<p><img src="https://lh6.googleusercontent.com/m_3-uN8zzQtE3xwg6j-7AwRL4X6ST_1tXRKxTffzVA3R6icm8Oes_5LJkejcilyCv5N4KsuagFPoXhP7eHFfueqlUBZjQC7qVz-D-RgQ1FCWRJ_17FnG2JqwRqWTuhMPxXkH2kC-" alt=""></p><p>We are going to investigate why the sample IA “The exponential nature of a bouncing ping pong ball” is graded as 5 with respect to each of the 5 IA criteria and it can be improved to achieve a grade 7. The original scores of the IA can be seen below:</p><p> </p><p>Personal<a target="_blank" rel="noopener noreferrer" href="https://docs.google.com/document/d/1vkv6m6hx6zXV2qrWk2qJa_r01vPNKXRM1orwyZ2hU58/edit#bookmark=id.j15x29mi7fgp"> </a>Engagement (1/2)</p><p>Exploration (4/6)</p><p>Analysis (4/6)</p><p>Evaluation (3/6)</p><p>Communication (3/4)</p><p>Total (15/24)</p><p> </p><p>Each criteria will include the original text followed by a list of reasons of why it got the grade that it did, and some improvement tips that would help raise the score for that criteria.</p><p>The texts modified using the improvement tips are included after that as well.</p><p> </p><p><strong>Personal Engagement (1/2)</strong></p><p> </p><figure class="table"><table><tbody><tr><td><p>(Original Level 5)</p><p>(Introduction I - Exponential Growth and Decay)</p><p>Having studied exponential growth in mathematics, I thought it was a very interesting phenomenon. Exponential growth is when some value grows by a fixed multiple each period of time. This is also known as geometric growth. Exponential decay is when the rate is a value less than one.<a target="_blank" rel="noopener noreferrer" href="http://en.wikipedia.org/wiki/Exponential_growth/"> </a>In the example of a barrel holding 100 gallons of water. A leak would reduce the level of the water exponentially until it empties. Exponential growth and decay are very important in nature, which is something I am very interested in. In biology this occurs with microorganisms in a culture, with the growth of a virus, sometimes in human population growth. In the field of economics exponential change occurs in finance; and it is found in computer technology. </p><p> </p><p>The research question for this experiment will be “How does the bounce height of a ball vary as the ball bounces”. In order to answer this question the bounce height will be measured and graphed to determine the relationship. </p></td></tr></tbody></table></figure><p> </p><p><i>Reasons for 1/2:</i></p><ol><li>The engagement portion of the seems insincere.</li><li>Explanation of the exponential growth is not rigorous in its definition. </li><li>Context of the concepts of exponential growth and decay do not include examples in physics.</li><li>Positive: Writer showed knowledge of a well known relationship then extended the investigation (shows interest, shows it's worth doing)</li><li>RQ is unfocused and lacks clarity. It does not really convey what is being investigated nor what the results of the investigation will be.</li></ol><p> </p><p><i>Improvement tips :</i></p><ol><li>Saying “I was interested”, “I was curious” does not really count as engagement: Use an interesting fact or quote to show that not only you did some research but also show exactly what it was that you found interesting or curious.<ol><li>Use a clear quote or citation that shows the interest: Reference to the extract from legion of a king and a chessboard</li></ol></li><li>Fix the explanation of the context of the experiment where exponential growth and decay are explained, and expanded the explanation of how the leaking barrel shows exponential decay using numerical value.</li><li>Include examples of how the topic is relevant and/or significant specifically in physics.</li><li>The research question should be rephrased so that it instead explains that this IA is an investigation on the exponential decay of the bounce height, and explicitly state that the goal of the experiment is to confirm the exponential decay mathematical model and the bouncing half-life. </li><li>The scope of the experiment should be defined as the vertical bounce including a brief explanation of how one of the main parameters of the experiment (drop height) was defined.</li></ol><p> </p><p>*Major changes highlighted in yellow</p><figure class="table"><table><tbody><tr><td><p>(Model Personal Engagement worth 2/2)</p><p> </p><p>(Introduction I - Exponential Growth and Decay)</p><p> </p><p>An amazing story about exponential grow in given in mathematics classes. There is a fascinated geometric phenomena described in the legend of a king and a chessboard.