Archery: Draw Weight vs Draw Force

Archery: Draw Weight vs Draw Force

Twang, thwapp! We’ve all heard this sound and know what it means, some archer has hit their mark. Bows are one of the most commonly known about items in our world despite their lack of use. We can thank characters like Legolas, Katniss, Hawkeye, or even Robin Hood for this, characters who have romanticized the bow to the extent that we all know what it is, but do we know what it can do, or how it works? I decided to set out and find this out using the physics we have learned throughout the past semester.

To first understand bows, we need to understand how they work. Bows are relatively simple constructions that, like most technology and creations, have gotten much more complicated over time. For well over 12,000 years bows have been a simple system that consists of a string attached to elastic limbs that store mechanical energy imparted by the user drawing the string. This energy would then be stretched over the string as the archer drew the bow, giving it its displacement. The elastic limbs themselves changed over time and cultures based on the materials used, traditional springy wood was preferred for its strength, and its elasticity or spryness allowed the bow to maintain its shape while in use. The arms would be produced with varying strength depending on the user, material, and would require different amounts of force to draw the bow and launch the arrow, this was then put into terms of weight giving us the term, “draw weight.”

Since then we have expanded further still on the bow and created the compound bow. The compound bow works under the same general principle as previous iterations of the bow, but is instead made from modern plastics, aluminum, and composite allows, and included pulleys, levers, and cables to enhance the draw and give a “let off point” at its max draw where the archer does not need as much force to maintain the bows draw. This type of bow also gives the strongest transference of energy to the arrow yet, normally between 70-85% of the stored energy is transferred to the arrow. This stored energy is referred to as potential energy. When transferred to the arrow it is referred to as kinetic energy. Thus I decided to test if the rate of the strength to force was the same no matter how much draw weight a bow had. I used two separate types of bows with differing draw weights, one of 15lb and one of 25lb  to see how the ratio of joules to weight ratio matched up, and if it was consistent. I hypothesized that the ratio would stay consistent no matter the draw weight of the bow as the size and power ratio should always be similar if not the same.

I then set up a target range and measured out 35 feet (10.667 meters) to use for the experiment and recorded myself launching the arrow so we would be able to accurately calculate the arrows velocity (v=distance/time) to get its force and joules. I found that the 15lb bow was able to cover the distance in .488 seconds, meaning that it had a velocity of 21.801 meters per second (48 miles per hour) and the 25lb bow was a full .1 seconds faster and went 28.074 meters per second (62 miles per hour). After plugging both of these into the kinetic energy equation (and taking into account the .076 kilogram training arrow) k=1/2 mv^2 we were able to calculate the force of each bow at 29.95 joules for the 25lb bow,(1.197 joules to pounds ratio) and 18.09 joules for the 15lb bow (1.206 Joules to pounds ratio).

Next we need to figure out the force. Using an energy bar chart we find that W=KE or work equals kinetic energy, thus our force will be equal to energy divided by the displacement, which in this case is the draw length of the bow. The 15lb bow has a draw length of .48 meters, meaning that our work equation is simply 21.801/.48, giving us 45.46 Newton’s of force (giving us a ratio of 3.03 Newton’s of force per pound of draw weight). The 25 pound bow had a .6m draw distance, making this equation 29.95/.6, or 49.9 Newton’s (a ratio of 1.996 Newton’s of force per pound of draw weight). This means that my main hypothesis of the ratio’s being relatively the same was incorrect, with a full 1.034 newton difference between them. However the two bows shared a very similar ratio of energy conservation, with only a .008 variance dependent on the quality of the bow and its size.

Thus we can see that my main hypothesis was incorrect, the larger a bow is the more its force drops off, likely due to its larger displacement causing it to have more time to loose energy, showing that a weaker bow actually has a better force to weight ratio. However we also found that both had an extremely similar ratio of energy to pound, and as a result we can consider that notable as well. Thus I will cede that I was wrong in my initial prediction, but was proven correct in a different test that shows that the total weight will always create the same amount of energy.

Possible problems

One of the most critical issues I found was my bows themselves. Both are years old, and very well used. My methods of containing them also are generally not the greatest, simply storing them in a shed in the backyard and keeping them relatively vulnerable to the cold but safe from the elements. The amount of time they have spent being used can also compromise the strength of the bow over time. For example dry firing a bow (loosening the string without an arrow in it) causes the entire energy (all 20+ joules for both bows) goes back into the bow arms from their point of weakest resistance, damaging the bow and compromising it structural integrity. I already know that this had some negative effects on the arrows themselves, as a direct weight to joules comparison places both at less force than they should, but this was much less that should worry about, as one can also consider the force loss to friction, cold, or the wind speed that could change its trajectory and time.