Equations of Motion - Lab report PDF

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Lab 2: Equations of Motion

Date: 9/13/21 Landon Harris and Manu Nair TA: Zane Ronau

Introduction and Objectives In this experiment, a constant velocity chart was compared against a constant acceleration chart. Velocity represents position over time and acceleration represents velocity over time. Various plots were constructed to show that acceleration is related to position over time via a quadratic equation while velocity is related to position over time via a linear equation. The experiment also explored that acceleration can be calculated from a velocity over time chart via a linear equation.

Data and Analysis In the first experiment, a trial was conducted via an online simulator. The velocity was set at 3 m/s and an acceleration of 0 m/s . The gas pedal was held for two seconds and the distance the 2

car traveled was measured from the front of the car. Although the distance was measured at tick marks at each meter, the distance was able to be reasonably measured to the nearest tenth of a meter with an uncertainty of ±0.1 m. The trail was conducted eight times each time increasing by around 2.0 s and the results are shown in table 1. Using the eight trials, a scatterplot was constructed. The graph is located at the bottom of the page and is labeled Graph 1. The trendline for the scatter plot had a slope of 2.9565. In a position vs time graph, the slope of the line represents velocity due to change in position over change in time. The percent error was calculated as (2.9565 - 3.0)/ 3.0 = -0.0435. Taking the absolute value times 100%, the percent was 4.4%.The slight discrepancy between the two values could be caused by approximation errors. Using the same online simulator, the velocity was kept the same and the acceleration was set to 2.1 m/s . Fifteen trials were conducted, each one being approximately one second longer 2

than the last (the first trial was 1.0 second, the last trial was 15.0 seconds), and position and velocity were recorded for each. Two scatter plots were constructed, a time vs. position plot, and a time vs. velocity plot. The acceleration determined by Excel from the position vs. time plot

(a_quad) was 2.137 m/s². The percent error was calculated as follows: (2.137-2.1)/2.1 = .037 x 100% = 3.7% error. Next, using the velocity vs. time plot, acceleration was calculated again using the slope (a_linear), and this was calculated to be 2.096 m/s² and the uncertainty value was .0084. The percent error was once again calculated as follows: (2.096-2.1)/2.1 = -.004. Taking the absolute value; .004 x 100% = .4% error.

Discussion 1. Equations 1, 2, and 3 are all supported from the data in Graphs 1, 2, and 3. Graph 3 shows the validity of Equation 2. Rewriting Eq. 2, a = (v/t) and this is shown in Graph 3. The slope is the change in the y axis (velocity) divided by the change in the x axis (time), which is exactly what Eq. 2 states. Equation 1 can be rewritten to v = x/t, which is shown by Graph 1. The change in the y axis (position) divided by the change in the x axis (time) gives the slope, which in this graph, is velocity. This is a visual piece of evidence of the validity of Eq. 1. Finally, Equation 3 is directly applied in Graph 2. The polynomial equation in the second order shows the validity of this equation. In this equation to get position, the acceleration side of the equation is divided by two. Doing the reverse to the leading coefficient in the quadratic equation gives the approximate acceleration used in the experiment. 2. The values recorded for velocity and position were most likely not accurate. They were recorded to the nearest tenth with an uncertainty of 0.1 m. If the initial velocity value and the position value recorded were calculated using any of Equations 1, 2, or 3, then the values would have had values slightly more accurate and not estimated to the nearest tenth. This was most likely the cause of the percent error calculations in the slope/acceleration values of Graphs 1, 2, and 3.

