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Experiment: Free-fallGuiherme Antonio Marczak GiorioThis report attempts to discuss the concept of free-fall and air resistance, looking at when the latter can be ignored, and present the results of a free-fall experiment using a steel and a plastic ball, by comparing their results with the theoretical value, g (9.8 m/s²). Using an automated system to calculate the time of fall for both balls, it was obtained an acceleration of 9.67 m/s² for the steel ball and 9.12 m/s² for the plastic one when using the gradient to calculate the acceleration. Results show that the values are somewhat consistent with the theory after considering the errors during the process.IntroductionThis experiment proves that the concept of free-fall provides underpinning knowledge to understand air resistance and, consequently, how fast objects fall. Without proper knowledge of these concepts, it would not be possible for people to use parachutes, for instance. The objective of the experiment was to neglect the drag force caused by air resistance and try to calculate the acceleration using SUVAT equations. The results were then compared with the actual acceleration from gravity, giving insight on how air resistance affects these bodies.The experiment also provides interesting insights on how different calculation methods can provide slightly different results.Upon research, it was also found that similar experiments have been conducted and their results were similar where the same height was used 1.Having a better understanding of free-fall is also helpful to understand basic Newtonian concepts and their applicability in real life scenarios.TheoryTo understand the concept of free-fall, it is first necessary to refer to Newton’s Second Law of Motion, commonly expressed by the formula:F ?=ma ?  (1)Where F is the force (N), m is the mass (kg) and a is acceleration (m/s²). In free-fall scenarios, however, a is commonly substituted by g (gravitational acceleration). In the scenario of any given object falling freely within the Earth’s gravitational field, its acceleration will always be the one due to gravity, amounting to approximately 9.8 m/s². This acceleration is independent of the mass of the object since gravity will act equally on each object.If there weren’t any other forces acting on the objects, then every object in free fall under the same conditions would fall at the same time as proposed by Galileo. However, this doesn’t happen due to an opposing Force exerted by the air, known as drag. In any body falling towards the Earth, the acceleration will be directed downwards and the drag upwards.This drag force helps deaccelerate the body and is expressed by the formula 2:F ?d=(?C_d A)/2?v^2  (2)Where ? is the density of the air (kg/m²), A is the area of the object that is in contact perpendicular to the air (m²), C_d is the drag coefficient (no units), which depends both on the object falling and its speed and v is the velocity (m/s).As the body starts to deaccelerate, it reaches one point where equation (1) will be equal to equation (2) and at that point the velocity will be constant. This velocity is known as terminal velocity.This concept is crucial as it helps to predict how fast an object will fall and what to do to reduce its landing speed. For instance, the typical terminal velocity of a parachutist is about 5 m/s whereas the one for a skydiver is 60 m/s (Halliday, Resnick and Walker, 2008). This means that accurate calculation of terminal velocity is vital in order to avoid accidents, and it plays an important role in the design of parachutes.On the other hand, there are cases where the drag force is too small and it can be ignored. In these instances, we can use SUVAT equations to determine either the time, the acceleration or the distance of something in free-fall.Given the SUVAT equation:s=u_0?t+1/2 a?t^2 (3)Where s is the total distance (m), u_0 is the initial velocity (m/s), t is time (s) a is acceleration (m/s²).To calculate the acceleration, given that the initial velocity is 0, formula 3 can be rearranged in terms of:t^2=2s/a  (4)  or a=2s/t^2  (5)By using equation (5), you can obtain the acceleration, however, there is also the possibility of plotting a graph of the time squared against the distance, which would mean that the slope of the graph would be half the amount of a, given that in equation (4) the double of the distance is used, but the plot is represented by s.Experimental methodTo calculate the acceleration of the two balls, a set of devices was used so that when connected between each other, it could precisely calculate the time between when the ball is dropped and when it reaches the floor.A magnet drop box was placed at the top in a way that, when it was on, it would hold the balls (a small magnet was added to the plastic ball to keep it in suspension). Once the timer was activated, the drop box released the ball and, when it reached the detector pad at the bottom, the smart timer would give the total amount of time taken. The height of the system could be adjusted with a clamp connected to the support which was holding the drop box, and a measuring tape was used to measure the height. This can be seen in more detail on the diagrams below:  Figure 1: The system set up with one of the balls being held by the drop box. Figure 2: Once the timer was pressed, the ball would instantly drop. Figure 3: Once the ball reaches the detector pad, the smart timer calculates the total amount of time taken for the given distance.After collecting the