The Relationship between the Peak Height of a Bouncing Ball and the Time at the Peak Height - Coursework Example

What is the relationship between the peak height of a bouncing ball and the time at the peak height? By + name University name City, State Date Design Aspect 1: Defining the problem and selecting variables The core aim of this study shall be in relation to the determination of the relationship between the peak height of a bouncing ball and the time at the peak height. In this regard, the dependent variable shall be the time at the peak height while the independent variable shall be the peak height obtained by the bouncing ball. Therefore, the study shall seek to determine how the peak height of a bouncing ball will influence the time at the peak height. This implies that other variables will be required to be either controlled or monitored as are illustrated in the table underneath, along with the dependent and independent variable highlighted above. Variable Type of variable Why and how it was controlled/monitored Peak height of a bouncing ball Independent The variable was independent Time at the peak height Dependent The variable was dependent Height drop Controlled This will be measured by the use of a ruler to determine the starting height of the ball. The subsequent height will be automatically calculated considering that the next bounce is half the first one. The ruler has been chosen in this context so as to ensure the efficiency of the results. Therefore, only one ruler was used to measure the height of the bounces of the ball. Surface material Controlled Given that the difference in the surface material from which the ball is bounced may cause variation in the obtained results for heights and time taken, a single surface was maintained throughout the experiment. The chosen surface for bouncing the ball was on a flat smooth concrete floor. Elasticity and size of the bounce ball Controlled/monitored The same ball was used throughout the experiment taking into consideration the aspects of equal pressure, size and elasticity to bounce back. A ball that does not bounce back would have distorted the results of the study for failing to meet the half bounce back height. Controlling Variables The controlling variables will be kept at constant rates so as not to have any effect on the outcome of the study. The height drop of the ball shall be kept constant by ensuring that only the same ruler is used as a measuring gadget. This will make the measurements obtained to be constant considering the calibrations used. The surface material variable will be kept constant by ensuring that the ball is bounced on a similar surface for the entire course of the experiment. This will ensure that there is no variations in the elasticity of the bounce by the ball. The surface to be used shall be a smooth flat concrete floor. The elasticity and size of the ball variable shall be maintained by ensuring that only one ball is used throughout the experiment. This aspect will be measured by gauging the pressure of the ball before each experiment as well as the elasticity of the ball to ensure that it is able to bounce at the same rate. Data collection methods Given the following variables in which some are being monitored, controlled or requiring to be measured as whether independent or not, the following set up was put in place in order to allow for the accurate recording and observation of the results. a. The measurement ruler was vertically mounted at perpendicular angels to the floor so as to ensure that the accurate heights of the ping pong ball during bouncing are noted. b. The floor was also assessed in order to be able to determine if it was flat and appropriate to be used for the bounding balls. c. I then set the stop clock ready in order to use it to record the time taken by a ball to achieve the peak height. d. The ball was then bounced and the recordings taken accordingly based on the peak height achieved after each bounce. The data that shall be collected shall be in relation to the time at bounce, the time at the peak, the falling time, and the uncertainty in the time. Subsequently, the data collection shall be in relation to the height obtained by the ball at peak, the uncertainty in height, the log of the height and the uncertainty in the log of height. The restrictions to the data collection process was in the determination of the uncertainties in the data, particularly in relation to the determination of the uncertainties in the time taken to attain peak in each bounce. However, considering the strength of the correlations, I was able to correlate the data and attain a line of best fit based on the log of the uncertainties. This is as presented in the graph below. The line of best fit was used to obtain the regression equation: y = -0.2796x + 0.4724 Data Collection and Processing Raw data The following table indicates the raw data that was collected in relation to this study. Time at bounce Time at peak Time between bounces Falling time Uncertainty in Time Height at Peak Uncertainty in Height Log of Height Uncertainty in Log of Height t(bounce) / s t(peak)/ s t(between) / s t / s Δt / s h / m Δh / m log(h) Δlog(h) ± 0.002 ± 0.002 ± 0.002 ± 0.002           1.320                 1.786                 2.533 2.502 0.466 0.233 0.009 0.543 0.009 -0.265 0.01 2.967 2.160 0.747 0.374 0.005 0.412 0.004 -0.385 0.00 3.467 2.750 0.434 0.217 0.009 0.318 0.006 -0.498 0.01 3.791 3.217 0.500 0.250 0.008 0.260 0.004 -0.585 0.01 4.167 3.629 0.324 0.162 0.012 0.213 0.005 -0.672 0.01 4.701 3.979 0.376 0.188 0.011 0.173 0.556 -0.761 0.01 4.956 4.434 0.534 0.267 0.007 0.150 0.002 -0.824 0.01 5.330 4.829 0.255 0.128 0.016 0.128 0.004 -0.893 0.01 5.542 5.143 0.374 0.187 0.011 0.109 0.002 -0.963 0.01 5.723 5.436 0.212 0.106 0.019 0.089 0.003 -1.051 0.02 5.956 5.633 0.181 0.091 0.022 0.082 0.004 -1.086 0.02 6.253 5.840 0.233 0.117 0.017 0.073 0.003 -1.137 0.01 6.502 6.105 0.297 0.149 0.013 0.059 0.002 -1.229 0.01 6.645 6.378 0.249 0.125 0.016 0.051 0.002 -1.292 0.01 6.782 6.574 0.143 0.071 0.028 0.045 0.003 -1.347 0.02 6.963 6.714 0.137 0.069 0.029 0.039 0.002 -1.409 0.03 7.201 6.873 0.181 0.091 0.022 0.034 0.002 -1.469 0.02 7.345 7.082 0.238 0.119 0.017 0.031 0.001 -1.509 0.01                                     Max Gradient Min Gradient           x-value y-value x-value y-value           2.502 -0.258 2.502 -0.273           5.436 -1.067 