本文是数学专业的留学生作业代写范例，题目是“How does the jump of a horse affect its performance in showjumping competitions?(在跳马比赛中，马的跳跃是如何影响其表现的?)”在本期Math IA中，我想谈谈不同类型的马，以及它们是如何比其他动物更好地跳跃的。我这个内部评估的目的是分析马匹的数学可以影响骑马跳跃。

我从小就对马和骑马感兴趣。我一直想知道为什么会有不同种类和品种的马，比如跳跃表演、盛装舞步、赛马、马术竞技等。所以在这个数学内部调查中，我将使用函数，三角函数和二次函数作为我的主要数学用途来找出我的目标。作为我思考的一部分，我将延伸到另一种动物物种，狗跳跃和奔跑，看看它们是否像马一样跳跃，看看动物的跳跃是否会影响它的表现。

** Rationale理由**

In this Math IA, I am wanting to talk about the different types of horses and their how they jump is better than other animals. My aim of this Internal Assessment is to analyse the mathematics of horses that can affect the jumps in horse-riding. I have been interested in horses and horse-riding since I was young. I have always wondered why there are different categories and breeds of horses, for Show jumping, Dressage, Racing, Rodeo, etc. So in this Math Internal Investigation, I will be using functions, trigonometry and quadratics as my main use of math to figure out my aim. As part of my reflection, I will be extending to another animal species, dogs jumping and running and to see if they have the jump as horses and see if the jump of an animal can affect its performance,

Research Question:

How does the jump of a horse affect its performance in showjumping competitions?

** 2.0 Introduction作品简介**

In the earlier days, horses are used as work animals to help with our everyday life. But as time goes by, a sport called horse-riding is a famous sport in many countries. In this sport, there are different categories and one of them is called Show jumping, where riders has to jump over a series of obstacles that varies in colours, height and shape by following a set course as quickly as possible, and also without knocking any over.

在早期，马被用作工作动物来帮助我们的日常生活。但是随着时间的推移，一项叫做骑马的运动在许多国家是一项著名的运动。在这项运动中，有不同的类别，其中一个被称为跳跃表演，选手必须按照规定的路线尽可能快地跳过一系列颜色、高度和形状各异的障碍，同时不撞倒任何障碍物。

An obstacle jump is created from two stands that support the poles. The shapes of jumps are very varied: from simple, double, triples, combinations, walls, verticals, oxers, or water. (Equi Supermarket, 2017) The vertical jump is where the poles are directly above one another. Oxer jumps are two vertical jumps placed together at a wider width, and triple bar jump is the poles of increasing in the heights. (Equi Supermarket, 2017) In riding competitions, riders and horses are required to complete a round that has around 10-12 jumps, and jumps are numbered for riders to find their way around the course. The main objective of the sport is not knock down or refuse any jumps so both horse and rider has to be in harmony in order to not receive a penalty. Each fault results in a penalty of 4 points. Going over the expected time for the course time allowed will also results in penalty points. Three refusals of a jump or a fall result in a straight elimination.

The different breeds of horses are also important when doing the specific sport. If you use a horse breed that is bred specifically made for stamina and endurance during racing to do a show-jumping sport. it will not perform at its best as the proportions and height of the horse are wrong. For example, the Arabian horse breed is a adaptable horse breed that can be used for many sports. Another example, the Thoroughbred and Quarter horse are very popular for horse racing. The Belgian draft is a draft horse breed originating from Belgium and is mainly used for draught power. The Lipizzaner is a breed originating in Central Europe and is very popular as a dressage horse. There are also ponies that are smaller in size than the average horse, that are used for leisure riding and sometime show-jumping events. (Kurtz Corral, 2018)

**3.0 Discussion讨论**

In the image on the left, it was taken from the video of me jumping over an obstacle course at 80cm in height. The horse I was on was a Dutch Warmblood, a horse known for their jumping skills and speed. For a Dutch Warmblood breed, the horse’s head is refined, the neck is long and arched. The shoulder is well-sloped and the girth is deep, with plenty of heart room.The forelegs are strong, with well-muscled forearms. The hindquarters are powerful and the hock joints are low to the ground. These factors combine to give great power to the “engine” of the Dutch Warmblood. And as we can see, horses have a parallel jump, where their legs and body will be parallel to the jump poles when in ‘mid-air’ or are over the obstavle.. This difference indicates that the Dressage horses selected for the specific sport are becoming overall larger and these findings show that the characteristics of these animals are being modified for this sport as a result of selective breeding. Stronger muscles in the hind limbs and taller hind limbs indicate greater power for jumping and ability to give a solo performance. This greater power is a desired trait in these horses.

