Lab #1: Linear Kinematics PDF

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Linear kinematics lab report....

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John Smith 2/16/21 PHY 133 L69 TA: Sergey Alekseev Linear Kinematics

Introduction: Linear Kinematics is the study of an object’s one-dimensional motion with respect to time. Specifically, it center’s around the object’s relationship with displacement, velocity, and acceleration. In this lab, we will use a cart to study motion by examining the relationships between position vs. time, velocity vs. time, and acceleration vs. time through the following 𝑑𝑥 equations: 𝑣𝑥 = 𝑑𝑡 , which states that velocity is equal to the change in position divided by the 𝑡

change in time or the slope of the position vs. time curve, 𝑥 − 𝑥 = ∫ 𝑣 𝑑𝑡, which states that 0

0

𝑥

the change in position (displacement) is equal to the area under the velocity vs. time curve, 𝑡

𝑣 − 𝑣0 = ∫ 𝑎𝑥 𝑑𝑡, which states that the change in velocity is equal to the area under the 0

acceleration vs. time curve, and 𝑎𝑥 =

𝑑𝑣 , 𝑑𝑡

which states that acceleration is equal to the change in

velocity divided by the change in time or the slope of the velocity vs. time curve. After measuring the displacement, velocity, and acceleration of the cart, we will be able to prove the equations that connect the three parameters by gathering correct data. Procedure: 1. Turn on the IOLab device and plug the dongle into the computer 2. Make sure the cart is facing wheels-down on a flat surface 3. Press the Record button to start recording data, give the cart a push, and then stop recording when the cart has completely stopped 4. Adjust the graph for maximal data collection before clicking the analytics button to measure proper values such as the average, area, and slope 5. Analyze the data to test the relationship between position vs. time and acceleration vs. time 6. Analyze the data to test the relationship between position vs time and velocity vs time Record data for hitting the cart back and forth between your hands

Figure 1: The picture shows me conducting step 7 where I push the cart back and forth between my hands.

7. Adjust the graph to compare the slope of position to average velocity and compare the change in velocity to the area under the acceleration curve 8. Make sure your cart is positioned facing upwards in the z-direction 9. Start recording data while the cart is at rest for 10 seconds 10. Continue recording while dropping the cart in freefall

Results:

Figure 2: The graphs show the results of position, velocity, and acceleration of the cart with respect to time over a certain time interval for the Linearly Decreasing Velocity experiment.

As seen in Figure 2, the slope of the velocity graph, -0.23 m/s2, is approximately the same 𝑑𝑣 as the acceleration, -0.220 m/s2. This supports the equation 𝑎𝑥 = 𝑑𝑡 , which states that acceleration is equal to the velocity’s average rate of change (slope). In addition, we can see that the area under the acceleration graph is approximately equal to the change in velocity. This helps 𝑡

prove the relationship 𝑣 − 𝑣 = ∫ 𝑎 𝑑𝑡, which says that change in velocity is equal to the 0

0

𝑥

integral of (area under) the acceleration curve.

Figure 3: The graphs show the results of position, velocity, and acceleration of the cart with respect to time for the Back and Forth Motion experiment (particularly when average velocity is positive)

I analyzed my graphs again to observe a time interval where the average velocity is negative. The tables below shows the data that was collected and the corresponding Error Analysis: Time Interval

Slope of x vs. t (m/s)

Average Velocity (m/s)

Error Analysis (m/s)

0.89s - 1.89s

0.17

0.144

± 0. 12

2.89s - 3.30s

-0.37

-0.319

± 0. 11

Figure 3.1: The table above displays and compares the data collected for the slope of the position curve and average velocity.

Time Interval

ΔPostition (m)

Area under v vs. t (m)

Error Analysis (m)

0.89s - 1.89s

0.052

0.144

± 0. 053

2.89s - 3.30s

0.093

-0.05

± 0. 066

Figure 3.2: The table above displays and compares the data collected for the change in position and the area under the velocity curve.

Figure 3.1 supports the relationship 𝑣𝑥 =

𝑑𝑥 𝑑𝑡

, which states that velocity is equal to the

position curve’s average rate of change, (slope) because the slope of the position vs. time graph 𝑡

is approximately equal to the average velocity. Figure 3.2 proves the equation 𝑥 − 𝑥0 = ∫ 𝑣𝑥 𝑑𝑡, 0

which says that the displacement is equal to the integral over a certain time interval of the velocity curve. The change in position (displacement) is approximately equal to the area under the velocity vs. time curve.

Figure 4: The graph displays the acceleration of the cart in the x, y, and z directions for the Accelerometer experiment.

Calculations: Error Analysis for row 1 of Figure 3.1: σ = 𝑣

2

2

(0 𝑚/𝑠) + (0. 12 𝑚/𝑠) = 0.12 m/s

Error Analysis for row 2 of Figure 3.1: σ𝑣 =

2

2

(0 𝑚/𝑠) + (0. 11 𝑚/𝑠) = 0.11 m/s

Error Analysis for row 1 of Figure 3.2: σ𝑥 =

2

2

(0. 053 𝑚) + (0 𝑚) = 0.053 m

Error Analysis for row 2 of Figure 3.2: σ𝑥 =

2

2

(0. 046 𝑚) + (0 𝑚) = 0.046 m

Discussion/Conclusion: Upon completion of this lab, we were able to prove all four equations of linear kinematics and show the relationships between position, velocity, and acceleration with respect to time. During the linearly decreasing velocity portion of the lab, our graphs proved that the slope of the 𝑑𝑣 position vs. time graph was equal to the average velocity (𝑎𝑥 = 𝑑𝑡 ) and that the change in 𝑡

velocity was equal to the area under the acceleration curve (𝑣 − 𝑣0 = ∫ 𝑎𝑥 𝑑𝑡) by getting values 0

that were very close to each other. The same could be said for when we proved that the average 𝑑𝑥 velocity equals the slope of position (𝑣𝑥 = 𝑑𝑡 ) and that displacement equals the area under the 𝑡

velocity curve (𝑥 − 𝑥0 = ∫ 𝑣𝑥 𝑑𝑡) in the back and forth motion of the lab. For example, during 0

the time interval of 0.89s to 1.89s, the slope of the position curve was 0.17 m/s while the average velocity was 0.144 m/s. The error was calculated to be 0.12 after using the addition/subtraction rule for error propagation, so the values were within ± σof each other. In addition, the data displayed in Figure 4 regarding the Accelerometer makes sense because the beginning portion shows that when the IOLab device was resting in the first 10 seconds, all three axes had constant acceleration, which was expected, and the acceleration in the z-direction was around 9.8 m/s2 (acceleration due to gravity). Then, when the device was dropped, there was a rapid shift in the z-direction since it was the direction that the device was dropped in. The results of our lab matched the expectations we had prior to experimenting, but they were not exact because of sources of error. Some possibilities are that the device was not calibrated perfectly and the service was not completely flat. Overall, linear kinematics was thoroughly tested during this lab and the values recorded were very close to the results predicted from the linear kinematics equations....