5.436 -1.034           In ascertaining the uncertainties in the heights as found out during the experiment, a measure of ±0.002 has been adopted as an adjusting value for the time periods recorded during the bouncing stages. However, the uncertainty values have a great effect in the results in that it would likely lead to experimental imperfections. As a result, the quantitative data may be affected; hence, an effect on the availability of the qualitative data. Processing Raw Data The raw data obtained above was processed in various forms, key among them, the time taken in each phase of the bounces, and the uncertainties. For instance, the uncertainty in time was found using the formula: Δt / s While uncertainty in height, Δh / m, and uncertainty in the log of the heights being, Δlog(h). Presenting processed data The graph below indicates the plotting that was done for the obtained results to determine the relation between height and the time at peak. Using the information contained above in relation to the maximum and minimum gradients for the values of x and y, the uncertainty for line of best fit may be calculated using the procedure: ½ range = (maximum-minimum)/2 For the x-value, the value for the uncertainty of the line of best fit will be zero. Conclusion This study’s main objective of determining the relationship between the peak height of a bouncing ball and the time at the peak height has been obtained with the results showing that there is a negative association between the dependent and independent variables. This yields in a gradient line of 1.6057 in the equation y = 1.6057e-0.544x. The independent value (peak height) which was started at 0.4m, was observed as decreasing with time in each bounce, and the recorded results are as plotted in the graph above. However, the answering of the research question is pegged on a lot of uncertainty which is indicated as having a huge impact due to its variations in the points. There is almost seen a straight horizontal line in the graph of uncertainties having the equation y = -0.2796x + 0.4724. The overall relationship seen through the graph of the experiment is one of a steady decrease in the height and the time taken to reach the peak height. However, there is no proportionality in the relationship given that the line does not pass through the origin. The uncertainties in both the time and the height may be regarded as insignificant given that in both instances they were less than 0.1%. However, the spread in the results was a result of the uncertainties which caused the weakness in the results. Moreover, the study did not employ the maximum and minimum gradient lines because placing them on the graph would have minimal effect given that the spread and variation of points make these of no worth (DeRugy, Wei, Müller & Sternad, 2003). Therefore, the relationship has been obtained through the accuracy in data collection, processing and analysis. Evaluating Procedures Source of error Effect on the results Improvements that can be made to limit the source of error Recording of the height of the bounce Even though the uncertainty in the height of the bounce was accounted for by ±0.002, and seemed marginally less, the effect could have been great. This is because the ball was not always dropping perpendicularly, and parallel to the ruler. Ideally, the height may have significantly varied, based on the Hypotenuses’ theory. This error can be minimized if an apparatus is set up that defined the source of the ball. Alternatively, the dropping point for the ball could be fixed at a single place; thus, making it definite that the variation in the height of drop and peak height are minimized. Initial speed when ball is dropped may not be zero The ball was released by the hand under the assumption that its initial velocity was zero. However, errors may arise due to external net forces that might alter the assumption of zero velocity. This would result in a larger height than expected. Additional apparatus (fixed release mechanism) such as mention in the first case should be used so as to hold the ball still before release. When being released, it should be ensured that no external forces are experienced that may exert pressure on the ball; thus, affecting the speed and height measurements (DeRugy, Wei, Müller & Sternad, 2003). Even though this may not clear the error, it may result in the consistency in the test values; hence, decreasing the error values. Velocity of release of the ping ball Even though it is assumed in the study that the release velocity for the ball is at zero, it cannot be effectively approved whether the ball is released at this speed. Given that in this experiment, the release of the ball was performed using the hand, in dropping the ball, it is expected that at certain points in time, the hand could have exerted an amount of pressure on the ball, thus, causing it to be released at a speed higher than zero. This is a variation that could not even be rectified by the uncertainty in the time and the adjustment value. Solving this problem would require the fixing of another apparatus or release mechanism for the ball so that the velocity is kept at zero or evidently determined at a constant rate throughout the experiment. This, even though may not solve the error problem in the velocity would ensure that there is uniformity in the start speed for all the tests to be conducted. Improving investigations If the suggested improvements mentioned above are done, there will be a considerable reduction in the errors and uncertainties to be encountered in the process. Subsequently, the experiment would be more accurate and the results would be reliable since making the three improvements indicated above would have resulted in the reduction of the probable errors to be encountered in the study; thus, resulting in the release of near accurate and predictable results that can be subjected to further analysis and discussion in different forums. Furthermore, this experiment can be improved further by looking at the controlled variables and their effect on the experiment such that this is limited to all extents possible. The only variables that should be at play are the independent and dependent variables. Reference List DeRugy, A., Wei, K., Müller, H., & Sternad, D. (2003). Actively tracking ‘passive’ stability in a ball bouncing task. Brain research, 982(1), 64-78. Read More