Math Exploration:

The Physics Behind Equestrian

Before this exploration begins, it is important to be informed of the definitions of many equestrian terms that will be used in this exploration. Equestrian: Of or relating to horseback riding or horseback riders. (Kirkland, Sarah) Walk: The walk has a distinct four beat rhythm. When the horse is walking, its movement is easily accounted for by the rider. (Kirkland, Sarah) Trot: The trot has a two beat rhythm to it. It is much harder to adjust to the trot than to walking because the rider is bounced up and down with each pace.

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This bouncing causes the rider to be thrown up and down, hitting the saddle pretty hard easily unseating them if they do not adjust properly to this movement. The horse exerts a force on the rider as its hooves make contact with the ground.

The rider in turn is bounced upward. To account for the bouncing, the rider can do something called posting, which is where for every other step, or beat, of the horse, the rider lifts themself (with the push of the horse) on their stirrups and misses the horses jerk.

For the second beat she sits down lightly and then is pushed up again. Posting is a controlled way of trotting. The rider synchronizes their posts with the horses, and it makes the ride much less bumpy. (Kirkland, Sarah) Canter: During the canter, which is a three beat gait, there is a point where the horse has all four hooves off the ground. This is a much smoother gait than the trot. The speed however is much greater than the other two gaits, and the important thing is to keep the rider’s weight distributed equally in both stirrups, and also to keep their center of balance above the horse’s. (Kirkland, Sarah) Gait: A particular way or manner of moving on foot. Any of the ways, such as a canter, trot, or walk, by which a horse can move by lifting the feet in different order or rhythm.

Show jumping: The competitive sport of riding horses over a course of fences and other obstacles in an arena, with penalty points for errors. (Kirkland, Sarah) Dressage: The guiding of a horse through a series of complex maneuvers by slight movements of the rider’s hands, legs, and weight. (Kirkland, Sarah) Strides: The number of steps taken between two jumps. (Kirkland, Sarah) Stirrups: Each of a pair of devices attached to each side of a horse’s saddle, in the form of a loop with a flat base to support the rider’s foot. (Wikipedia, 2012) Outside leg/hand: The arm or leg of the rider that is faced alongside the fence. The outside leg is used to ask the horse to transition from a walk/trot/halt to a canter. Two point: This is the position in which riders take when jumping over a jump. It is called two point as two points of your body (Feet and knees) are in alignment with each other. By the rider taking this position, it allows for the horse to carry the rider’s weight easier.

The reason as to why I decided to choose this topic is due to my interest in equestrianism. For almost 4 years, I have dedicated many weekends to this very demanding sport. Being a committed rider, it is important that one understands the physics behind the sport in order to achieve their best possible performance. I figure that by further investigating the physics behind equestrian, I will be able to apply these new found findings to my riding and become a better rider. I would also like to demonstrate that equestrian is not an easy sport and that it does involve more thinking and human involvement than many may think. I have also chosen this topic because it is very easy for me to communicate my ideas to others as I know this subject very well. With my audience and their comprehension of the topic in mind, I have chosen to pursue this interesting topic of equestrianism.

The sport of equestrianism is an ancient sport and the date of in which it began is controversial but it is believed that humans domesticated and rode horses as far back as 6000 B.C. Horses have played an important role in human history as they were used in warfare, transportation, trade and for agricultural purposes.(Wikipedia,2012) In historical times, it was crucial that one learned how to ride a horse as they were heavily used for transportation purposes (Riding and carriages) before the invention of the automobile in 1886 by Karl Benz. This exploration will cover the physics involved in two types of equestrian disciplines, show jumping and dressage. Both these disciplines involve physics as certain requirements must be met in order for the task to be carried out properly. This exploration will also cover the formulas and equations that riders must calculate on a daily basis in order to achieve success whether in a show or a simple ride. Newton’s Laws of Motion

