### Arrow Speed

One of the most important parameters in defining archery performance is arrow speed.   The first velocity measurements of projectiles were made in the early 1700s using a ballistic pendulum.  The principle is simple, a projectile is fired at a pendulum and the velocity is calculated from swing of the pendulum arm.  Today, ballistic pendulums are only used in high school physical labs.  Modern instruments measure arrow speed by measuring time.

This post compares three modern methods for measuring arrow velocity; optical chronographs, doppler chronographs, and a new laptop-based audio application that measures arrow speed by recording the time between the sounds of arrow launch and target contact.  This post includes a link to the free Technical Archery Arrow Speed application.  I will begin with a bit of theory on each method and then provide more details on the new audio speed application.

1) Optical Chronographs are the gold-standard for speed measurement.  The optical chronograph measures speed using two light sensors separated by a precisely known distance.  The white plastic arcs at the top of the chronograph produce a diffuse light field that is sensed by upward looking detectors at the front and back of the chronograph.   The shadow of the arrow or bullet breaks the rear light field, and then a fraction of a second later breaks the forward light field.  Velocity is computed by dividing sensor distance by sensor time.  Optical chronographs used to cost many hundreds of dollars, but are now available for as little as \$120.   The advantage of an optical chronograph is a the ability to measure projectile speeds from 30-7000 feet per second.  A disadvantage is that shooting errors can destroy the chronograph, particularly when measuring downrange velocities.

2) A relatively new speed measurement device is the doppler chronograph.  Speed is measured by the doppler shift of microwave light produced by the chronograph.  Most of us have experienced the doppler effect when listening to an emergency vehicle pass our stationary location.   Approaching vehicles cause the frequency of a siren to increase as the sound waves compress due to the speed of the vehicle.
The effect is reversed as the vehicle passes causing a decrease in frequency.  (See diagram from Wikipedia shown to the right.)  The speed of the object can be computed from the magnitude of the frequency shift.  The doppler chronograph measures the frequency shift of reflected microwave light from an arrow passing the chronograph.  The faster the arrow flies the bigger the doppler shift.   Doppler
chronographs cost around \$100 and can measure speeds between 105-450 feet per second.   These new devices are light and compact.  An advantage of doppler chronographs is the ability mount the sensor on the front of the bow for easy velocity measurements.  A disadvantage is that it is very hard to measure downrange velocities.  The currently available doppler chronographs can not measure bullet velocities.

3) The final speed measurement tool is computer-based digital stop watch.  Most laptop computers have remarkably good microphones capable of recording the sound of arrow launch and target contact.   An arrow traveling 300 feet per second takes about a third of a second to travel 30 yards.   It is a very easy for a computer to make a 300 millisecond (0.3 second) time measurement.  If fact, the biggest error in making these measurements is caused by errors in distance measurement from the archer to the target.   The Technical Archery Arrow Speed application runs on a Macintosh or Windows based computers.   Details on running the application is presented in this post.    The advantage of the Technical Archery Arrow Speed application is the price (\$ free \$) and the ability to measure BOTH arrow speed and aerodynamic drag.  Of course, the disadvantage is capital cost of a reasonable laptop computer.

#### Technical Archery Arrow Speed Application

The image to the right shows the basic principle of the application.  Sound is recorded as a function of time to better than two tenths of a millisecond.  The green vertical line marks the sound of the bow string release.   The red vertical line marks the sound of the target contact.  The difference in time is 0.38 seconds, which is the arrow flight time for a 100 foot shot.   Diving 100 feet by 0.38 seconds equals an average arrow speed of 257 ft/second.  Like the timing for any sport, precisly starting and stopping the timing device is critical.  The program makes these measurements to better than one thousenth of a second  by continuously recording sound and automatically triggering the arrow start and stop times based on the twang of the bow at arrow release and the pop of the arrow hitting the target.

A difference in this method compared to the chronographs is that the measured speed is the average speed of the arrow over the entire flight distance.   As the arrow leaves the bow it is traveling at its maximum velocity.  As the arrow flies aerodynamic drag slows the arrow until it hits the target.   Since the app times the entire arrow flight it captures the average arrow velocity.  The longer the arrow flies the more it will decelerate and the slower the measured average arrow speed.   We can take advantage of average arrow speed measurements to determine both the arrow launch speed and aerodynamic drag.

A plot of average arrow velocity versus shot distance yields an (almost) linear plot that can be extrapolated to zero shot distance.  (At zero distance the effect of drag is zero.) The Y intercept of this plot is the launch speed of the arrow.   Equally interesting, the slope of the plot yields the aerodynamic drag on the arrow.   Using these two measurements it is possible to accurately calculate the trajectory of an arrow at any distance.    The TA Arrow Speed App makes it easy to measure the average speed of your arrow at several shooting distances and post this data to an analysis window where the linear fit to the data is performed.

