Kinematics lab report PDF

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PHY 133 Linear Kinematics

Introduction:

Linear Kinematics is the study of how a particular object moves related to time, in a straight line. It is best described by a series of equations. Kinematics focuses on the motion of an object relative to its relationship with velocity, acceleration and displacement. In this experiment, the cart is used to study motion, utilizing those relationships. Velocity, displacement and acceleration are all related to each other through certain equations that are 𝑑𝑣𝑥 demonstrated in this experiment. Considering they’re all related, we use the equations, ax= 𝑑𝑥 𝑡 which basically says that acceleration is equal to the slope of velocity over time, v-v0=∫0 ax 𝑑𝑡 which states that the change in velocity is equal to the area under acceleration vs. time, x-x0 = 𝑑𝑥 𝑡 ∫0 Vx 𝑑𝑡 , which states change in position is equal to the area under a velocity curve and Vx= , 𝑑𝑡

which states that velocity is equal to the change in position with respect to time. By measuring the velocity, acceleration and displacement of the cart, the previous equations will be proven correctly by extracting correct information Procedure: • Turn the iOLab on and plug the dongle into the computer. Check if the accelerometer and wheel sensors are visibly present on the top of the screen. • Make sure the cart is facing down on a flat surface with the wheels facing down. • Press the record button and push the cart, further allowing it to come to a stop on its own. Stop the recording once the cart has come to a full stop • Adjust the graphs for maximal data collecting. • Once zoomed into proper segments of the 3 graphs, click the analytics button to measure proper values such as the average, area, and slope. • Analyzing the data recorded, test the relations of the values achieved to the equations 𝑡 𝑑𝑣𝑥 ax= 𝑑𝑥 and v-v0=∫0 ax 𝑑𝑡, further allowing a validation of the hypothesis. • Record the back and forth motion of the device between your hands in the y direction. • Once the recording is done, adjust the graph so it only shows 2 hits from one hand to the other. • Use the analytics button to highlight the graph that shows the slope of displacement and average velocity portrayed on the same interval. • Find the velocity wave and located the area under the curve • Locate the starting and ending positions along with the previous data’s intervals. • Calculate the area under the acceleration curve and its corresponding average velocities for before the curve starts and after it ends. • Compare the rate of change in the position curve to its average velocities. • Compare the change in position values (y) to the area under the velocity curve corresponding to the same intervals. • Compare the area under the acceleration curve to the change in velocity. • Ensure your accelerometer is visible on the screen and position the cart, so the positive z is facing upwards.

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Click the record button and allow the data to read 10 seconds of the device lying flat on the table, and then record data in the same graph while dropping the device on a pillow in the same direction. Results: Figure 1: Linearly

Decreasing Velocity

Figure 2: Linearly

Decreasing Velocity Equation Verification

Figure 3a: Linearly Position Change

Figure 3b: zoomed in

Figure 4: Area Under a Peak

Figure 5: Acceleration

Calculations derived from data

Discussion: According to figure 2, the data extracted from figure 1 proves the hypothesis correct that acceleration is equal to the slope of velocity with respect to time. This can be proven by 𝑑𝑣𝑥 using the equation ax= and plugging in the values that were found. Acceleration which 𝑑𝑥 reporting back to figure 1 is equal to 0.247 m/s2 and the rate of change of velocity over time, which is equal to the slope of the velocity curve is equal to 0.26 m/s2. So being that these are very close to each other, it is verified that acceleration is equal to the slope of the velocity curve with respect to time. The second relation that can be proven by the found data, is that the change in velocity is equal to the area under the acceleration curve. This relation is proved 𝑡 when the corresponding values found in figure 2 are plugged into the equation v-v0=∫0 ax 𝑑𝑡. Referring back to figure 2, the change in velocities is found to be 0.78 m/s and the area under the acceleration curve is reported to be 0.778 m/s. When calculating the percent error by plugging in ((v-v0)-a(dv/dt))/(v-v0) x 100 is equal to a 0.2571 % error which means it is valid to say that this experiment supports the idea that the change in velocity is equal to the area under an acceleration curve with respect to time. Referring to figure 2 the data observed from the experiment coincides with the previously mentioned hypothesis, which of that states that velocity is = to the slope of position with respect to time. This can be mathematically derived with the data from figure 3 by the 𝑑𝑥 equation Vx= . Stated in the calculations sheet, average velocity was found to be 0.274 m/s 𝑑𝑡 and the change in displacement is S=0.29 m/s. These two values are fairly close and therefore agree with the equation that states velocity is equal to the slope of position. Another relation that can be proved right by the data found in this experiment is that change in position is equal to the area under a velocity curve with respect to time. This is found with the data in the 𝑡 calculations sheet by the equation x-x0 = ∫0 Vx 𝑑𝑡. The change is position is calculated to be x (0.248m) –x0 (.001m) = 0.247m. The area under the curve can be extracted from figure 3, which shows a= 0.244m. This extraction of data from the experiment agrees with the hypothesis that the change in position is equal to the area under a velocity curve with respect to t. The last 𝑡 thing that can be proven by figure 3 and its data, is the equation v-v0=∫0 ax 𝑑𝑡 . Found in figure 4, the change in velocity is calculated to be 0.560 m/s according to the findings in the experiment, and the area under the acceleration curve corresponding to the same interval is a= 0.511 m. Although this data is not extremely close in values, it is very similar to each other and is valid enough to support the hypothesis that change in velocity is equal to the area under an acceleration curve with respect to time. Lastly, according to figure 5, the experiment demonstrates what happens in different situations of acceleration and positions. When the device is sitting on the table for the first 10 seconds of the graph in figure 5, the acceleration is equal to zero because it is not moving anywhere. But when the device is dropped and in freefall the acceleration actually does begin to equal a value. This makes sense because when an object is in free fall in the y coordinate, gravity takes responsibility for the acceleration, whereas when the object is not falling at all, gravity has no effect on it. Errors that may have occurred in this experiment can be due to improperly highlighting the correct intervals for some of the values, thus leading to a bigger percent error.

Conclusion: In this experiment, kinematics was properly supported by the data extracted from the performed experiments. The values calculated by the graphs almost perfectly prove that the relationship between velocity, acceleration, and position are correct. It can be stated that the relationship acceleration= the slope of velocity, change in velocity= area under acceleration curve, velocity= to the slope of position and the change in position= area under a velocity curve are all mathematically and experimentally proven correct....