在左边的图片中，这是我在80厘米高的时候跳过障碍训练场的视频。我骑的是荷兰温血马，这是一种以跳跃技巧和速度著称的马。荷兰温血马的头部很精致，颈部长且呈拱形。肩部倾斜，周长深，有足够的心脏空间。前腿强壮，前臂肌肉发达。后腿有力，飞节离地较低。这些因素结合在一起，为荷兰温血族的“引擎”提供了巨大的动力。正如我们所看到的，马有一个平行跳跃，当它们在“半空中”或越过障碍物时，它们的腿和身体将与跳跃杆平行。这一差异表明，为特定运动选择的盛装舞步马整体上变得更大，这些结果表明，这些动物的特征正在为这项运动的选择性育种的结果。后肢更强壮的肌肉和更高的后肢表明跳跃的力量更大，也更有单人表演的能力。这种强大的力量是这些马所渴望的特质。

A quadratic equation is the algebraic function of a parabola. A parabola is bell shaped that is either facing upwards or downwards depending on the equation. In this case our parabola is facing downwards since horse-jumping is going over an obstacle. In order to find the equation of the parabola for the horse’s jump, we must use the two coordinate pairs. A quadratic equation is expressed as

Tangent =

100 cm – Triangle on the Left (Takeoff):

Tangent =

Tangent =

110cm – Triangle on the Left (Takeoff):

Tangent =

Tangent =

**5.0 Extended Research扩展研究**

After all my research and calculations, my research can finally be answered, how does the jump of a horse affect its performance in showjumping competitions? The angles of takeoff and landing and distance and height of the jump of the horse does affect its performance in competitions. When the horse is jumping, math is extremely important. To further extend my knowledge, I chose to focus on the shape of the jump using Golden Ratios and like many things in Nature, the jump takes the shape of a parabola. In a correct jump of a horse, the peak is the highest part and height of the fence that is usually measured in the middle. The distance of takeoff and landing of the horse’s jump should be the same from the base of the jump. As seen in the picture below, the horse, no matter what heights, all stayed within the Golden Ratio Swirl and has a strong parabola curve when going over the obstacle.

经过我所有的研究和计算，我的研究终于得到了答案，一匹马的跳跃是如何影响它在跳马比赛中的表现的?起落角度、起跳距离和起跳高度对比赛成绩有一定的影响。当马在跳跃时，数学是非常重要的。为了进一步扩展我的知识，我选择使用黄金比例来关注跳跃的形状，就像自然界中的许多事物一样，跳跃的形状是抛物线。在正确的跳跃中，马峰是栅栏的最高部分和高度，通常在中间测量。起跳和起跳的距离应该与起跳的起点相同。如下图所示，不管马有多高，都保持在黄金比例旋涡内，并且在越过障碍物时具有很强的抛物线曲线。

To compare, comparing with dogs is something dogs these days, there are dog shows, where different dog breeds dogs also have to jump different obstacles to test their agility. And this thought made me think about that does the golden ratio affect dogs’ jumps as well? So I did some calculating using the Golden Spiral and then getting the image below for both dogs and horses. The Golden Ratio was the same for both species of animals, during take-off and landing to be specific.

相比之下，和现在的狗相比，在狗展上，不同品种的狗还必须跳过不同的障碍来测试它们的敏捷性。这个想法让我想到黄金比例也会影响狗的跳跃吗?所以我用黄金螺旋做了一些计算，然后得到下面的狗和马的图像。这两种动物的黄金比例是一样的，具体来说是在起飞和降落的时候。

Figure above is the take-off, jumping and landing of a dog with golden spiral

Figure above shows 6 different horse breeds jumping the obstacles and the golden ratio.

** 6.0 Evaluations评估**

In this IA, I based my research on a small sample of jumps which will contribute to mistakes. For example, is that the take-off and landing points of the horses could not be told that thy were all equally distanced from the jump. Though I believe that my investigating to be quite successful, it has one major drawback. And that it is just a simple theory. To some people, it’s just a rider, a horse and an obstacle jump and they will not even think about the take off point. But this kinds of research can benefit professional riders that will look at every detail of jumps. Also, I have to mention about the external factors that can determine the jump, such as the confidence in rider or horse, horse strengths, since not all horse breeds are always able to go across a big distance.

在这个IA中，我的研究基于一个小的跳跃样本，这将导致错误。例如，马的起落点与起跳点之间的距离是相等的。虽然我认为我的调查是相当成功的，但它有一个主要的缺点。这只是一个简单的理论。对一些人来说，它只是一个骑手，一匹马和一个障碍跳跃，他们甚至不会考虑起飞点。但这类研究可以让专业车手受益，他们会仔细观察跳跃的每一个细节。此外，我还必须提到决定跳远的外部因素，比如对骑手或马的信心、马的力量，因为并不是所有的马品种都能跨越很长的距离。

** 7.0 Conclusion结论**

Even though horses do have the same golden ratio as dogs when jumping, the parabola will still be different due to the height and distance of jump. But it will not change the way the animals jump as there are the golden ratios present. Even with the golden ratios present that shows that the horse breeds jump at almost the same way as each other, horses these days are being bred to contain more muscles in their body or to approve their agility during a competition. But as much as they breed specifically, a horse is a horse. The jumping of a horse will stay the same.

尽管马和狗在跳跃时有着相同的黄金比例，但由于跳跃的高度和距离，抛物线仍然是不同的。但这不会改变动物跳跃的方式，因为现在有黄金比例。即使现在的黄金比例表明，不同品种的马跳起来的方式几乎相同，如今的马被饲养是为了让它们的身体拥有更多的肌肉，或者在比赛中表现出它们的敏捷性。但就像它们繁殖的特殊性一样，马就是马。马的跳跃将保持不变。

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