The raw data that we collected is duplicated below as well as the analyses of the raw data, including two graphs, the Linearised data, justifications for our uncertainties, and explanations and examples of the calculations used during this lab.
Raw Data Table and Accompanying Graph:
This graph shows that there is an inverse power relationship between the total mass of an object and the time it takes to hit the ground because the variable x is to a negative power close to –0.5. Furthermore, the relationship shown on the graph is not a perfect inverse square root relationship because the exponent is not exactly –0.5. This anomaly will be analyzed further in the conclusion.
Linearised Data Table and Accompanying Graph:
After finding the inverse of the square root of mass and graphing it against the time, it can be seen that the non-linear relationship becomes linear. This further proves that there is a proportional relationship between the inverse of the square rot of mass and the time, or in other words, an inverse relationship between time and the square root of the total mass. Furthermore, the fact that this graph has a positive slope shows that as the inverse of the square root of mass gets large (or the total mass gets smaller), so does the average time it takes the helicopter to fall with that mass.
The calculations that were done in this lab are represented below through both explanations and examples. The justifications for our uncertainties can also be seen below.
Uncertainty of Mass:
The uncertainty of the mass is plus or minus 0.1g. The Uncertainty is low because the measurements were done using an electronic scale. Therefore, there is only a very small margin or error involved.
Uncertainty of Time:
The uncertainty of time is plus or minus 0.1s for each of the trials and was based off of the human reaction time. Any deviation from the true value would be a result of human error—the time it takes for the button to be pressed after the helicopter touches the floor or is released.
Average Time:
Add up the total of all the trials and then divide by the number of trials (5)
Example: (2.15s+2.21s+2.14s+2.15s+2.14s)/5=2.158s
Average Time Uncertainty:
Find the difference between the trial that is farthest away from the average time and the average time itself. Then, add that difference to the established uncertainty of each trial—in the case of this lab that uncertainty represents the human error or stopping the stopwatch and is 0.1s.
Example: 2.21s-2.158s=0.052s 0.052s+0.1s=0.152s
Linearising the Data (finding the inverse of the square root of mass):
First, find the square root of the mass. Then, divide one by the square root of mass.
Example: √1.5g≈1.2247g0.5 1/1.224≈ 0.8156 1/g0.5
Inverse of the Square Root of Mass Uncertainty:
First, divide the uncertainty of mass by the total mass to find what percentage of the total mass the mass uncertainty is. Then, multiply that decimal by the inverse of the square root of the total mass to find the uncertainty.
Example: 0.1g/1.5g≈0.0667 0.0667(0.8165 1/g0.5) ≈0.0544 1/g0.5
In completing this lab, we found that time and the square root of mass are two inversely proportional variables. This can be concluded because when the raw data was graphed on a set of axis, the best fit line was a power function where x was to the power of about –0.7, which can be rounded –0.5. This error of –0.2 can be explained by random error, which is compensated for by our uncertainties. Therefore, the equation that includes x to the power of –0.5 is within our uncertainty. Furthermore, when the inverse of the square root of mass was then taken, the graph became very close to linear. The positive slope of this linear line, 3.3568, shows that as the inverse of the square root of mass gets larger, so does the average time it takes the helicopter to fall by a factor of about 3.35. Furthermore, because the inverse of the square root of mass is taken to make this relationship linear, it can be concluded that as the square root of mass increases the average time it takes an object to fall decreases. The fact that mass and time have an inverse relationship makes perfect sense because as the mass of an object increases so does the downward force of gravity on that object causing the object’s acceleration to increase, which decreases the time it takes that object to fall. Although, the inverse relationship between mass and time makes sense, the fact that it is the inverse of the square root of mass does not make sense. Because the equation for the force of gravity on an object is the acceleration of gravity times the mass of an object, is not a power function, but rather a linear equation, poses the question as to why the square root of mass had to be taken to linearise the data. We believe that our data was changed more by random error than systematic error. We believe that mainly random error was involved because the parts of our experiment that may encounter error all involve the human element. For example, the release of the helicopter may not have been in perfect unison with the starting of the timer. We will further discuss the various types of error that may have occurred in the following paragraph. Furthermore, some systematic, or random systematic error must have occurred because theoretically the line of the inverse of the square root of mass vs. time should have crossed the y-axis at (0,0), but in fact crossed the axis at (0, -0.5703) skewing our data down by about 0.5. In fact, there should have been no y-intercept because you cannot divide one by zero, and therefore cannot get an x-value of zero. Although there was some error in our data, we did come to the undeniable conclusion that there is some kind of an inverse relationship between mass and time, which proves our hypothesis correct, which was that if the mass increased the time would decrease.
The experiment as a whole was conducted very smoothly and without incident. The only noteworthy item to mention would be the effect of the random and systematic error on the data. The only anomaly that could have resulted in systematic error would have been the height of the measured piece of PVC pipe. Because the piece of PVC had been set up by hand, it is not known whether or not the pipe was at a perfect ninety degree angle with the floor, which would cause the true height of the drop point to decrease, causing the time for the helicopter to fall the measured distance to be less overall. The various types of random error are all forms of human error. As previously stated, the release of the helicopter may not have been timed perfectly with the start of the timer. Similarly, the stopping of time may not have been in perfect unison with the helicopter touching the floor. Also, the drop height may have randomly changed with each trial because the helicopter was held by hand and was also lined up with the top of the PVC pipe by eye. All of these random errors could have affected the results by either increasing or decreasing time. If time was started late or stopped early, the time it would take the helicopter to fall the distance would lessen and visa versa. Furthermore, if the drop height were increased the time it would take the helicopter to fall would be increased as well and visa versa.
There are two important improvements that could be made to this lab that would dramatically increase the accuracy of the results. The first would get rid of the systematic error encountered. By simply using a level or a protractor to check the angle of the PVC pipe with the floor and make sure it is ninety degrees, then the height of the PVC pipe would be certain. The second would be to set up an infrared laser system. We could set up two lasers, one above the other, and measure the distance between the two beams, and then connect a timer to the system that would start when the first beam was broken and then stop when the second beam was broken. This would not only get rid of the systematic error of not knowing the true height of the drop zone, but it would more importantly eliminate the random error of starting and stopping the timer and even the release of the helicopter. With the lasers, the start and stop times of the fall would be much closer to exact, and the helicopter could be dropped above the first beam and be allowed to fly through it because the time would not start until the first beam is broken, and there is a measured distance between the two beams.
