This is the technical section of the book. It is written assuming that you know no trigonometry, geometry or calculus. Whenever trig, geometry or calculus are used, the idea that guides its application will be explained. Physics, Math and Riding What no one ever tells us in school is the fundamental REASON for using math. We all get told the practical and aesthetic reasons: math is useful and it's great mental gymnastics. Well, dandy. But why, oh why, should math apply to the real world, when it has its own little virtual universes of assumptions? Why should Euclidean geometry, trigonometry and calculus apply to the physics of motion? The answer is subtle: wherever patterns in math correspond to real world patterns, we can use it to calculate directions, forces and other cool stuff. A pattern is a reapeated motif, a design, a template. In short, we make analogies when we use math. 

Biomechanics and Dressage at Michigan State University Links HERE INTRODUCTION to DIAGRAMS Tyrannosaur movie (In what way is a horse is similar to a tyrannosaur?) Sets of equations can be arranged as a model to analyze the behavior of physical or biological systems. We will be less formal than this, and only use math to inspect some interesting aspect of equine, human and carnosaur motion. Why carnosaurs? They have qualities (besides legendary appetites) which help us understand the forces we activate when we ride. Why can we treat a horse or any animal as a machine? Because we can make analogies with legs or other body parts with simple machines like levers and wheels. Simple machines MAGNIFY FORCES and simple machines can be interconnected to MAGNIFY DISTANCE. We can compare and contrast horses and humans as athletes on this basis. Some curious things become evident as we do this! A hint of the result of this comparison comes from the ancient Armenian saying: "A horse need not worry how fast a man may run." But both might worry how fast a predator may run... Geometry and trig let us calculate directions of forces and magnitudes (sizes) of forces. Something that has direction and magnitude is called a vector. When a horse puts a foot or a diagonal pair of feet on the ground, it pushes its body a certain distance in a direction FORWARD <> BACKWARD with an effort of a certain MAGNITUDE (M). If the force pushing the horse is rotational, it is called TORQUE. If the horse pushes iself into the air for a moment of SUSPENSION, it falls to earth according to the acceleration of gravity and lands with a Magnitude can be shown by geometric argument to correspond to a trigonometric ratio called a sine of an angle (length of the side opposite divided by the hypotenuse in a right triangle). You don't always have to use right triangles, but they make calculation easier because they are special in ways you can look up in a geometry text. So if you're smart, you look for ways to put right triangles somewhere in your working diagram so you can connect magnitudes by clever argument. Calculus is about rates, so we can use it when rates are the pattern in the real world we wish to calculate. Geometry, analytic geometry, trig and calculus are all related math disciplines, so the patterns they share are related and we can make analogous matches in the real world. Nifty. Oh well...why bother with the physics of equine motion? Because we need to know the reasonable physical limits on what we ask of ourselves and our horses. In the math we can find explanations for why we get tired just standing up. What forces affect a horse because of demands for "suspension in trot and canter? There are ethical reasons as well as practical reasons for respecting limits: pushing a willing animal or ourselves too hard leads to injury. Not only is this not nice, it's expensive. In addition, competition is not necessarily about "normal," it is often about pushing limits. A look at Olympic disciplines makes this clear. This is not an argument against Olympic ambitions. Rather it is a set of guides for probing limits carefully and systematically. Should our probing lead to problems, we can back away and figure another route to the desired goal. If some goals are inspected and found to be unrealistic, we have the means to figure out which goals are reasonable in order to make our rules and judging criteria accordingly. Training to higher skill levels is also about pushing limits, so calculation can help us decide where we are within a biologically sensible range with our efforts, or where we are inviting injury. As riders, what forces may we activate that build gymnastic capabilities such as strength, flexibility, range of motion and endurance?
Tyrannosaurs and horses have hind limbs which are similar. However, the rib cages are not, and that turns out to be a crucial difference. In addition, the horse does not take its entire mass on the hind limbs. Even with these differences, there are interesting similarities between the two species. The comparison illuminates the adaptations of both for speed. 

