Lab 2: Free Fall Lab

                                                                       Free Fall Lab
                                            Ian, Mo, and Caesar 
                                                   3/1/17

The fact that in the absence of any air resistance and other external forces, any object in free fall will accelerate at a constant rate of 9.81 m/s^2

When objects are relatively close to the surface of the earth, they undergo a constant acceleration of g when in free fall. In order to measure physically the effects of g on objects, a measuring apparatus must be used. This apparatus is a 1.5 m long column that is anchored to a heavy tripod base with an electromagnet and a spark sensitive tape. When the object is released from rest, its displacement is recorded with on the tape at 60 frames/ second. Recording and graphing this data gives us the position vs time graph for the object. Then, in the velocity vs time graph is obtained by using the formula of  Δx / the time due to the frequency = 1/60th of a second. This will give us the desired result. Finally, the slope of the velocity vs time graph as determined through the " equation on chart" on excel is the value of g because acceleration is change in velocity per time.

The measurement/ observation portion of the lab consisted of setting up the column with the tripod base with the tape. Since all of the actual processes of recording the spark on the tape are already completed. This initial step is just for reference. With the tape already marked, the next step is to measure the distance of each mark from the 0-cm mark. Then, we proceed to find displacement, mid-interval time, and mid-interval speed. All of these data are recorded on an excel spreadsheet.

TIME(s)	DISTANCE (cm)	∆x (cm)	Mid-Interval time (s)	Mid-Interval Speed               (cm/s)
0	0	0	0.008333333
0.01666667	1.2	1.2	0.025	72
0.03333333	2.8	1.6	0.041666667	96
0.05	4.6	1.8	0.058333333	108
0.06666667	6.7	2.1	0.075	126
0.08333333	9	2.3	0.091666667	138
0.1	11.5	2.5	0.108333333	150
0.11666667	14.4	2.9	0.125	174
0.13333333	17.3	2.9	0.141666667	174
0.15	20.9	3.6	0.158333333	216
0.16666667	24.4	3.5	0.175	210
0.18333333	28.6	4.2	0.191666667	252
0.2	32.6	4	0.208333333	240
0.21666667	37	4.4	0.225	264
0.23333333	41.7	4.7	0.241666667	282
0.25	46.7	5	0.258333333	300
0.26666667	52	5.3	0.275	318
0.28333333	57.3	5.3	0.291666667	318
0.3	63.2	5.9	0.308333333	354
0.31666667	69.3	6.1	0.325	366
0.33333333	75.8	6.5	0.341666667	390
0.35	82.3	6.5	0.358333333	390

Mid-Interval Time Sample Calculations

                                                  Mid-Interval Time=time+(1/120);  

                                                  0.025 s= 0.01666667 s+(1/120)s

                                                 

                                                   Mid-Interval Speed

                                                  

                                                  Mid-interval speed= Δx/(1/60 s) 

                                             

                                             72 cm/s= 1.2 cm/ 0.025 s

The two graphs above depicts velocity vs time and distance vs time. From the slope of the basic vs time graph is the velocity, and from the slope of the velocity vs time graph is acceleration. Therefore, the graphs allow the data to display visually the relationship between distance, velocity, and acceleration. These graphs were obtained through the use of scatter plot on excel document, and the line and equation were further obtained through the " display equations on chart" and " Display R-squared value on chart."

This lab was fairly straightforward with only the " hard " part on measuring the distance of each of the marks from  the 0-cm mark. The rest of the lab was filling in an excel spreadsheet and graphing the results of the position vs time and velocity vs time. The only major source of error in this lab is the precision of the measurement of the distances from the 0-cm mark. This human error includes the discrepancy in measurement of the uncertainty value of 0.1 cm. This initial uncertain value of distance can affect the value of the mid-interval speed. 

Questions/Analysis:

1. In the case of constant acceleration, the average velocity during that time interval is equal to the mid-time interval velocity because the graph of velocity vs time is a straight line, and the average of the initial velocity and final velocity just so happens to lie right on the midpoint between initial and final. 

2. I can get the acceleration due to gravity from my velocity/time graph by calculating the slope of the line or by looking at the equation for the line of best fit. My result is 9.63 m/s^2 which is 0.18 m/s^2 below the accepted value of 9.81 m/s^2. That is about 1.8% relative difference. 

3. I can get the acceleration due to gravity from my position/ time graph by taking the derivative of the slope of the line. Since the slope of the position vs time graph is velocity, and the derivative of velocity with respect to time is acceleration. The result I get is 9.67 m/s^2 which is 0.14 m/s^2 less than the accepted value of 9.81 m/ s^2. That's about 1.4 % relative difference. 

Additional Questions

1. There aren't any particular or noticeable pattern in the values of the values of g. 

2. The class' average value is below the accepted value by 0.18 m/s^2 or 1.8% relative difference. 

3. There aren't any patterns in the class' values of g. 

4. The factors of measurement of the distance on the ruler and air resistance. The human measurement of the distance on the ruler is a random error, and the air resistance is a systematic error. 

5. Part 2 of the free fall lab is to investigate and analyze the average of each group and the average of the class as well as to calculate the standard deviation of the mean for the class data. Introduction to the concept of the deviation of the mean is with the coffee machines that delivered variable grams of coffee. The two coffee machines exhibited different variability in each trial of the amount of instant coffee delivered. This gives rise to the theory of deviation from the mean value and how to calculate the mean so that it's value is positive. Then the formula of the standard deviation of the mean is introduced with the sigma symbol. The standard deviation of the mean describes the spread of the data. We were supposed to learn the definition and concept of the mean and the various deviations of the mean. In addition, the graph ofthe standard deviation of the mean was also introduced as a bell curve.