Classical Mechanics of Archery Part 1

Classical Mechanics of Archery Part 1

Archery is a sport that is good for the mind and the body which benefits me greatly for my health and wellbeing. But it’s also got something for me in other ways. One of the reasons why I choose archery over many of the other Olympic sports is because it is also a combination of physics and craftsmanship. I am a science geek as a well a sportsman and I have a broad range of interests in science which includes physics and engineering. At the time I took up archery after the 2012 Olympics I was studying for a degree in physics at the Open University. One of the topics I covered was classical mechanics which includes the physics of the mechanism of the bow and arrow and I spent some time examining the science of archery. For this post I have decided to combine my love of science and my love of archery to show that archery is a prime example of brains and brawn working together in harmony. This should set the stigma aside that geeks like me are not cut out for sports.

First of all let’s start with the mechanics of motion. When you fire a projectile like an arrow do you expect it to go straight forward towards the target or up and then down in a curved line known as a parabola? Well for thousands of years people used to assume the former as the great philosopher Aristotle, tutor to Alexander the Great, told of how projectile motion worked without experimenting with it. He lived in 4th century BC Macedonia, but he was originally from Greece where his tutor was Plato. The physicists of Aristotle’s day and age didn’t experiment with nature very much because they were theorists who used common sense from observation where the logic is impeccable. Aristotle believed that force was always needed to make an object move until it comes to ‘a natural state of rest’ as he called it. Arrows are made of naturally heavy materials and so their natural state of rest is on the ground. By that logic when we fire arrows they will continue to go on in the direction of the target and then land on the ground in a straight direction. Just aim at the target and once you release the arrow it will fall to the ground. He later tried to explain how come the arrow actually flies through the air when it doesn’t fall directly to the ground upon release as well. Once again without experimentation but by observation.

According to Aristotle the air around the arrow experienced a force from the firing of the bow. The arrow would travel the straightest path that it was set to take by the archer and in that case the arrow would always hit the target if aimed in a straight direction. That is not true, if it were then I would be able to hit the gold ring and score a perfect round and everyone could be an Olympian. An Islamic scholar and a French philosopher later trained to expand on Aristotle’s science by explaining that the force on the arrow was it’s impetus and it fell to the ground once it lost it’s impetus. This was supposed to explain how an arrow can fall to the ground when shot directly into the sky at an angle and then fall directly to the ground. Later it was realised that this impetus is actually called the momentum, which is best described as the force when the arrow’s mass is multiplied by the speed it is travelling. It wasn’t until the 17th century when scientists like Galileo Galilei finally applied experimentation to all the teachings of the ancients that the ground rules for physics were established. This meant that from now on experimentation as well as observation were compulsory for testing scientific theories and technological innovations.


One of the most significant discoveries about projectile motion that Galileo discovered would also become Issac Newton’s first law of gravity: inertia. That is the reaction an arrow gets when you release the string, the string will bounce back and the limbs will flex back into their natural shapes but the arrow can separate away from the string and fly off by the force of the potential power of the limbs and the string. Remember that the power of the bow is in the draw weight of the limbs. The stronger the tension they have when pulled the heavier the force they have on the arrow when released back into their normal shape. Now Galileo was perhaps most famous for proving that the Earth orbited around the Sun and disproved that the Earth was not at the centre of the Universe. Part of this science involved studying the behaviour of falling objects and seeing how gravity affects the path they take when they fall to the ground. He found that all projectiles like arrows don’t actually travel in a straight line at all, they travel in a parabola. A parabola is a curved line that starts from one point, goes up and then comes back down at another point at a distance from the starting point on the same level. You fire the arrow and it travels at speed towards a target upwardly and then as soon as it slows down gravity makes it fall back down again and it hits the target just before it hits the ground altogether. Providing the poundage is right for the distance you are shooting at. This is an important factor in target archery when you have to aim your bow using the sight and anticipate where it’s going to hit based on how level the target centre is with your line of fire. That’s why the common bowsight that we have on recurve and compound bows are adjustable for all sizes of archer as well as how far the target is and the poundage of the bow.


Now that we understand how arrows fly the way they do I think it might be worth looking into how bows behave. Bows are cleverly crafted pieces of engineering in style and simplicity. Da Vinci had a saying ‘Simplicity is the ultimate sophistication’. The mechanics and the physical behaviour of a bow is to be understood as an elegant tool of beauty and precision. A bow is a device that converts slow and steady human force over a distance into stored mechanical potential energy. Energy that is stored in the limbs of the bow. This energy is converted into kinetic energy upon release of the bowstring and a great deal of that kinetic energy is transferred to the arrow. Potential energy is the energy that an object has based on it’s position and kinetic energy is energy through motion. When you pull back on your bow, you apply a force to the bowstring which in turn bends the bow as it adds elastic potential energy. Thus a bow is basically a sprint which stores energy to be put into the arrow. The stretched shape of the bow is an example of potential energy and then there’s the kinetic energy which is what it becomes when the string springs back into it’s normal shape. As the second law of aerodynamics states that energy cannot be created nor destroyed, it just goes somewhere and that energy has now gone into the arrow. When the arrow leaves the bow it’s full of energy which is taking it into flight on a path towards the target where the only thing that can affect it’s flight is air resistance and gravity.

The draw weight of the bow is often associated with just the limbs of the bow. But in actual fact it’s more to do with the length of the bow itself as well. Draw weight is directly proportional to draw length, as the bow behaves just like a spring. Robert Hooke was a scientist who was studying the laws of motion using springs and he provided some inspiration to Isaac Newton. In 1660 he discovered the law of elasticity which bears his name: Hooke’s Law. Hooke’s Law explains that the amount of stretch in a spring is proportional to the force pulling on the spring. This can also be applied to bows where it’s known as elastic potential energy. When you pull the bow outwards the length of the stretch is the same as the force of you pulling it outwards. The force upon the bow is recognised as the draw curve. The draw curve is apparently a slight curve because of the shape of the bow, but when you combine it with other factors is becomes defined graphically to help understand the relationship between the draw length and the draw weight.


The relationship between the draw length and the draw weight is graphically defined as a straight line relationship where the energy stored in the bow is calculated by the draw weight multiplied by the draw length divided by the number of limbs (2). For example my bow has 28 lbs limbs and I have a draw length of about 28 inches. So 28 lbs x 28 ins/2 = 392 pounds per square inch.


For a recurve bow the draw weight is defined graphically as a straight line, but for a compound bow it is defined as a curved line. This is because compound bows use levers and cams, which decreases the draw weight with the draw length. Thus allowing the bow with the same amount of energy to require less force to pull on it. This is why compound bows feel lighter to pull than a recurve bow.


This is about as much as I can cover for now on the topic. I could go on but it would be too much for one post and so I have decided to make it in parts. I enjoyed writing this one because it gave me a chance to go over my old textbooks and reading material on physics and engineering principles. This could be a chance to show science in sport is not just for the physical attributes of the athlete but also for the performance of the equipment that they use. When you understand the laws of physics you can blend in with the nature of the competition.