Table of Contents
Introduction
Determination of variables affecting the time of pumping a ball
I got the idea for the topic of measuring the time to pump the balls on the second day of the break when it was necessary to pump a basketball. As always, we wanted to go out and play basketball, but the ball didn’t bounce enough, which was a clear sign that it needed to be pumped. My younger brother started doing this and it took a lot of time. At that point, I asked him – How much time do you need and will you ever be done with it? He really could have done it faster , but he had said that the cause was a pump too small for the ball. The next day at the beach, a similar question occurred. I had a ball from a few years before, that had to be pumped. I had a feeling that pumping it lasted forever. That ball was not as liked as before, but the job had to be done. Later on, when I was thinking about my IA topic, I though – “How long does it actually take to pump one ball, and why does it last so long?” There are several factors that influence the time required – the volume of the object that has to be pumped, the volume of the pump, the pressure to which a ball should be pumped to, the speed of pumping… I thought it would be a good idea to come up with a formula to determine the pumping time. Perhaps I could also show diagrams that exhibit dependence of pumping time on the factors aforementioned. I liked elaborating on this topic because I could test my formula with an experiment and the equipment I needed for the experiment was mostly available to me. Also, I find that I have enough knowledge to select other shapes of balls besides the round ones, and based on their cross-section, determine the function that defines the shape of the ball closest. After that, using integral calculus, calculate their volume. Of course, I know that I will have to make some simplifications in defining the shape, but I believe that the calculation will be accurate enough not deviate significantly from the experimentally obtained results.
BASIC INFORMATION ABOUT THE BASKETBALL
Basketball is a spherical ball used in basketball games. Basketballs typically range in size from very small promotional items only a few inches in diameter to extra large balls nearly a foot in diameter used in training exercises. In order to measure a basketball, we have to understand the differences in types of basketballs existing. A difference in men’s and women’s basketball is significant in its weight and size. The table below show all the sizes and weights of different types of basketballs. What is interesting is that there are rules on which age groups can use certain balls. For example, the Men Ball is appropriate for men and boys ages 15 and up, and that is the official size for men’s high school, college and professional basketball. On the other hand, Women Ball is used by not only women and girls ages 12 and up, but by boys from 12 to 14 years old too. All other types are used by boys and girls, the only difference is for which age group the ball is for, so size 5 is aimed to be for girls and boys from 9 to 11 years old, size 4 is for kids from 5 to 8 years of age, and size 3, also known as “mini basketball” is for kids between 4 and 8 years of age. Before committing to pumping any ball, you first have to know its size, air pressure recommended for that type of a ball, and which type and size of pump to use. Finding the proper PSI is essential for pumping a ball the correct way. The proper PSI is usually printed next to the valve on the ball by the manufacturer. The PSI is the recommended number of pounds of air per square inch, which tells you what pressure works best with a certain ball. Apart from the PSI unit of measurement for air pressure, there is also the BAR, a metric unit of pressure using kilograms, and it occurs mostly in the continental Europe. In general 1 BAR is equal to 14.5 PSI. SizeTypeWeight and size7Men620 g, 75 cm;6Women570 g, 72 cm;5Youth (N. America) Mini (FIBA)480 g, 70 cm;4Youth (N. America)400 g, 65 cm;3Mini (N. America)280 g, 56 cm.