3. If the stated velocity did not fall within the bounds of uncertainty, then it would invalidate the claim that they represent the same value. Velocity represents position over time which is given by the slope of the line. The only way the stated value could fall out of the bounds and not invalidate the data would be if there was a lot of error in the experiment. The same applies to the slope for Equations 2 and 3. Acceleration is strictly defined as the change in velocity over time which is what is graphed in Graph 3. If the slope does not fall within the bounds, this invalidates the claim that they represent the same value. Graph 2 presents the slope of the acceleration in a parabola as this is how acceleration is represented in a position over time graph. 4. If five seconds were used as the time interval, then slight changes between individual seconds would not be accounted for. However, the time interval should not change the values of the slope because the y-axis will still change at the same rate. The only possible difference would be the smoothing of the curve in Graph 2, or a slight straightening of the line in Graphs 1 and 3, since the smaller differences would be less represented on the graph. 5. The R value for this data was 0.9985, indicating that the data fits well with the trend of 2

the parabola. The visual quality of the graph shows the trend of the parabola, although the curve is very slight. However, this is probably due to the small range of x values, so if the range was larger, the visual parabola would be much more evident. 6. From the leading coefficient, the value for acceleration is 25.855 x 2 = 51.77 cm/s² and in SI units, this can be converted to .518 m/s².

7. The two most significant sources of uncertainty in this experiment would be the initial start speed being corrupted when it should have been 0 m/s and the improper placement of the photogates at the exact positions. 8. According to the equation provided by the graph, the initial speed would be 12.58 cm/s² or .1258 m/s². This is close to zero, but still indicates that the cart had some initial velocity. This was most likely due to a slight forward force being applied by the experimenter in order to start the motion of the cars. Since the car was being kept in place by a magnet, it needed to be pulled off of it, and this may have added some starting velocity. Also, the length of the car may have factored into this since the photogate recorded when the first part of the car crossed the laser. This means the initial position and speed of the car may have been slightly off.

Conclusion For the first part of the experiment, an online simulator was used to compare the differences between a car traveling at constant velocity and a car traveling at constant acceleration. The slope of the constant velocity graph represented the velocity the car was traveling. For the constant acceleration trials, both a position over time graph and a velocity over time graph were created. The slope of the 2nd order polynomial graph represented the acceleration of the graph as well as the slope of the linear equation of the velocity over time graph. The next part of the experiment was conducted in person and the experiment again showed how position over time graphs correlate to acceleration via a quadratic equation.

Table 1

Trial Time (s) Position (m) Uncertainty (m) 1

1.8

5.3

0.1

2

3.7

11.1

0.1

3

5.5

16.4

0.1

4

7.5

22.4

0.1

5

9.6

28.5

0.1

6

11.4

33.8

0.1

7

13.4

39.7

0.1

8

15.4

45.6

0.1

Graph 1

Table 2 a = 2.1 m/s² v(initial) = 3 m/s²

Trial Time (s) Position (m)

Pos. Unc. (m) Velocity (m/s)

1

1.0

5.1

± 0.1

5.1

2

1.9

9.7

± 0.1

7.0

3

3.1

19.8

± 0.1

9.8

4

4.0

28.8

± 0.1

11.3

5

4.9

40.7

± 0.1

13.2

6

5.9

53.9

± 0.1

15.1

7

7.1

73.9

± 0.1

17.9

8

7.9

89.9

± 0.1

19.8

9

9.1

113.9

± 0.1

22

10

9.9

132.4

± 0.1

23.8

11

11.0

160

± 0.1

26

12

12.0

188.7

± 0.1

28.2

13

12.9

212.8

± 0.1

30

14

13.9

245.8

± 0.1

32.2

15

15.0

282

± 0.1

34.5

Graph 2

Graph 3

Table 3 Time 1 (s)

0.3143

Time 2 (s) Time 3 (s)

0.3181

0.3258

Time Avg. (s) Position (cm)

0.3194

10

0.5729

0.5687

0.5788

0.5735

20

0.7436

0.7509

0.7709

0.7751

30

0.9843

0.9909

0.9956

0.9903

40

1.1267

1.1062

1.0933

1.1087

50

1.2637

1.2487

1.2390

1.2505

60

1.3614

1.3360

1.4135

1.3703

70

Graph 4...