</p><p><i>“According to legend, a courtier presented the Persian king with a beautiful, hand made chessboard. The king asked what he would like in return for his gift and the courtier surprised the king by asking for one grain of rice on the first square, two grains on the second, four grains on the third, etc. The king readily agreed and asked for the rice to be brought. All went well at first, but the requirement for </i>2n–1 <i>grains on the </i>nth <i>square demanded over a million grains on the 21st square, more than a million‐million (or trillion) on the </i>41st <i>and there simply was not enough rice in the whole world for the final squares.”</i></p><p>http://www.dr‐mikes‐math‐games‐for‐kids.com/rice‐and‐chessboard.html</p><p>The chessboard story is an example of exponential growth. This occurs when the growth rate of a mathematical function is proportional to the function’s current value. This is also known as geometric growth. Similarly, there is an exponential decay when the rate of decrease of a mathematical function is proportional to the function’s current value.<a target="_blank" rel="noopener noreferrer" href="http://en.wikipedia.org/wiki/Exponential_growth/"> http://en.wikipedia.org/wiki/Exponential_growth/</a></p><p> </p><p>Consider a barrel holding 100 gallons of water. A leak occurs and after 5 minutes only 50 gallons remain. After an additional 5 minutes the 50 gallons is reduced to 25 gallons. And, after another 5 minutes, the 25 gallons is reduced to 12.5 gallons. This continues with the same mathematical pattern, that is, half the remaining water leaves the barrel in the same amount of time no matter how much or how little water is in the barrel.</p><p> </p><p>Exponential growth and decay are fundamental properties in many aspects of nature. In biology this occurs with microorganisms in a culture, with the growth of a virus, sometimes in human population growth. In the field of economics exponential change occurs in finance; and it is found in computer technology. In physics the charge and discharge of capacitors follow exponential functions, and this relationship is well known in nuclear reactions and radioactive decay, and heat transfer. Many other areas of natural phenomena can be described as exponential. </p><p> </p><p>(Introduction II - Exponential Decay in This Investigation)</p><p> </p><p>In this investigation the exponential decay of a bouncing ball will be considered. The exponential mathematical model of the bouncing ball will be confirmed, although it was a previously unknown before the experiment. The bouncing half‐life (the number of bounces before the rebound height reaches one‐half a previous height) will also be determined.</p><p>To clearly focus this approach to the motion of a bouncing ball, only vertical bounces will be considered. The drop height will also limit the investigation; only one initial drop height will be used and only one set of data taken. Too high of an initial drop introduces noticeable spin and too low an initial drop does not provide enough data.</p></td></tr></tbody></table></figure><p> </p><p><strong>Exploration (4/6)</strong></p><figure class="table"><table><tbody><tr><td><p>(Original Level 5)</p><p>(Introduction II - Bouncing Balls)</p><p>Under normal bouncing conditions (a reasonable height and a uniform surface) balls rebound to a height that is less than the drop height because kinetic energy is lost on the rebound. The ball never experiences an elastic collision. Some heat and sound are always produced. We also know that there is a characteristic unique to a given ball for the amount of height lost between the drop height and the rebound height. This might be called the ‘bounciness’ or the percentage of rebound. It is related to the coefficient of restitution.</p><p> </p><p>These percentages of rebound express an exponential property for a bouncing ball. A golf ball, for instance, will bounce back to 36% of the drop height. See the information found on the web site:<a target="_blank" rel="noopener noreferrer" href="http://www.exploratorium.edu/baseball/bouncing_balls.html"> http://www.exploratorium.edu/baseball/bouncing_balls.html.</a> Each consecutive bounce will reach 36% of the previous height, and so on.</p><p> </p><p>(Methods I - Preliminary Work)</p><p> </p><p><strong>Ball Selection. </strong>A variety of balls were found in the physics lab, and each was dropped and the bouncing was observed, both in terms of keeping along the vertical and making a clear sound or well‐defined sound upon impact. I decided that a Ping‐Pong ball was the best ball to use.</p><p><strong>Drop Height. </strong>The Ping‐Pong ball dropped from about 60 cm made clear impact sounds and produced sufficient repetitive bouncing. This value was arbitrary and does not play a role any of my calculations, only for collecting sufficient data.