In the table below, listed are all the values of PSI (BAR) recommended for various types of sports balls. So, how long does it take to pump each of these balls? For every ball, there is a set of information needed, in order to calculate the time correctly. That information is: the perimeter of a basketball, perimeter of the ball (where it is the widest) ,pump length, internal perimeter. Now we are going to examine every ball listed above, and list all of the information. Fig. 1Fig. 2Fig. 3Fig. Type of ballRecommended PSI Value in BAR American footballFrom 12.5 to 13.5From 0.86 to 0.93BasketballFrom 7.0 to 9.0From 0.48 to 0.62Soccer ballFrom 8.5 to 15.6From 0.59 to 1.10Rugby ballFrom 9.5 to 10.0From 0.67 to 0.70Ball typeWidthLengthDiameterCircumference1 – American Football/7” or 17.78 cm5.9” or 15 cm22” or 55.9 cm2 – Basketball///29.5” or 75 cm3 – Soccer///68-70 cm4 – Rugby11.8” or 30 cm24.4” or 62 cm30.3” or 77 cm/
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BALL PUMP CHARACTERISTICS, UTILITY COEFFICIENT
Let’s assume that we already have a certain pump, which length is 8” or 20.3 centimeters. (Fig. 5) The useful volume of the pump can be calculated by its length and perimeter. For the sake of calculation, we will assume how much air will enter the ball in a single motion, its volume. (1) r – pump cross-sectional radius, H – length of the pump Fig. 5 BASIC CONCEPTS AND CHARACTERISTICS OF SPORTS BALLS WITH CIRCULAR CROSS SECTION What is a cross section? Cutting straight into an object, results in a shape we usually refer to as a cross section. If, in this case, we use a soccer ball, and cut right through it, we end up with a circular shape, or a circular cross section. If we would to cut into a chocolate, we would probably see a rectangular shape, depending on the width of the chocolate. Calculus enables us to calculate the volume of an object by dividing the object into a certain or infinite number of circular cross section (circles) and adding their volumes up through integration. (Fig. 6) In order to calculate the volume of a soccer ball, we need to know its radius. Radius is calculated through the calculation of the perimeter.
Volume is thus: Which sums to: After calculating the volume of a soccer ball, we can proceed to calculating how many times it takes to pump air into a soccer ball for it to be full. This will be done by dividing a volume of a ball by a volume of a pump. This means, the required amount of moves is approximately 62 in order to fill up a soccer ball. Assuming it takes a second to pump air into the ball once, it’s only 62 seconds required to fill up a ball. Of course, this is a relative statement. In order for this to be correct, a person filling the ball, would have to pump nonstop without any delays, and at a certain speed. This might take longer if a person filling the ball is slower.
Calculating the volume of a ball intended for American football is much more complicated, since this object doesn’t have a circular cross-section. (Fig. 7) A football is in the shape of a prolate spheroid , which is simply a solid obtained by rotating an ellipse about its major axis. An inflated football averages 28 centimeters in length with a 15 centimeters diameter. If the volume of a prolate spheroid is , how much air does the football contain? In this example, we are going to neglect the material thickness. Keep in mind that the dimensions used are actually dimensions of a professional NFL ball.
First off, we are going to calculate the a and b values: – a value can be calculated by dividing the length value by 2: – since we know the value of center circumference, and that formula for it is – 2πb, we can find the b value through: By finding these values, we are now able to calculate the volume of a prolate spheroid. Using the aforementioned formula for finding the N value, and assuming that we are using the same pump, that has a volume of 94 cm 3 , we will be able to determine the number of moves it takes in order to fill up an American football.
It takes approximately 48 moves for the ball to be filled with air. If we assume it takes a second for each pump, we can say that it will take around 48 seconds for pumping the ball. Of course, this statement is relative, due to a speed of pumping, and other causes. Another object which I was interested in was an inflatable water toy. Which formulas would be used in order to find the volume of a floaty? After a careful analysis of different ways to get the volume of this object, I came to the realization that the most efficient way of calculating this value was by dividing this floaty to a number of object, and then adding up their values. In this case, I used already existing dimensions which were listed on the box, which this floaty came in. First step is to divide the base of this inflatable water toy, so the first object would actually be similar to a ring or a tube. In order to find the volume, we have to know the perimeter first. Section 1 – Inner diameter (ID) is 30 centimeters long; – Outside diameter (OD) is 80 centimeters long; – Thickness diameter (TD) is 25 centimeters long. – Hight of the tube is 15 centimeters.
For us to know the accurate perimeter, we need to calculate P1 or outside perimeter , from which we will subtract P2 or inside perimeter. And finally, the volume would be calculated in a similar way: My idea for the “neck” and “head” part is to divide it into five cylinders, at points in which dimensions are changing. Dimensions are known and height is the same in all cases: h=3cm Section 2 Fifth, and the last part was the beak, which is cricked, and there is actually not a formula for this type of object, and the most similar one was a cone, that’s why the following formula is actually for a cone. r5=2cm, h 1=5cm We will find the volume of the section 2 by adding all the values up.
