The
most powerful Nerf gun title is held by
Giaco Whatever. In his YouTube video, he
built a cannon which shoots foam Nerf
darts out of the barrel at approximately
800 m/s or Mach 2.351. He calculated the
speed by tracking how far the dart flew
in a frame with his camera. His camera
is rated about 1000 FPS (frames per
second) which means it takes a picture
every 0.001 seconds and the dart
traveled 80cm in one frame. After some
simple unit conversions, he came out
with the final speed of the dart that is
powerful enough to break through
aluminum soda cans and glass. The
maximum pressure capabilities of his
stainless-steel cannon far exceed
anything that can be built with PVC.
However, we can still model the
pressures necessary to accelerate a foam
dart to that speed. Now using a similar
approach to Giaco Whatever to measure
velocity, I gathered the data in the
graph below with multiple trials and
averages of all the values.
One slow motion video used as a trial in
measurements for velocity
Giaco Whatever's cannon shooting a piece
of wood with a Nerf dart
In the
graph, it shows that the graph best fits a
natural log curve which is to be expected, since
the natural logarithm can function be used for
data sets similar to these. The natural
logarithm can essentially be defined for any
positive real number, a, as the area under a
curve y= 1/x from 1 to a when the area is
negative if a is less than 1. The pressure vs
velocity plot can be modeled with the natural
log formula in the figure. In order to find the
pressure necessary to make a Nerf dart travel
Mach 2.0 with my PVC rifle, we can plug Mach 2.0
(or 686 m/s) into the x of the equation and
solve for y, which gives 1169 kPa. This pressure
value is about double the maximum pressure of
the garden hose solenoid used on the rifle, and
best left for the much more expensive metal high
pressure solenoid valves.
Theoretical
Maximums
This is
an image of the air tank which extends
from the solenoid to the bottom of the
stock.
We
can see that a lot of energy is wasted
from the air tank energy to the final
dart energy, but there are many factors
which cause that. First, the dart is not
the same size as the barrel so a lot of
air escapes around the dart instead of
transferring energy to the dart. The
dart also has a very low mass for the
accelerations it is getting, if it had a
larger mass, it would accelerate with
more energy. The measurements of the
final velocity were done after the dart
left the barrel so the velocity and
acceleration at the very end of the
barrel would be higher since the dart
had stopped being accelerated by the
compressed air. Although the maximum
power is only 11.840 J, shooting foam
darts at people leaves red marks on skin
through cloths at lower initial
pressures. After much experimentation, I
have found that a pressure of 275 kPa
(40 PSI) is a reasonable pressure which
does not cause too major of damage to
the dart, but still stings through
cloths.