</p><p><strong>Rebound Height. </strong>This will be calculated based on the time interval between two consecutive bounces.</p><p><strong>Time Measurement. </strong>The sound of the bounce impacts is recorded using a microphone, data logging interface and a computer. I used the <i>Vernier </i>(see<a target="_blank" rel="noopener noreferrer" href="http://www.vernier.com/)"> http://www.vernier.com/)</a> Lab Pro interface and the Logger Pro software with my Mac computer. This is illustrated in below in Graph 3.</p><p> </p><h2>(Methods II - Experimental Design) </h2><p> </p><h2><strong>The purpose of this investigation is to find out how bounce height varies with time</strong>. I will limit my study to one type of ball, and I will limit the study to just one initial drop height and the resulting bouncing heights. Because the ball’s rebound height decreases with time this is the same as having a number of different drop heights in sequential order; that is to say, for example, that rebound number 4 is the drop height of bounce number 5, and so on.</h2><p> </p><p>Using the microphone, the <i>LabPro </i>data‐logging interface and computer system, the sound of bounces was recorded. Heights were calculated by the spreadsheet. The ball bounce events were simply counted; the first bounce as <i>n </i>= 1, the second was <i>n </i>= 2, etc. When a rebound height became one‐half the height of some previous height then it will have bounced a number of times <i>n </i>(not necessarily a whole number, perhaps <i>n </i>= 3.33) indicating its half‐life. With radioactive decay, the half‐life is a function of time (seconds, minutes, years) but with a bouncing ball the event of a bounce is the measure.</p><p> </p><p>(Methods III - Experimental Variables)</p><p> </p><p>In order to determine the consecutive heights, I recorded the sound of the bouncing ball. Upon impact, the ball made an impact sound. This is the same as saying the air pressure is constant while there is no sound, but changes or is offset from this constant when there is sound. Sound is longitudinal wave, a compression and rarefaction of air pressure.</p><p> </p><p>The <strong>independent variable </strong>is the bounce number. The life of a bouncing ball is measured as the 1st, 2nd, 3rd, etc. bounce number. This is a counting number, a pure number with no units and no significant uncertainties. However, identifying the moment of impact is limited to the precision of the sampling rate, so there is a minimum uncertainty, about 1 ms in this study.</p><p>The <strong>dependent </strong>variable is the rebound height, <i>H</i>, the maximum height reached between bounces. This was calculated in units of metres (m). To measure <i>H</i>, the time ΔT<i> </i>between consecutive bounces was determined (from a graph of air pressure offset as a function of time) and <i>H </i>calculated from <i>H </i>= 1 <i>gt </i>2 where <i>t </i>= 1 ΔT<i> </i>. The 1 comes from the fact that ΔT<i> </i>is the time up to the rebound height plus the time down from the rebound height. It is far more accurate to measure this time interval, and then calculate the height, than it would be to try to measure the rebound height of a moving bouncing ball. There is no significant uncertainty in the calculated height as it is based on a very precise timing mechanism with the computer and interface. The offset of air pressure is recorded at a rate of 1000 measurements per second.</p><p>The <strong>controlled </strong>variables included using the same surface and the same ball. This was obvious as I made only one data set. Time limited repeated measures. Nonetheless, I made a number of trials and selected the cleanest data set, and one where the bouncing was more or less always along the vertical. This is good scientific practice thus demonstrating the nature of science at work. If the ball moved off the vertical while bouncing, then the data was rejected. A controlled variable was that the bouncing stays more or less along the vertical.</p><p> </p><p>(Methods IV - Four Experimental Assumptions)</p><p> </p><p>(1) The evidence for exponential decay has already been discussed and it is a reasonable model for the bouncing ball. This was confirmed in Graph 1. I then select one drop height and let the ball bounce again and again and then see if this motion is indeed exponential. This is <strong>the first mathematical model assumption </strong>I made.</p><p>(2) The value of free‐fall gravity will be assumed in the calculations. If this is in error it will be a systematic error, a constant, and hence will make no difference to the results.