This lab measures the linear motion of a cart on an air track with a motion sensor. In the first experiment the air track was level. Two data plots were taken of this motion and stored in the Logger Pro program for analysis using the supplied data table. The track was tilted in the second experiment. Three data plots were taken of the motion and stored in Logger Pro. Some of the graphs created by the data in Logger Pro will appear later in this analysis.
Analysis of the motion along the air track helps the beginning physics student understand motion in both the horizontal and vertical direction. Slanting the air track at a slight angle allows for the easy calculation of the cart acceleration and allows the student to see how the acceleration affects the cart along the horizontal plane. This lab reinforces the use of derivatives and integration to understand the relationship between distance, velocity and acceleration.
The first part of the experiment required the cart to be accelerated along the air track by hand while the sensor collected the data in the Logger Pro program. Two experiments were done with the data collected and saved in the program. The second part of the experiment required the track to be tilted and three experiments done with the data saved in the Logger Pro program. The data was analyzed according to the instructions and put in the data table which appears at the end of this report. During the analysis fourteen questions were asked while compiling the data for the table. The answers to these questions are the basis for the results of this analysis.
A: Distance vs. Time
5. Evaluate the derivative at tmax. What does this value represent?
B: Velocity vs time
Yes! The mean value is the same average velocity derived from part one using the slope of the distance vs. time curve.
7. Find the area under the velocity vs. time curve. What does this represent? Is it a reasonable value?
A. When you integrate (v dt) it gives the distance value back. As shown in the distance vs time data table the distance traveled by the cart was 0.314 m. The integral given in the velocity vs. time chart indicates 0.313 m. The difference is most likely from the program rounding in different graphs and tables.
B. Yes. This represents the distance traveled over the 1.2 seconds the integral was evaluated at that average velocity.
B. Yes
A: Distance vs time
t = .75 s
This derivative gives the velocity of the cart as it slides down the air track. Note that the velocity is negative (-0.321 m/s) indicating the cart is sliding down the track.
This is the acceleration over t = 0.75 s.
Part II
B: Velocity vs time
a. This derivative represents the acceleration of the cart as it slides down the air track.
b. Yes, although it seems to be slightly low at 0.77 m/s2 for a six degree angle.
a. The integral of the velocity equation returns the distance traveled by the cart or 0.599 m.
b. Yes-This is the distance traveled over the 0.75 seconds along the track.
Part II
C: Acceleration vs time
a. The integration of the acceleration returns the velocity of the cart over the selected time span of 0.75 seconds.
b. Yes.
3. Integrating the values gives the area under the acceleration curve which equates to the distance traveled for the cart over a set time period.
Agreement between theoretical and experimental values:
In this experiment the sin of was given as 0.10452 or six degrees. The acceleration values for this angle should approach approximately 1 m/s2, but only reach values between 0.827 to 0.835 m/s2. These values are off by approximately 18%, leading me to believe that the person responsible for measuring the angle did not accurately measure it. The angle was given as ± one degree. At minus one degree the angle does approach the values given by analysis of the graphs. This certainly would account for the uncertainty of the theoretical value.
Other areas of decreasing acceleration values could be some slight air resistance by the sail attached to the cart, although it is expected that value would be minimal over the short distance traveled by the cart. There was a lot of trouble getting the sensor to work properly in these experiments. This will surely induce error into the values derived by the sensor data and the motion recorded by the sensor.
Overall the experimental values given by the sensor and Logger Pro graph analysis are consistent with the expected distance, velocity and accelerations values.
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