time measurements for different distances for each ball, two methods could be used to calculate the acceleration: plot a graph of time squared against distance using equation (4) and multiply the gradient of the plot by two to obtain the acceleration; rearrange the equation (4) in terms of a and calculate the acceleration from that.For this experiment, however, the first method was used for both balls. Since the plot was done as time squared against distance instead of the opposite, the gradient formula used to obtain the acceleration was:m_g=(x_1-X_0)/(y_1-y_0 )  (6)  Where x_1 is the final value in the x-axis and x_0  is the first one (both in m). The same pattern follows for y_1 and y_(0 )(s²).The m_g  was then multiplied by two. With that data, it was possible to compare the results with the expected acceleration due to gravity.Results Looking at the graph, it is noticeable that both lines are close to each other, but their gradient and, consequently, their acceleration, are different.The total average time for the steel ball was 0.3213s for the lowest height (0.5m) and 0.6430s for the highest (2.0m). For the plastic ball, it was 0.3122s at 0.5m and 0.6530s at 2.0m.The calculation of the gradient and of the acceleration was done using equation (6). For the plastic ball, g was 9.12m/s², and for the steel ball, it was 9.67 m/s².The error for the height (m) was ±0.05 given the precision of the measuring tape. The error related to time for the steel ball ranged from ±0.0005 to ±0.002, while for the plastic ball, it ranged from ±0.0001 to ±0.0003. For t² (s²), the errors ranged from ±0.0003 s to ±0.003 for the steel ball, and ±0.0001 to ±0.002 for the plastic one. For that reason, the error bars were too small and could not be shown in the graph.The errors for the time were calculated using SEM (Standard Error on the Mean) and using error propagation formulas 3. The final formula used to calculate the error on g can be found below:?z=m_g??((?x/x)^2+(?y/y)^2 )   (7)Where ?z is the final error (± m/s²), ?x is the error on x (distance) and ?y is the error on time squared. x and y are the mean values for distance and time squared.The error for the plastic ball was ± 0.23 m/s² and the steel ball was ±0.52 m/s².Using the second method mentioned in the experimental method and equation (5), the average g for the steel ball was 9.70m/s² and, for the plastic ball, 9.74 m/s². In this method, although the acceleration for the steel ball remains similar to the first method, there is a considerable change for the plastic ball.The second method also shows that, on average, both results are reasonably closer to the expected result of 9.8 m/s².DiscussionThe results are somewhat consistent with the acceleration due to gravity, with the steel ball achieving a closer result to the expected value of 9.8m/s² when using the first method.The results show that the steel ball seems to have been less affected by air resistance, resulting in an increased net acceleration from the system. Given equation (2), it is fair to assume that the plastic ball could have suffered from more drag force as it was bigger and, thus, it had a larger contact area with the air.It is also possible to see that, although it seems that the steel ball starts to descend more slowly, as the height increases, the time of descent for the plastic ball surpasses that of the steel one.It is also likely that the fact that the steel ball was heavier, while still smaller than the plastic one, influenced the result due to the former suffering from less air resistance, as mentioned by SILVEIRA, F. L. 4 ”For spheres of the same shape and different densities in free-fall, the effect of the air resistance will be lower on the body that has the higher product of density x radius” (free translation). For the distance, the error calculation included the error from the precision of the measuring tape, however, it did not account for reading errors or possible full reliability of the height, as height had to be adjusted manually and it was easy to miss the exact measurement by a few centimetres.When calculating g for each height using equation (5), it is also possible to see a distinct difference in the result, with the acceleration constantly decreasing as the height increases (10.25m/s² at 0.5m and 9.38m/s² at 2.0m).Overall, the results seem consistent and indicate that in some circumstances, such as very low height, air resistance can be ignored while still achieving reliable results. It also shows that smaller, higher density spheres achieve a better result.ConclusionOverall, both methods proved to be reasonably effective to calculate the acceleration due to gravity, with the steel ball getting more reliable results.The fact that the steel ball had better results is also consistent with the theory. Finally, it is possible to say that the results obtained indicate that in a controlled environment, spheres, when released from a small height will suffer little air resistance, and their acceleration can be calculated using simple SUVAT equations.References1 Valero, D. (2010). RELATÓRIO FINAL: MEDIDA DA ACELERAÇÃO DA GRAVIDADE POR QUEDA LIVRE E SENSORES.. Campinas: Unicamp, pp.6-7.2 Halliday, D., Resnick, R. and Walker, J. (2008). Fundamentals of physics. 8th ed. Hoboken, N.J.: Wiley, pp.122-123.3 Fellows, J. (2017). Foundations of Physics Laboratory Handbook 2017/18. Bristol: University of Bristol, pp.15; 22-24.4  Silveira, F. L. (2015). Efeito da resistência do ar na queda de corpos. online Available at: 

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