Finally, volume of this inflatable animal is: And if we were to fill it up, with our pump, it would take around 696 moves to do it. We would spend around 11 ½ minutes in order to fill the floaty with air. In this case, the pump is too weak for an object this size, and in order to fill it up faster, we would need a pump with a higher capacity (volume). HOW MUCH AIR IS IN A BASKETBALL BALL? There are certain rules and regulations which have to be followed in every basketball game. Some rules suggest the size, amount of air, and even the manufacturers of balls used. Often, in the movement analysis of a basketball, its rotation must be taken into account, which immediately raises the question of the unknown moment of inertia. It is clear that the ball is filled with air. Now let’s consider what is the contribution of air to the moment of inertia of the ball. According to the official rules of a basketball game, the ball should have a radius of 12 cm and a mass of 600 g. Assuming the thickness of the material the ball is made from, is 5 mm, it follows that the volume of air in the ball is 6370 cm 3. The ideal gas equation is written as: Assuming that the molar mass of air is M=28.8 g/mol, at room temperature T=20°C, we get that the ratio of pressure to mass of air in the ball: (one atmosphere corresponds to a pressure of 101325Pa, which is the value of atmospheric pressure).
As mentioned before, the pressure of a properly inflated ball is printed on each ball, marked “7-9” LBS or 1.54 ATM. Using the formula above, we can calculate the mass of air in a properly inflated ball. It amounts to 11.8 g, or 2% of the total mass of a ball. In order to confirm these calculations, we have to determine the mass of both balls with and without air in them. Thus, it is necessary to take into account the Archimedes’ principle. However, the effect of the buoyant force can be eliminated in only two cases: by measuring the weight of a ball when fully inflated, or when it is pumped to atmospheric pressure. Namely, in the second case, the radius and therefore the volume of a ball decrease very little, hence the change in buoyant force is negligible. Therefore, the difference in these two measurements will represent the weight of the air which has entered the ball by pumping it. If the mass of air in a ball at atmospheric pressure amounts to 7.7 g. If m a denotes mass of the air in an inflated ball, and Δm represents difference in the aforementioned measurements: mv By using this formula we will come to understanding that the mass of the inflated ball is 590.3 g, while the mass of the ball inflated to the atmospheric pressure is 586.3 g. Based on the formula above, we can now calculate the total mass of air in the fully pumped ball: mv =4.0g+7.7g=11.7g. Based on these measurements, we can also assess whether the ball we w e re m ea s u r i n g w a s c o r re c t l y pumped. Since the mass of air in the ball is directly proportional to the pressure, we can observe the results, that show that the ball has been adequately pumped.
CONCLUSION
After concluding this analysis, I can fully understand why pumping cannot be instantaneous, and has to be a process that requires a lot of movements if done manually. As seen and analyzed, factors are numerous, and all of them influence the process one way or another. What I grasped from this research is that I would strive to always use a higher volume pump to do the job. Mostly because of the impatience and eagerness to play. I can’t imagine working in a factory where balls are manufactured and tested for issues. What would it look like for my brother to work there, with his slow pumping pace, and all of the breaks he takes until he gets the job done? I now fully understand the process of filling a ball with air, why it takes that amount of time, and how to decrease the time required. Honestly, I didn’t even assume this much physics and mathematics went into an everyday obligation. I would now agree with the manufacturers, and definitely spend a few euros more in order to buy appropriate pumps, which would deliver the recommended PSI level, and have a higher capacity so I would be able to pump any type of object efficiently, and not spend that amount of time and energy. And I would definitely suggest the same solution to everyone. Nowadays, even having a compressor would have a big impact. But as we all know, you can’t always bring a compressor to the beach, pool or a lake. Having at least two pumps with different volume, or a pump and a compressor would increase the efficiency enormously, and give us more time to play or enjoy the vacation on our inflatable toys.
REFERENCES
- Breakthrough Basketball. (2014.). Basketball size chart. Available at: https://www.breakthroughbasketball.com/basketballs/size-chart.html M P Trivedi, P Y Trivedi. (2008).
- Consider Dimension and Replace Pi. Available at: https://www.ebsco.com Stephen Lee. (2014.)
- An Introduction to Mathematics for Engineers: Mechanics. Available at: https://www.ebsco.com Clements, Douglas & Battista, Michael. (2001).
- Length, perimeter, area, and volume. Available at: https://www.googleschoolar.com Jeremy H. (2018).
- What Is the Official Size of the NFL Football? Available at: https://www.sportsrec.com/6560043
