I am a math teacher in a small high school in Alaska and we began exploring a concept this year for an idea I had a while back.  The general thought is – if you make a laser site and have it mounted above the arrow can you align it to approximate the trajectory of the arrow?  If you can, how far would it approximate that trajectory?  We took that idea a step further and said, if we add a second laser to start approximating the trajectory where the first one leaves off we can really extend the range of the site.  In fact multiple lasers could be used to approximate the trajectory as far out as desired.  Multiple lasers would project multiple dots on the target but the lowest dot would always be the one to use.

We did in fact develop the theory behind this and built a working prototype.  It works.  The students won best of show at the local school wide district science fair.

It only took two lasers to approximate the trajectory on a Bowtech Allegiance out to 50 yards with a maximum 2″ of error.  The following video is of us testing the site shortly after we set it up.  The first clip show 5 shots at random distances out to 50 yards and the second clip shows popping balloons at 10, 20, 30, 40, and 50 yards on a bright sunny day.

If you are interested in the theory behind the development we established the following procedure:

This procedure includes a fair bit of math – that was of particular interest to me as a teacher using this as a learning project – but this can be automated with the computer and the initial setup of this site is actually quite simple, fast and effective.

Procedure

Since the flight of an arrow follows a decaying parabola, its trajectory can be approximated with a quadratic equation in the form of y=ax2+bx+c where y=drop and x=distance from the target.

1. Record arrow drop from three distances covering the effective range of the bow by shooting groups of arrows from each distance and recording average arrow drop from aim point.
2. Create three different equations using the known x and y values, with x=distance from target and y=arrow drop from aim point.

Y1=ax12+bx1+c

Y2= ax22+bx2+c

Y3= ax32+bx3+c

1. Solve for the unknowns: a, b and c.  Do this by using a graphing calculator and setting the numbers up into a matrix and transforming the matrix to reduced row echelon form.
2. Once coefficients a, b and c are solved for they can be plugged into the quadratic equation ax2+bx+c.  This will create the quadratic equation that predicts the arrow trajectory.
3. Use Excel and the quadratic equation to graph the predicted arrow trajectory.
4. Once graphed, use lines of best fit over different ranges to follow the trajectory of the arrow with an acceptable margin of error. Ultimately, these will be the lasers.
5. By using multiple lasers, or lines of best fit, we should be able to approximate arrow trajectory out to the effective range of the bow. Multiple lasers will project multiple dots on the target, but the bottom laser dot will always be the approximating arrow trajectory.
6. Construct a laser mounting apparatus that can be mounted onto a bow and which allows lasers be adjusted in elevation as well as fine tuned left, right, up or down. This laser mount must be rigid enough to maintain its position on the bow while sustaining the shock of repeated shots.
7. Take the first line of best fit and find the equation of the line in slope-intercept form. B, or the y-intercept, will be the distance between the laser and the arrow.  Mount the laser at this distance above the arrow.
8. The line of best fit will cross paths with the arrow trajectory at two places on the parabola.  Solve for the x values, or distances, where this occurs by setting the equation for the line of best fit and the quadratic arrow trajectory equation equal to each other and solve for x.
9. Site the first laser in at the previously solved for x values by shooting a group of arrows at the two distances and adjusting the laser accordingly. After this step your bow should be striking your aiming point at the two distances.
10. The next laser can be aligned without shooting the bow at all.  The two lasers will cross at a specific distance. This distance can be solved for by setting the equations of the lines of best fit equal to each other and solving for x.  Simply adjust the top laser so it is on top of the previous laser. Ultimately, at these two distances you will see only one dot.
11. Repeat the previous step to align any additional lasers.

Now you can test-shoot the bow from essentially any distance that your bow is effective to and see if the lasers allow you to shoot within the predicted margin of error at these distances.

The following is the actual implementation of the procedure on the test bow (Bowtech Allegiance) with the real numbers and generated formulas.

Step 1: Record Arrow Drop.

 Distance from Target Arrow Drop (Inches) Group 1 15 Feet or 5 Yards 0.4375 Inches Group 2 60 Feet or 10 Yards -5.3125 Inches Group 3 150 Feet or 15 Yards -46.8125 Inches

Step 2: Create Equations.

0.4375=a(32400)+b(180)+c

-5.3125=a(518400)+b(720)+c

-46.8125=a(32400000+b(1800)+c

Step 3: Using spreadsheet program utilizing rref solve for a, b and c.

a= -0.0000171467764060

b= 0.00478395061728

c= 0.1319444444444440

Step 4: The quadratic equation predicting arrow trajectory is:

-0.0000171467764060x2+0.00478395061728x+0.1319444444444440

Step 5: Use Excel to make a graph of projected arrow trajectory using the previously found quadratic formula.

Step 6: By graphing trajectory over shorter distance ranges and using line of best fit on Excel, we were able to come up with a combination of two lines of best fit that approximates the projected arrow trajectory from zero out to 50 yards with an error of + or – 2 inches.

First Line of Best Fit:

Second Line of Best Fit:

Step 7: We were able to use 2 lasers and have a margin of error of 2 inches and were able to approximate an arrow strike point out to 50 yards. The top laser mount location is 25 inches above the arrow.  With a top laser mount of 33 inches we, we were able to approximate arrow strike point out to 60 yards.

Step 8: We chose 1 inch extruded aluminum display rail since it was readily available, rigid, lightweight and laser fixtures could be mounted anywhere along its length. This was mounted to the bow utilizing the bow’s standard site mounting holes.

We modified a generic green laser pointer to use as our laser sites. To allow for windage and elevation adjustment of lasers we mounted one end of the laser on a horizontal threaded bolt and the other end of the laser on a vertical threaded bolt in an aluminum square tube.

For our power source we made a battery pack using standard plumbing supplies and screwing it into the stabilizer-mounting hole on the bow. The bow was used as the ground and we routed one positive wire through a momentary push-button switch on the bow handle up to each laser.

The lasers were mounted onto the bolts by soldering a nut onto a ½ inch copper pex crimp fitting and crimping it onto the laser.

Step 9: The equation for the first laser line is y=-0.0137x+4.028.  The laser should be mounted at four inches above the arrow.  Mount second laser at 25 inches above the arrow, the equation for this laser is y=-0.0384x+25.132.

y-intercept=4.028=distance laser is mounted above the arrow.

y-intercept=25.132=distance laser is mounted above the arrow.

Step 10: Find where the first laser crosses at both places on the parabola.  See below.

Step 11: We adjusted the laser fairly close at 8.7 yards and then adjusted it to be right on at the next distance: 21.2 yards.  A quick check showed that the laser was right on at 8.7 yards as well.

Step 12: We solved for the distance that the laser crossed.  See Below.

We then aligned the lasers as to make one solid dot at 23.7 yards.

Step 13: There were no additional lasers.

Step 14: We tested the site by shooting arrows at random distances out to 50 yards, and all the arrows were within the predicted margin of error (+ or – 2 inches in elevation). See video. Further testing was done to demonstrate both the accuracy of the site out to 50 yards and the visibility of the green laser on a bright sunny day by shooting balloons at 10, 20, 30, 40 and 50 yards. See video.

I would certainly like our students to receive feedback on your thoughts about this concept.  We did file a provisional patent on the idea.  I can be contacted at [email protected]