The physics behind archery

     Archery is a well known sport that displays many physics principles. In archery, a bow is pulled back and propels an arrow towards a target. Archery has been used for many years for hunting, war, and as a sport. 

    The main physics principle displayed through archery is the conservation of energy. Just as a ball at rest on the top of a hill has only potential energy then only kinetic energy at the end of the hill, a bow and arrow follows a similar principle. Before the arrow is shot and the bow string is untouched, there is no energy in that system, but as you put in work by drawing back the bow string, there is an increase in potential energy. This is a special potential energy, and as we learned in class, this is spring potential energy. Firstly, we will deal with the work of drawing the bow string back. As we know, the equation for work is W=Fd cosθ. The force in this case would be the force that you pull the bow string back, d would be the value that you displace the string, and assuming that you are pulling back the string with an angle of 90 degrees relative to the bow, the cosθ would be zero. 

For the potential energy the bow acts like a spring, so we know the equation is ½ kx^2. The value of k, or the spring constant coefficient is different for each bow, with a thicker, stronger bow string having a greater value for k and a child's bow having a much lower value for k. The x in this equation would be the displacement of the bow string. Additionally, if the bow is held above the line you make y=0, there would be gravitational potential energy, but for this purpose we will set y=0 level to the bow and arrow. When the bow is drawn back and the arrow is released, according to the conservation of mechanical energy, all energy will be turned into kinetic energy (½ mv^2). This is assuming that there are no other forces acting on the system, which we know that there are such as air resistance and friction of the arrow on the bow, but for these purposes we will ignore those values. Thus, if you know the spring constant of the bow, the displacement of the bow string, and the mass of the bow, you can solve for the velocity of the arrow. With these equations you cannot determine whether or not the arrow will perfectly hit the target; however, you can tell how fast the arrow will leave the bow.

https://www.wired.com/2014/12/much-energy-bow-goes-kinetic-energy-arrow/ 

http://ffden-2.phys.uaf.edu/webproj/212_spring_2015/Addis_Gonzalez/Addis_Gonzalez/bow.html https://arxiv.org/pdf/1511.02250.pdf

By: Zack Kraushaar