Past posts have detailed Excel-based tools to display arrow trajectories.   The TA Arrow Speed App will also compute and display trajectory calculations as a function of shooting speed, aerodynamic drag, and shot angle.  These results are shown as an arrow flight profile (height versus shot distance) and as a target profile.  Details on how to use these tools are provided with the TA Arrow Speed App download and the on-screen instructions.

A final feature of the program that will be of interest to the technically oriented archer is the ability to record and analyze the sound of the bow and arrow.   Since the program measures time by recoding the sound of the arrow release it can also capture sound data.   The sound of an archery shot is a complex summation of the many vibrating components of the bow and arrow.  We add lots of fancy accessories to our bows to make them quieter by dampening these vibrations.   The sound analysis tools allow the archer to quantify the effectiveness of bow noise dampeners.

Using the TA Arrow Speed App
you can can make an audio recording of your bow.   You can zoom in and out on this data to see the finer details of the sound of your shot (or any other sound source that you want to study).  The figure to the left is the sound of a typical bow release.   It has a characteristic shape which starts with the click of my arrow release, a build up of sound as the string and limbs start to move, and then a long (50 milli second) tail due to the ringing of the bow limbs and string after release.   The frequency of the oscillations recorded on the trace is directly related to the frequency of the mechanical ringing of the bow.   In this example, the ringing at the end of the trace has a primary frequency of 350 Hz (oscillations per second)  The height (amplitude) of the oscillations are directly related to volume.   The faster the parts of your bow vibrate the higher the frequency of sound produced.  The more the parts of your bow vibrate the louder the sound produced.  By comparing the sound of a shot with and without a string dampener or limb silencer it is possible to determine if these products are making a difference on shot noise.

Now the scientists and engineers reading this post are probably thinking that it would be really cool to perform a Fourier Transform on the data to calculate the specific frequencies of bow and string vibration.   I agree.   The TA Arrow Speed App lets the user select any part of the sound trace and compute the Fourier Transform of the data (also called the Power Sepctrum) to determine the frequencies and amplitude of each bow vibration.   How does this work?  Lets consider a music analogy.  The musical score to the right defines the frequency (notes) and volume (intensity) that a musician should play to create a specific musical sound.  Pressing the keys on a piano causes a hammer to strike a string causing the string to vibrate and produce sound.  Simple sounds are produced by pressing one key, while complex musical chords are produced by pressing two or more keys at the same time. The Calculate Power Spectrum function in the TA Arrow Speed App takes the sound data for your bow and transforms it into the
fundamental frequencies (notes) of bow movement.   The figure to the left shows the sound trace of playing a single C major cord on a piano (C4 - 261.6 Hz, E4 - 329.6 Hz, G4 - 392.0 H).  Notice the regular oscillations of the signal with time that slowly decrease as the volume of the cord fades.   The pleasent sound of the cord is due to the ringing of the piano stings at three fundamental frequencies (261, 330, and 392 Hz) and also doublets of these frequencies.   The sound we hear or measure is the sum of each of these individual vibrations.  This trace produced on a piano looks a lot like the sound trace from a bow.

Using the Calculate Power Spectrum function, the WA Arrow Speed App performs the
Fourier Transform of the data with the results shown in the figure to the right.  Notice the six major peaks from left to right at 261, 329, 392, 522, 658, and 784 Hz.   These peaks correspond to the primary frequencies and doublets of the C major cord.   It is interesting that that the intensity (height) of the 329.6 vibration is less than the others.  Perhaps I pressed this key a bit more softly when I played the cord or dust on the string is attenuating the vibration.   We would expect that vibration dampeners that we place on our bows will be more or less effective at attenuating sound depending on the frequency of the sound produced.   The Power Spectra tool allows the archer to test these ideas.

The Technical Archery Speed Application may download from our website.   The current version only works on a PC, although the Mac version will be released soon.   The program is written using National Instruments LabView so the installer will install both the NI runtime module and the TA Speed Application.

Windows Application:  TA Arrow Speed Application
Macintosh Application: Coming soon.

Both the Window and Mac programs require a computer with a microphone and may require an OS specific LabView runtime module available free from National Instruments (The TA Arrow Speed App installer does not support the runtime files for older OS).

Comparison of Speed Measurements:

I have used all three systems to measure the speed of my arrows.   The TA Arrow Speed App and the optical chronograph produce results that agree within +/- 1 foot per second at arrow speeds of 250 ft/second.   Critical to accurate measurements using the TA Arrow Speed App is accurate archer to target distance.   A 2% error in distance measurement will produce a 2% error in speed.  Two percent of a 30 yard shot is 1.8 feet - so measure distance carefully.   The optical chronograph is insensitive to shooting distance, but don't hit the chronograph with your arrow.   The doppler chronograph is fast and easy to use.   The model that I tested produced arrow velocities that were slow by about 4 feet/second (1.6%) compared to the other methods.   In general, the differences in measured speed for the three methods are small enough that it really doesn't matter which method is used.   The TA Arrow Speed App is the only method that can easily produce aerodynamic drag coefficients.

- Winter 2014.