Newton’s three laws of motion are relevant to the equestrian sports as they explain how certain actions are possible and carried out. 1. Every object in a state of uniform motion tends to remain in that state unless an external force is applied to it. (Georgia State University, 2012) 2. The relationship between an object’s mass (m), its acceleration (a), and the applied force (F) is F= ma. Acceleration and force are vectors; in this law, the direction of the force vector is the same direction of the acceleration vector. (Georgia State University, 2012) 3. For every action, there is an equal and opposite reaction. (Georgia State University, 2012) An example of Newton’s first law in equestrian terms is a moving horse. By this law, a moving horse will remain in motion until the rider applies pressure on the reins to tell the horse to stop or this law can also be seen if a moving horse is stopped by the external force of an object. (Godden, B.) This law of motion can be demonstrated by the following diagram:

This formula allows us to calculate how velocities change when forces are applied (Henderson, T., 2012), e.g. a change of gait. The second law of motion relevant to the sport of equestrian can be demonstrated by the example of how it is much more difficult to stop a heavier horse moving with much more acceleration because their force is greater. The formula can be seen in the following diagram. Figure 1 A)

Newton’s third law of motion in terms of equestrian can be seen when a horse must apply the same equivalent of a force to slow to a halt. It can be seen in the following diagram: Figure 1 B)

Newton’s laws are an example of linear systems and algebra as one often finds themselves isolating variables in order to solve the equation. The Gaits

In the equestrian world, there are four main gaits in which a horse moves: The walk, trot, canter and gallop. The trot, canter and gallop each have certain positions that a rider should take when in this gait in order to optimize speed and comfort. Trot

Figure 2A)

The trot can be seen in the image above. The trot is the most bouncy of all gaits because of the diagonal movement of the legs. The force of the opposite legs landing at different times creates an opposite reaction which is a thrust of the rider forward. In order to counteract the force, riders perform a technique called “posting”; a rise out of the saddle when the horse’s outside leg moves forwards. If the rider rises out of the saddle while the inside leg moves forward, they are on the wrong post and the ride will be bouncier. This allows for the trot to be smoother and easier to ride to. The following diagram shows the technique of posting. Figure 2 B)

The Canter

The canter and the phases that occur within it can be seen in the following diagram.

The canter is used very frequently in show jumping and dressage as it is an in-between speed between trot and gallop and is much easier to control. Riders must be able to estimate how long their horse’s stride is in the show jumping discipline. Riders must increase or decrease their horses speed in order to get the correct set number of strides. In order to determine how many strides that a horse will get at a certain speed or the speed of a horse or the distance covered, formulas must be followed:

Here are some example problems that a rider could be faced with at a show jumping show. Example: In order to meet the judge’s request, Eleanor must complete a line (Two jumps separated by a flat distance) measuring 25 feet in a maximum of one minute. How fast should Eleanor go? Answer: Total speed: 25 ÷ 1 = 25.

25 feet in one minute can be transferred into an equation of feet/hour by multiplying the answer by 60. Eleanor will have to travel at a speed of 1500 feet/hour. Show jumping and formulas

There are three steps involved in a jump in the sport of show jumping. 1) The approach: Involves the first three strides taken before the take-off in front of a jump. 2) The jump: The air time as the horse rises in the air, experiences all feet off the ground and then descends back down on the ground. 3) The landing: Includes the impact of the horse hitting the ground and the follow through as both horse and rider return to their normal positions. The whole jumping part of this sport is not a simple process. The most important part of this process is to regulate the speed of the horse’s approach so that the horse has enough speed to clear the jump. The process will be explained in formulas in the following paragraphs. The Approach: Upon approaching the fence, the horse and rider only possess kinetic energy (Energy possessed by being in motion). The formula for kinetic energy can be seen in the following diagram. (Erin R., March 2003) Figure 4)

The formula for kinetic energy can be explained using Newton’s laws. First, let us have a constant force, F, acting over a distance s acting on a particle of mass m, then, Fs = KE. Now we use Newton’s law KE = mas where a is the acceleration of the horse. Now we use v2=u2+2as where v is our final velocity and u is our initial velocity. If we assume that the horse was initially standing at a halt and then accelerated over the jump, then u=0 and we define the KE at velocity zero to be zero. Then figure four is equal to an arbitrary velocity for v. (B., A. September 2011) The Jump: Over the top of the fence, the horse and rider reach maximum height and their velocity is reduced to zero; thus, they possess only potential Energy, PE. (Erin R. March 2003) This potential energy is expressed by: Figure 5)