</p><p>(3) It is true by definition that the distance up equals the distance down. However, if the ball spins and moves off a purely vertical path then the data trail was rejected. Again, in this experiment a number of trials were done but only one set of data was used to determine the results.</p><p>(4) If the distance up equals the distance down then assuming uniform acceleration we can say that the time up equals the time down. This assumes that air friction plays no role in the motion of the ball. From a practical point of this, air friction can be ignored, but technically the time up is not equal to the time down if we account for air friction. On the upward journey the weight is directed downward and the air resistance retards the motion; air friction is directed downwards. The net force causing the ball to decelerate is then <i>F</i>weight + <i>F</i>air. When the ball falls from the maximum height to the ground the weight is again directed downwards but the air resistance, which retards the ball’s motion, is directed upwards. Hence the force causing the ball to accelerate is <i>F</i>weight – <i>F</i>air. This means that the time going up is less than the time coming down when air friction is accounted for. In my experiment the balls maximum speed is rather small, and the height distance is small too, and I assume air resistance is negligible. My <strong>second mathematical model assumption </strong>then is the basic equation of uniform accelerated motion, relating height <i>h</i>, gravity <i>g </i>and time <i>t</i>, as <i>h </i>= 1 <i>g t</i>2 .</p></td></tr></tbody></table></figure><p> </p><p><i>Reasons for 4/6:</i></p><ol><li>The explanation of how bouncing balls provides the context and background of the experiment, but not enough explanations are provided on the mechanisms of energy loss </li><li>The coefficient of restitution is mentioned but not defined</li><li>The explanation as to how the bounce height relates to the coefficient of restitution is not given.</li><li>Scientific data of bounce height is not entirely appropriate since it is a different ball. Link included provides source of doubtful reliability.</li><li>Details of the surface used to bounce not found.</li><li>Part of the preliminary work should be to gather data that pre-validates the model used, shows calibration methods if applicable, or shows the reasoning for experiment design choices, etc. </li><li>Control variables are stated, but why they are controlled is not sufficiently discussed.</li></ol><p> </p><p><i>Improvement tips for 6/6:</i></p><ol><li>Add an explanation of the mechanisms of energy loss.</li><li>Add an explanation on the coefficient of restitution and how it is related to bounce height.</li><li>The data on the rebound properties of a golf ball should be replaced with more relevant data on ping pong ball with a more credible and serious source.</li><li>Include the surface selection in the preliminary work section as it is an important variable.</li><li>Add a short trial experiment using bounce data to verify the model, and as a reference to preliminary experiments.</li><li>Research question should be replaced with the investigation and purpose.</li><li>Add an explanation of why the controlled variables need to be controlled.</li></ol><p><br> </p><figure class="table"><table><tbody><tr><td><p>(Model Exploration worth 6/6)</p><p>(Introduction III - Bouncing Balls)</p><p>Under normal bouncing conditions (a reasonable height and a uniform surface) balls rebound to a height that is less than the drop height because kinetic energy is lost on the rebound. The ball never experiences an elastic collision. Some heat and sound are always produced during these collisions because at an atomic level deformations are never truly elastic since some of the atoms will be randomly rearranged and imperfections introduced. The collisions will also cause the objects’ shape to undergo damped harmonic motion, thus producing sound and some thermal energy. We also know that there is a characteristic unique to a given ball for the amount of height lost between the drop height and the rebound height. This might be called the ‘bounciness’ or the percentage of rebound. It is related to the coefficient of restitution.</p><p><img src="https://lh5.googleusercontent.com/qH5iuchlm3eef4u-FhR1Nfo3u_cJQqSQ_omLP-r_iyIdvF-KMqC5-54i7I59YLu10JJUmknWgCSZIjf8pu5w3Cm_VdmDr_95UDdOjAqxtWo3nHLTAa7BJ4Q1dAd9PYa9njDxHVjX" alt=""></p><p>In cases where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to:</p><p><img src="https://lh5.googleusercontent.com/mGnwuHEvT6A6SA6-zaAoYOtX3upayAIn_zDSMko70VWmaGgB8Y3PeQs5YVjutwFJP0FQW2tl9Eh_KOpcGmHilKoh06Gwu2UIeaLYKIRpLbZcTExv697J-aZdTx0vQPyOi0tKSBnY" alt=""></p><p>Therefore if the coefficient of restitution is assumed to be constant, then the “bounciness” is also constant, allowing for exponential decay behavior of bounce height.