The Physics of Paintball

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I made this website to help aspiring paintball players understand and exploit the physics of paintball. