Arrow Velocity vs. Draw Lengths

Background

Energy Storage

Arrow Velocity

Description of the Questions

Hypothesis

Procedure

Data Table & Graph

Conclusion

Biography

Bows work by storing Elastic potential energy in the arms of the tiller or bow from tension in the strings. This potential energy is then (mostly) transferred to kinetic energy. Bows come in varying pounds and sizes, and there are many different types of arrows. Each factor in the bow, from the weight of the arrow, to the type of string, to the draw length, and to the lubrication all play a role in how fast the arrow will go, and how much force the arrow will have in terms of KE. However, none of these factors play major roles in determining the arrow’s speed except for draw length. “The additional arrow weight is usually not enough to offset the increase in stored energy” (1). The stored energy being the potential energy from pulling back on the string. Also “an important quantity for the archer is the bracing height. This distance determines the amount of energy to be stored in the bow during drawing the arrow until the full draw” (2). Which means that although the type of material used for the string has little effect, the string may give the more it is fully pulled back, which could cause potential problems in determining a consistent relationship between draw length and the velocity of the arrow. The graph (5) below illustrates this idea of bracing height. The chart (3) shown also accounts for the arrow slowing down the longer it is in the air, but this portion will also not play too much of a factor in the experiment. Lastly, the example (4) below also shows that the poundage of the bow isn’t as much of a factor in fps of the arrow compared to the power stroke or length at which the string was drawn.

(5)

Work Done = Force X Displacement in direction of force

Work Done = Force (constant)

Draw force 444.82216 (N)

Draw length 40.64 (cm)

Force = 444.82216 N

Distance = 40.64 cm

F = -kX

Elastic Potential = 1/2*k*X^2

Work Done = 12 444.82216 * 0.4064 Work Done = 90.38786  J

Kinetic energy = 12 mass X velocity^2

Task 2: Calculate the velocity of the arrow (mass 22.5g), assuming efficiency of energy transfer of limbs to arrow 0.7 (a reasonable efficiency for bows (see sources))

Mass = 22.5g

Work done = 90.38786J Efficiency = 0.7

Kinetic energy = 0.7 X work done

Kinetic energy = 12 mass X velocity^2

Therefore velocity = √(2 X kinetic energy/mass) = √(2 X 63.2715 / 0.0225) = √ 5624.13351 = 74.994 m/s

The purpose of this investigation is to see if the dependant variable of velocity of a arrow will be manipulated by the varying independent variable of draw length of a 20 lb.  Hoyt Sprint recurve bow and will thus show how much of an impact the draw length of bows have on the velocity of the arrows fired, using the control of the calculated velocity via maths.

Hypothesis:

If the draw length of a 20 pound bow is increased by 5 cm intervals starting at 40 cm and ending at 65 cm, then the velocity of the arrow fired from the bow will increase by an average of 20% when compared to the previous velocities.

1.    Gathered materials needed (20 pound bow, arrows, camera, meter measuring wheel, meter markers, cardboard)

2.    Set up bulls eye

3.    Measure out 15 meters from the bulls eye

4.    At every 5 meter mark place the meter markers (sticks with white flags)

5.    Set up camera far enough away to see meter markers and the bulls eye and start recording

6.    Have archer stand at the 15 meter mark

7.    Have someone measure out 41 cm from the handle of the bow to the top of the archer’s fingertips

8.    Repeat step 7 four more times

9.    Repeat steps 7 and 8 with 45 cm

10.Repeat steps 7 and 8 with 50 cm

11.Repeat steps 7 and 8 with 59 cm

12.Repeat steps 7 and 8 with 63 cm

14.Change the videos to slow motion

15.Check the time from when it passes one meter marker to another.

16.Repeat step 15 with all trials

 Velocity (M/S) Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Average 95% Confidence 41 cm 25.9 30.0 28.8 25.0 30.0 27.9 2.93 45 cm 36.6 34.1 30.6 34.9 35.3 35.3 4.51 50 cm 37.5 36.6 33.3 39.5 34.9 36.3 2.94 59 cm 41.6 38.5 36.6 42.9 46.9 41.3 4.97 63 cm 44.1 46.9 48.4 41.7 51.7 46.6 4.81

Data file: text

Graph:

In conclusion our hypothesis of “If the draw length of a 20 pound bow is increased by 5 cm intervals starting at 40 cm and ending at 65 cm, then the velocity of the arrow fired from the bow will increase by an average of 20% when compared to the previous velocities” was proven accurate over the entire data range despite some differences between individual data points. The general increase across the data is 4.3m/s per inch of draw length. This lab has surprisingly small error bars. This would be due to the amount of errors present in the lab ie. release of arrow, angle of arrow, and the bad measurement tools. This could be fixed by using a crossbow set in a position and using better data collection tools.

Related sites

Seven ways to increase arrow velocity. Describes how different mechanics work together in the launching of the arrow

http://www.real-world-physics-problems.com/physics-of-archery.html

http://mrfizzix.com/archery/

Talks about all the mechanics used in archery and in certain bow types as well as giving some history to archery.

https://arxiv.org/ftp/arxiv/papers/1511/1511.02250.pdf

Super in depth analysis of all the finer points of archery physics.

http://www.learn-archery.com/proper-draw-length.html

Gives a reason for draw length differences and adds understanding of what draw length is.

Bibliography:

1.    Winke, Bill. "Seven Ways to Increase Arrow Speed." Seven Ways to Increase Arrow Speed. Midwest Whitetail, 13 Feb. 2009. Web. 30 Nov. 2015. <

2.     "The Measurement of Arrow Velocities in the Students' Lab." The Measurement of Arrow Velocities in the Students' Lab. Sagittarius Twente University Archery Club, n.d. Web. 30 Nov. 2015. <

3.     "Carbon Arrow University - Speed & Kinetic Energy." Carbon Arrow University - Speed & Kinetic Energy. Hunter's Friend, n.d. Web. 30 Nov. 2015. <

4.     "Draw Weight." Crossbow Men. Cabela's, n.d. Web. 30 Nov. 2015. <

5.     "Additional Discussion On Draw Lengths." Hunter's Friend. Hunter's Friend, n.d. Web. 30 Nov. 2015. <

6.     "Arrow Speed." Arrow Speed. A & H Archers, n.d. Web. 30 Nov. 2015. <