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lab 2 Free Fall Lab- determination of g and some statistics for analyzing data

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Purpose
Determine the gravity coefficient by determine the velocity of a falling object with spark generator and spark–sensitive tape. Compare the result with the actual value to calculate the error from systematic nature.

Procedure
The spark generator is connected to a cylinder which would fall with a distant of 1.5 m and was linked with the spark-sensitive tape. When the cylinder released, the spark generator left sparks on the tape with a frequency of 60 per second. After measuring the distant between sparks and analyzing those data, we can finally get the local gravity coefficient.

Method of Fitting
From the sparks we can derive the time of each spark. To ensure the velocity’s accuracy, we should choose a mid-interval time and the corresponding mid-interval velocity. When we plot velocities in excel and plot the fitted curve, the slope of the fitted curve is the gravity coefficient.

Conclusion and Analysis




The gravity coefficient determined by the slope of the velocity verse time curve is 9.62852 m/s^2, which is smaller than the value 9.8 m/s^2 we often use. The reasons of this error are multiple. First, the tape has friction when it falls, which may make the acceleration smaller than the gravity coefficient. Second, the frequency of the spark generator may be bigger than the 60 Hz, which will make the actual time interval smaller than 1/60 second and result in the decrease of gravity coefficient. Besides the 9.8 m/s^2 is an ideal value which is determined in vacuum. So the resistance from air can be one of those reasons. Though under the low velocity of the cylinder the resistance is quite small, the resistance cannot be ignored.





Questions:
1.      Show that, for constant acceleration, the velocity in the middle of a time interval is the same as the average velocity for the time interval.
Assume that the time interval is [t1, t2] and the acceleration is a. The velocity in the middle time equals the product of the middle time (t1+t2)/2 and the acceleration a. The average velocity equals the average of a*t1 and a*t2, which equals to the velocity in the middle time.
2.      Describe how you can get the acceleration due to gravity from your velocity/time graph. Compare your results with the accepted value.
The relation between velocity and time is V=a*t. So the slope of the fitted curve is the acceleration, which is 9.6358 m/s^2. It is smaller than 9.8 m/s^2 due to the reasons above.
3.      Describe how you can get the acceleration due to gravity from your velocity/time graph. Compare your results with the accepted value.

The relation between velocity and time is X=0.5*a*t^2. So the quadratic term coefficient of the fitted curve is the half of acceleration. So the acceleration is 9.6188 m/s^2. It is smaller than 9.8 m/s^2 due to the reasons above.

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