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

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




1.0 Rationale:


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:


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.

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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:

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.

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:

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.



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.

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:


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.


7.0 Conclusion:


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.



  • Equi Supermarket. (2017). A Guide to Show Jumping Fences. Retrieved on January 1 2019, from
  • Mathmaticsofhorsebackriding. (2016). Jumping. Retrieved on January 1 2019, from
  • Kurtz Corral. (n.d.) Horse Breed Descriptions. Retrieved on January 1 2019, from


Golden Ratio at extension can be better, saying that its only for performance while dogs do not as its for hunting or leaping purposes. s


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