電梯問題
本研究主要是研究「電梯問題」： 令某棟建築內任兩層樓都至少有一部電梯連接的建築稱為全能建築，若某棟建築內有m部電梯，每部電梯停n層，f (m, n)為使這棟建築成為全能建築的最高層數。對於不同的m, n，f (m, n)的值為何？ 本研究一開始先以一些參考資料為基礎，試著整理出其中未完成的部份。接著，我們針對了一些個別的f (m, n)，具體求出其值。令某棟建築內任兩層樓都恰有一部電梯連接的建築稱為完美建築，我們引入了一個新的函數g (n, k)=m，得到了一個與完美建築有關的定理。另外，我們還利用一些構造方法，求出了所有g (3, k)的值。 在這次的研究中，我們成功運用了各種不同的構造手法，得到了一些相關的結果。至於是否能將這些構造方法運用在求出一般的f (m, n)之值？這是我們將繼續探究的課題。
Fig Preservation
Figs have become an expanding industry here in New Zealand and are a current export fruit which could potentially provide a large amount of profit to both growers and the New Zealand market as a whole. Nicola’s family has about 10 acres of fig trees. They sell the figs locally and as an export. They generally sell for about $13 per kilogram here in New Zealand and $26 in the USA. However, figs only have a shelf life of about 7 days. This is because at present there is no proven pre or post-harvest treatment or method of storage that helps to decrease the rate of decay of the fig fruit. After researching post-harvest treatments for figs, Nicola found a report which claimed to have developed treatments that increased the shelf life of figs by about 5 weeks. With this kind of increase, it would be possible to transport, store and export figs over longer periods of time without running the risk of losing large amounts of produce, or delivering unsatisfactory fruit to customers. Nicola developed 7 different post-harvest treatments based on the ones that had shown promise in earlier research. These were hot-water baths of different temperatures, both with and without different bleach concentrations. To test these on the fruit she set up four experiments – a dry matter test, a firmness test (using a penetrometer), a colour test and observation of detrimental features of the fig. She tested these treatments at 0, 7, 14, 21 and 28 days from harvest. Nicola found that after 7 days, the firmness of all of the figs that had been treated had decreased to a large degree. The only figs that did not have a massive decrease were the untreated fruit. However after about 14 days, the firmness of all of the fruit became about the same and after this 14 day mark, she would not have considered any of the figs to be edible. However, in the appearance tests, it seemed that the treated figs that had the least amount of mould and rot were the ones that had been treated with higher levels of bleach such as the 55 degree Celsius water bath with 0.003L of bleach to every litre of water, and the 35 degree Celsius water bath with the same concentration of bleach. Overall, Nicola’s results showed that the hot water bath, and hot water bath and bleach post-harvest treatments did not slow the decay of the fruit in the earlier weeks after picking. In effect, Nicola’s research showed that the information she had relied on to help plan her study had claimed too much and that the treatments were less effective than had been stated. More research will be needed to find a more reliable way to improve the shelf life of figs.
The use of Square shaped wheels in ship harbouring using an inverted catenary surface
Riding around on a flat tire is no fun. It feels really bumpy. But a square wheel may be the ultimate flat tire. There's no way it can roll over a flat, smooth road without jolting the rider again and again. Here, I have constructed a bicycle with square wheels. It's a weird contraption, but you can ride it perfectly smoothly. My secret is the shape of the road over which the wheels roll. A square wheel can roll smoothly, keeping the axle moving in a straight line and at a constant velocity, if it travels over evenly spaced bumps of just the right shape. This special shape is called an inverted catenary. A catenary is the curve describing a rope or chain hanging loosely between two supports. Turn the curve upside down, and you get an inverted catenary--just like one of the bumps in my road. Make the road out of a whole bunch of those bumps all in a row, and you can take your square-wheeled bike for a quick spin. Just as a square rides smoothly across a roadbed of linked inverted catenaries, other regular polygons, including pentagons and hexagons, also ride smoothly over curves made up of appropriately selected pieces of inverted catenaries. As the number of a polygon's sides increases, these catenary segments get shorter and flatter. Ultimately, for an infinite number of sides (in effect, a circle), the curve becomes a straight, horizontal line. In the end, I conclude with possible enhancements in the project that might take us to a whole new world.
Project Motion in Sports
A projectile refers to any body that is thrown in space and falls under the influence of gravity and the motion of such a body is called projectile motion. In this context we will ignore the effects of air resistance to make calculations easier. Through the usage of trigonometric ratios and vectors it is possible to accurately predict the position of a body after a certain time, the maximum height attained by it and the horizontal distance it covers from the point of projection. Horizontal displacement or range of a projectile is the main index of performance in many cases of projectile motion. If air resistance is negligible, there is no net force in the horizontal direction (ΣF = 0; ax = 0) Through this topic we aim to explain the science behind the performed actions and movements in sports such as Golf, Football, Basketball and Javelin throw. Factors Affecting Distance traveled by a projectile: 1. Relative height of release 2. Speed of Release 3. Angle of release Projectile Motion: Theory v/s reality Theoretically optimal angle is about 45° however taking air resistance into consideration the angle reduces to about 42°. Long jumpers use angles of 17-23°. This is because when traveling at ~10 m/s, there is not enough time to generate a large takeoff angle. The game of Golf is based on the trajectory followed by the golf ball as it moves through the air and in this sport we have addressed issues such as the required club face angle and swing speed for the ball to go in the hole. For instance if we have a ten degree driver it will carry the ball lower than a 60 degree wedge and hence it can be deduced from the above statement that a greater angle of the club face launches the ball at a greater angle. Effects of Air resistance can be very large in case of golf. Therefore, the golf ball has dimples on its surface to negate the effect of air resistance. To depict the application of projectile motion in football, we have shot a video on our school’s football field showing the trajectory followed by a football and have addressed issues like horizontal and vertical velocity required depending on the nature of the kick. In the sport of Basketball we shot a video showing a student shooting a 3 pointer. Furthermore with the help of charts, we have calculated the velocity required for a basketball to go inside the hoop at different angles of projection such as 30, 45 and 60 degree. Finally we have included a question to determine whether a ball hit by Sachin Tendulkar will be a six or not using kinematical equations as well as equations related to projectile motion. Hence by shedding light on this wonderful topic we attempt to reveal how an athlete’s brain functions and through years and years of practice and hardwork he is able to accurately predict distances and achieve his goals.