This equation is much simpler than the equation for kinetic energy. In this equation mass represents mass of the horse, g represents the gravitational field strength (9.8 N/kg on Earth), and height represents the height of the horse. (Henderson, T.) The Landing: When the horse returns to the ground, horse and rider possess only kinetic energy. (Erin R., March 2003) Example of an equation:

If a rider wishes to jump a 5 foot fence, how fast will they need to be going on ‘approach’? Also, If horse and rider do clear the fence, how fast will they be going on ‘landing’? Part One:

KEi = PE (I is for initial) 1/2 mv2 = mgh h= 5 ft. = 1.52 m because 1 meter is equal to 3.281 ft.

In calculation, one should assume that the horse may jump up to six inches higher than the fence, depending on their perception, thus y = 1.52m + 0.15m or h=1.52m Due to non-conservative forces, such as air resistance and heat, the potential energy reached at the top of the jump will only be about 80% of the kinetic energy present on approach. (Erin R., March 2003) 1/2v2 = 0.8(9.8m/s2*1.52m) hlow = 6.1m/s hhigh = 6.4m/s Part Two: mgh = 1/2 mv2

Again, due to non-conservative forces, the kinetic energy present on landing will be only approximately 80% of the potential energy present at the top of the jump. 9.8m/s2*1.52m = 0.8(1/2v2) vlow = 4.88m/s vhigh = 5.12m/s (Erin R., March 2003) Another important motion in show jumping is to consider the horse’s movements in terms of projectile motion. In order to calculate projectile motion, you must have the range equation and the height equation. Range equation: R = (vo2÷g) (sin2θ) Height equation: H = (vo2 sin2θ) ÷ (2g)

Example Question: With what initial velocity will a horse need to take off in order to jump a 1.5m high fence? Solution: In order to have some values to start off with, let’s assume that the horse leaves the ground at an angle of 45° at a speed of 9.8 m/s. H = (vo2 sin2θ) ÷ (2g) In this solution, “vo” is equal to the initial velocity that the horse needs in vo= √((Hx2g) ÷(sin2θ)) order to take off. H is equal to the height that this horse must reach. Θ is equal vo= √((1.5m x2x(9.8m/s)) ÷ (sin245) to the angle at which the horse took off. “g” is equal to the initial speed vo= 7.7m/s the horse was travelling prior to the jump.

The stages in which a horse approaches, jumps, and lands over a jump can be seen in the following diagram:

Equestrianism is a difficult sport that without the laws of physics; wouldn’t exist. These formulas and equations can also be applied to several other sports and the knowledge and understanding of these formulas will help one further understand other concepts in mathematics. I believe that it is easier to understand these concepts with help of a visual aid, such as a horse jumping. This topic explores many mathematical concepts and formulas that help us to understand the logic of everyday happenings around us. It is very important that we study and investigate the mathematics involved in our daily lives as it helps us to understand why. With the knowledge I have gained by exploring all the mathematical formulas involved in this sport, I will be able to have a more precise and more thought out performance in the show ring.

Bibliography

Wikipedia. (2012). Retrieved from http://en.wikipedia.org/wiki/Equestrianism Kirkland, S. (n.d.). The physics of horseback riding. Retrieved from https://sites.google.com/site/thephysicsofhorsebackriding/horseback-riding-terms-1 Georgia State University. (June 2012). Astronomy 161: Newton’s Laws of Motion Retrieved from http://csep10.phys.utk.edu/astr161/lect/history/newton3laws.html Godden, B. (n.d.). Retrieved from http://www.horseforum.com/horse-riding/riding-physics-newton-47176/ Henderson, T. (2012). The Physics Classroom. Retrieved from http://www.physicsclassroom.com/class/newtlaws/u2l1a.cfm B., A. (2011, September 17). Explaining the formula of kinetic energy. Retrieved from http://www.scienceforums.net/topic/59887-help-in-explaining-formula-of-kinetic-energy/ Erin R. (March 2004). One Giant Leap: UNC Physics. Retrieved from http://www.unc.edu/~erinr/