</p><p> </p><p>The<a target="_blank" rel="noopener noreferrer" href="https://en.wikipedia.org/wiki/International_Table_Tennis_Federation"> International Table Tennis Federation</a> specifies that the ball shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block thereby having a COR of 0.89 to 0.92.<a target="_blank" rel="noopener noreferrer" href="https://en.wikipedia.org/wiki/Coefficient_of_restitution#cite_note-8">[8]</a> This means the bounce height percentage should be between 94% to 96% on a steel surface. This gives an upper limit to the “bounciness” for the linoleum floor used.</p><p><a target="_blank" rel="noopener noreferrer" href="https://en.wikipedia.org/wiki/Coefficient_of_restitution#cite_note-9">https://en.wikipedia.org/wiki/Coefficient_of_restitution#cite_note-9</a></p><p> </p><p>(Methods I - Preliminary Work)</p><p> </p><p><strong>Ball Selection. </strong>A variety of balls were found in the physics lab, and each was dropped and the bouncing was observed, both in terms of keeping along the vertical and making a clear sound or well‐defined sound upon impact. It was determined that a Ping‐Pong ball was the best ball to use.</p><p><strong>Surface Selection </strong> A hard marble stone surface was selected in lieu of a steel block. Although a steel block is used as the standard surface for ping pong ball testing by the International Table Tennis Association, limitations on availability forced the experiment to be done on a marble tabletop as it was the hardest surface available.</p><p><strong>Drop Height. </strong>The Ping‐Pong ball dropped from about 60 cm made clear impact sounds and produced sufficient repetitive bouncing. This value was arbitrary and does not play a role in any of the calculations.</p><p><strong>Rebound Height. </strong>This will be calculated based on the time interval between two consecutive bounces.</p><p><strong>Time Measurement. </strong>The sound of the bounce impacts is recorded using a microphone, data logging interface and a computer. <i>Vernier </i>(see<a target="_blank" rel="noopener noreferrer" href="http://www.vernier.com/)"> http://www.vernier.com/)</a> Lab Pro interface and the Logger Pro software was used to this end. This is illustrated in below in Graph 3.</p><p><strong>Percentage of Rebound Test. </strong>A preliminary test was made by dropping the Ping‐Pong ball from a variety of heights and measuring the rebound height using the <i>Vernier </i>sonic motion detector. If the percentage of rebound is consistent then the exponential model may well be appropriate. The results are shown below in Graph 1.</p><p><img src="https://lh6.googleusercontent.com/8N6_xaoqjbYS2oXqHdvHuP6Ia-7Q4oulJUxK81MA2C0WOcpYjlVop9JWnESb0L3zG_c6mrPnvrjK5z8Gfd35nUdzR_rBWlEXJ6mjRtMjEsZZ6fmTt__SV4TXCucNcqc1rSl6YM9b" alt=""></p><p>The data reveals a linear function with a gradient of 0.853. This is the same as saying that the ball experiences 85.3% efficiency on its bounce for a variety of drops heights. The systematic shift in all the data is simply an artifact of the measuring technique and does not affect the gradient value. It is safe, therefore, to assume that this rebound percentage is a characteristic of the Ping‐Pong ball and hence the exponential model is appropriate.</p><p> </p><h2>(Methods II - Experimental Design)</h2><p> </p><p><strong>The purpose of this investigation is to confirm the exponential nature of a bouncing ball and thereby calculate the half‐life of the bouncing ball</strong>. The experiment will be limited to one type of ball, and at a one initial drop height. Because the ball’s rebound height decreases with time this is the same as having a number of different drop heights in sequential order; that is to say, for example, that rebound number 4 is the drop height of bounce number 5, and so on.</p><p> </p><p>Using the microphone, the <i>LabPro </i>data‐logging interface and computer system, the sound of bounces was recorded. Heights were calculated by the spreadsheet. The ball bounce events were simply counted; the first bounce as <i>n </i>= 1, the second was <i>n </i>= 2, etc. When a rebound height became one‐half the height of some previous height then it will have bounced a number of times <i>n </i>(not necessarily a whole number, perhaps <i>n </i>= 3.33) indicating its half‐life. With radioactive decay, the half‐life is a function of time (seconds, minutes, years) but with a bouncing ball the event of a bounce is the measure.</p><p> </p><p>(Methods III - Experimental Variables)</p><p> </p><p>In order to determine the consecutive heights, the sound of the bouncing ball was recorded. Upon impact, the ball made an impact sound. This is the same as saying the air pressure is constant while there is no sound, but changes or is offset from this constant when there is sound. Sound is longitudinal wave, a compression and rarefaction of air pressure.