LEFT Click on the image to view at full size. A tyrannosaur of only six tons at the run (individuals of about 12 tons are known). For torque, think of the animal with a wheel with a radius of about 11.5 feet, the hip height. The length of the femur is 4.25 feet (line AD) which moves through an angle of 61 degrees in a run step. Notice that while the femur moves the distance B'D' along an arc, the foot goes from B to D. This is an example of the magnification of force by the levers of the legs. The foot cannot follow an arc because of the ground, so the joints have to flex as they adjust each section of leg to the terrain. There has bee interest in tyrannosaur jaws: but the foot performs an extraordinary role in propelling this massive animal. Every bone in every toe has a role to play and had to be considered in the animation. We can calculate the arc traveled by the femur by calculating the circumference of a circle of radius 4.25 feet (C = pi x 2r = 26.7 feet). There are 360 degrees in a circle and 75 degrees is 20.8% of that so the knee end of the femur goes 5.6 feet (26.7 feet x 0.208). Meanwhile the foot is at the end of a 11.5 foot effective leg (it has been argued that tyrannosaurs ran with the knee bent in order to keep the knee from dislocating). The circumference of the wheel/circle is 144.5 feet with the same angle and same percentage as the femur. If the leg were straight and followed the lines AE and AB', it would travel So the T. rex foot on a leg with a bent knee goes 26.4 feet (24.4 feet/sin 75 degrees or 0.924) while the femur only went 5.6 feet! This is a theoretical mechanical advantage of about 4.7 for the tyrannosaur leg with the bent knee. Notice how the bend in the knee contributes to the step length compared with the straight leg (a stride is two steps). A world class running human with a 1.7 foot femur, a 3.5 foot leg and the same 75 degree range of motion covers 20.8% of 44 feet or 9 feet in a step. The femur makes an arc about 1.8 feet. With a step one third (26.4/9 = 3) of the tyrannosaur, a person would have to run with a step frequency about 3 times that of a tyrannosaur to cover the same ground in the same time. Outrunning such an animal (especially if it had warm blood and the aerobic adaptations for breathing that some paleontologists think it did) would be just exercise before lunch (with the person as lunch). Given the uncertainty about how fast large (small ones have been estimated at 12 m/sec) carnosaurs moved, a human sprinter at 11 m/sec might outrun a T. rex moving at 6 or 7 m/sec. Gregory S. Paul, a paleontologist, estimates a tyrannosaur at the run to be as fast as a race horse, because they have similar hind legs. Twenty miles per hour is 8.97 m/sec. A 16 hand (5 foot 3 inch) trotting horse with a 2.2 foot femur, a 5 foot effective hind leg and a 75 degree range of motion covers 20.8% of 62.8 feet in a step or 21 feet. The femur makes an arc about 2.4 feet. Horse races are won at between 15 and 17 m/sec. 

Calculation of TORQUE involves the downward force on the leg (12,000 pounds) and the arc of 4.25 feet traveled by the femur (or the foot), which is the bone where the hamstring group and the gluteals act. For the human, the numbers are 150 pounds and 1.8 feet traveled by the knee (pi x 2(1.7') femur x 0.169). A horse can move its femur through a 70 to 75 degree arc degree arc (0.194 to 0.208 of a circle circumference) torque = power ÷ speed power = force x distance x rate For the tyrannosaur (femur) stride: For the tyrannosaur (foot on a straight leg) stride: For the human (femur) stride: For the horse (femur) stride assuming 40% of mass on hind legs:


Click on the image at left to view at full size. Hind legs of a 1200 pound horse compared with a 12,000 pound tyrannosaur at the same point in a step. The horse is at suspension, having just pushed off of the right hind. I have shown the tyrannosaur landing on tip toe rather than spending time with both feet off the ground. Gregory S. Paul compares the horse's hind limb and tyrannosaur leg with the horse at a gallop, an asymmetric gait. However, the equine trot is symmetrical and in animation comparison (the movie is under construction) shows more similarity to the movements of the tyrannosaur. Another difference is that the horse hind limb bears only about 40% of its mass while the tyrannosaur hind limb has to bear its full 12,000 pounds. Elastic response for the horse's steps comes from its Achilles tendon system as well as from its compressible rib cage. For the tyrannosaur, the rib cage is a rigid bony box protecting viscera from battering during movement. It is shown dimmed to indicate this. I have not shown the carnosaur's sternal ribs because they interfere with viewing the hind limbs (also missing are the hyoid apparatus and patella). It shares with the horse just one source of elastic response (potential to kinetic energy) in the stride: the Achilles tendon system. In the tyrannosaur, this is connected to the remarkable foot mentioned above. 

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