I've seen a lot of people, mostly on the internet, using aerodynamics equations to attempt to explain the physics of Nerf darts. For example: This article mentions that some shorter darts might actually have better drag properties, and mentions a coefficient of 0.67. This project page assumes a coefficient of 0.82.
But why should I believe what I read on the internet, when I have the equipment to test it myself?
I decided to set up as follows:
- I put a bunch of paper taped to the wall as a scale. I know paper is 8.5"*11", which I can convert to metric later.
- I attached my Nerf Mega Magnus Blaster to the wall, also using tape.
- I set up my LG G3 smartphone to record video at 120 frames per second and 720p.
- I taped my smartphone to the opposite wall and made sure it was recording.
- I then fired the Blaster (several times) to collect data.
- I took the video off my smartphone, chopped out the unimportant bits, and converted the rest to individual frames.
- I then used GIMP to create this image:
- More importantly, I used the paper as a ruler. 11 inches is apparently 96 pixels on all of the papers. I thought that the physical size would be smaller near the edges, but apparently, cameras assume that the viewer is at a particular finite distance from the screen such that they would cause no distortion, as opposed to assuming that the screen is curved with each pixel having equal angular size. This simplified my task somewhat.
- As it turns out, capturing the dart in flight means it becomes fairly blurry (who knew!) and difficult to discern. What I decided to do was to look at the front and back points of the dart's "nucleus" on the screen and average their positions to find the center. I selected everything that was near-identical between frames, and then inverted my selection and I had the nuclei of each dart selected.
- After plugging the data into a curve fit for a logarithmic growth (such as that of the position of aerodynamic projectile), the following results were discovered:
- The muzzle velocity was estimated as 21.09 m/s. This translates to a velocity of 69 ft/s, which is almost exactly what one would expect, and matches proper chronograph data perfectly.
- The acceleration due to drag at time zero was estimated as -14.42 m/s^2
- Combining these facts with an air density of 1.225 kg/m^3, a projectile diameter of 0.75 inches, and a projectile mass of 2.5 grams, all from reputable sources, we obtain a drag coefficient of 0.464.
- Plugging this data into an external ballistics simulator (GGDT), I get results that, while not EXACTLY at Nerf's advertised 85 feet, are a respectable 77 feet in an arc (And several feet more if the gun starts above head level). This probably means that Nerf's advertised maximum range is accurate.
- By comparison, a drag coefficient value of 0.82, as would be found in a long flat-tipped cylinder, yields 58 feet of range. Nerf apparently knows what they are doing by streamlining their darts, as they seem to have cut drag by 43% from what a flat-tipped cylinder would have. In fact, my result says that Nerf Mega Darts fly equally well to a smooth sphere. (typically considered to have a drag coefficient of 0.47).
- Having done this experiment, I wonder if I can beat Nerf at their own game? I wonder if I can design some sort of modified dart that still fires in a spring-powered gun, but has a better drag coefficient, and remains safe? Regardless, what this means is that if you want to have decent range, use Streamline or Mega darts, not suction-cup darts. Those probably have a terrible drag coefficient. Even most homemade darts might lack the same aerodynamic performance as Nerf Megas.
Also, video of gratuitous fire tests in slow motion. A few are duds due to hitting the paper ruler.
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