 -0.19273 -0.02253 1.998 0.900855 -0.19204 -0.02539 2.0313 0.89427 -0.19502 -0.01451 2.0646 0.887684 -0.19365 0.008401 2.0979 0.881373 -0.1925 -0.01222 2.1312 0.875062 -0.19548 -0.00878 v a -0.1893 Table 1: A sample of the data taken from Procedure B. Table 2 compares the slopes recorded to the average values. Procedure B Slope Value Average Value x-curve -0.1802 m/s v-curve -0.1552 m/s 13.87% v-curve -0.133 m/s a-curve -0.123 m/s 2 Percent difference 2 7.52% Table 2: The slopes of the x-curve and v-curve as they are compared to the averages of the v-curve and a-curve. The percent difference was calculated using Equation 5. Table 3 shows two samples of the data taken at Region 1 and Region 2 during Procedure C. Procedure C Region 1 time Region 2 m v a time m v a 0.666 1.126412 -0.12337 -0.29595 1.8981 0.749935 -0.48938 -0.30684 0.6993 1.122296 -0.13276 -0.29614 1.9314 0.733197 -0.49853 -0.28602 0.7326 1.117631 -0.14329 -0.29118 1.9647 0.716733 -0.50769 -0.28411 0.7659 1.112692 -0.15244 -0.2803 1.998 0.699446 -0.51799 -0.26999 0.7992 1.107478 -0.16137 -0.28583 2.0313 0.682158 -0.52623 -0.24898 0.8325 1.10199 -0.17167 -0.28946 0.8658 1.095954 -0.17991 -0.30875 2.0979 0.646761 -0.54248 -0.32039 0.8991 1.090191 -0.1925 -0.31982 2.1312 0.628376 -0.55667 -0.31867 0.9324 1.083057 -0.20326 -0.28068 2.1645 0.609442 -0.56491 -0.28335 0.9657 1.076471 -0.20944 -0.28564 2.1978 0.590783 -0.57384 -0.28621 2.0646 0.664322 -0.5331 -0.26655 Table 3: Samples of the data taken during Procedure C. Table 4 compares the slopes recorded at both regions to the means recorded at those points. Procedure C Region 1 Region 2 Slope Value Average Value x-curve -0.1669 m/s v-curve -0.1670 m/s v-curve -0.2924 m/s a-curve -0.2934 m/s x-curve -0.5308 m/s v-curve -0.5311 m/s v-curve -0.2821 m/s 2 2 a-curve -0.2934 m/s % difference 0.05% 2 0.34% 0.06% 2 4.01% Table 4: The slopes of the x-curve and v-curve as they are compared to the averages of the v-curve and a-curve. The percent difference was calculated using Equation 5. Table 5 shows three samples of the data taken at three different points (Regions 1, 2, and 3) during Procedure D. Table 5: Samples of the data taken during Procedure D. Table 6 compares the slopes recorded at Regions 1, 2, and 3 to the means recorded at those points. Procedure D Region 1 Region 2 Region 3 Slope Value Average Value x-curve 0.4326 m/s v-curve 0.4338 m/s v-curve -0.2791 m/s 2 a-curve -0.2729 m/s 2 2.22% x-curve 0.0001498 m/s v-curve 0.0007782 m/s 419% v-curve -0.3503 m/s 2 a-curve -0.2729 m/s 2 22.10% x-curve -0.3188 m/s v-curve -0.3219 m/s v-curve -0.2434 m/s a-curve -0.2729 m/s 2 % Difference 0.27% 0.97% 2 12.12% Table 6: The slopes of the x-curve and v-curve as they are compared to the averages of the v-curve and a-curve. The percent difference was calculated using Equation 5. To calculate the acceleration, the length (L) was needed. The distance between the two supports of the track measured 114.6 cm. The height (H) of the incline was equal to the height of the block that was placed under the track, which measured to be 3.5 cm. Equation 4 was used to calculate the acceleration using the given values. Table 7 shows the calculations for Procedure C. Acceleration Time (s) 2 A (m/s ) 0.666 -0.295952721 0.6993 -0.296143658 0.7326 -0.29117929 0.7659 -0.280295867 0.7992 -0.285833047 0.8325 -0.289460854 0.8658 -0.308745516 0.8991 -0.319819875 0.9324 -0.280677741 0.9657 -0.28564211 avg a=-0.2934 m/s 2 calculated a= 0.2804 m/s 2 Table 7: A sample of the acceleration values and the average acceleration from Logger Pro (avg a) and the acceleration calculated using Equation 4. Table 8 shows the same calculations, but for Procedure D. Acceleration Time (s) A (m/s 2 ) 1.2654 0.15007667 1.2987 -0.270176193 1.332 -0.37347324 1.3653 -0.370227307 1.3986 -0.319247064 1.4319 -0.285451172 1.4652 -0.321920185 1.4985 -0.392566964 1.5318 -0.370800118 avg a= -0.2729 m/s 2 calculated a=-0.2804 m/s 2 Table 8: A sample of the acceleration values, with the average acceleration from Logger Pro (avg a) and the acceleration calculated using Equation 4. VI. Conclusion From the results of the Procedures A, B, C, and D, it can be concluded that the goals of the experiment overall were met. The relationships between acceleration, position, and velocity were experimented under different conditions and the resulting graphs appeared the way they were predicted to. Based on the percent differences, the values that were equal had no statistically significant difference. The group encountered some problems with the motion detector and sensor where it would record the person’s hand instead of the cart, which made the group repeat some of the procedures, but that only benefited the data in the end. When position versus time is graphed, it yields a quadratic curve for Procedure C and Procedure D due to how the cart is manipulated. In Procedure B, the position versus time graph is linear since the cart is just pushed towards the sensor. Acceleration versus time in Procedure C and Procedure D both yield a linear, constant line.