</p><p>The <strong>independent variable </strong>is the bounce number. The life of a bouncing ball is measured as the 1st, 2nd, 3rd, etc. bounce number. This is a counting number, a pure number with no units and no significant uncertainties. However, identifying the moment of impact is limited to the precision of the sampling rate, so there is a minimum uncertainty, about 1 ms in this study.</p><p>The <strong>dependent </strong>variable is the rebound height, <i>H</i>, the maximum height reached between bounces. This was calculated in units of metres (m). To measure <i>H</i>, the time ΔT<i> </i>between consecutive bounces was determined (from a graph of air pressure offset as a function of time) and <i>H </i>calculated from <i>H </i>= 1 <i>gt </i>2 where <i>t </i>= 1 ΔT<i> </i>. The 1 comes from the fact that ΔT<i> </i>is the time up to the rebound height plus the time down from the rebound height. It is far more accurate to measure this time interval, and then calculate the height, than it would be to try to measure the rebound height of a moving bouncing ball. There is no significant uncertainty in the calculated height as it is based on a very precise timing mechanism with the computer and interface. The offset of air pressure is recorded at a rate of 1000 measurements per second.</p><p>The <strong>controlled </strong>variables included using the same surface and the same ball as changing the surface or ball would change the mechanics of the bounce collision, and therefore change the bounciness. A change in the bounciness would result in non-exponential behavior.</p><p>This was obvious as only one data set was used. Time limited repeated measures. Nonetheless, a number of trials were conducted and selected the cleanest data set, and one where the bouncing was more or less always along the vertical. This is good scientific practice thus demonstrating the nature of science at work. If the ball moved off the vertical while bouncing, then the data was rejected. </p><p> </p><p>(Methods IV - Four Experimental Assumptions)</p><p> </p><p>(1) The evidence for exponential decay has already been discussed and it is a reasonable model for the bouncing ball. This was confirmed in Graph 1. One drop height was selected and the ball allowed to bounce to see if this motion is indeed exponential. This is <strong>the first mathematical model assumption </strong>made.</p><p>(2) The value of free‐fall gravity will be assumed in the calculations. If this is in error it will be a systematic error, a constant, and hence will make no difference to the results.</p><p>(3) It is true by definition that the distance up equals the distance down. However, if the ball spins and moves off a purely vertical path then the data trail was rejected. Again, in this experiment a number of trials were done but only one set of data was used to determine the results.</p><p>(4) If the distance up equals the distance down then assuming uniform acceleration we can say that the time up equals the time down. This assumes that air friction plays no role in the motion of the ball. From a practical point of this, air friction can be ignored, but technically the time up is not equal to the time down if we account for air friction. On the upward journey the weight is directed downward and the air resistance retards the motion; air friction is directed downwards. The net force causing the ball to decelerate is then <i>F</i>weight + <i>F</i>air. When the ball falls from the maximum height to the ground the weight is again directed downwards but the air resistance, which retards the ball’s motion, is directed upwards. Hence the force causing the ball to accelerate is <i>F</i>weight – <i>F</i>air. This means that the time going up is less than the time coming down when air friction is accounted for. In my experiment the balls maximum speed is rather small, and the height distance is small too, and air resistance is assumed to be negligible. Moreover, a test with the sonic motion detector (<i>Vernier</i>) revealed that any asymmetry between the up and down times is equal the period of the sampling frequency. That is, no difference is detected other than the precision of 1 ms and any noise or random effects, artifacts of the measuring process. The <strong>second mathematical model assumption </strong>then is the basic equation of uniform accelerated motion, relating height <i>h</i>, gravity <i>g </i>and time <i>t</i>, as <i>h </i>= 1 <i>g t</i>2 .</p></td></tr></tbody></table></figure><p> </p>
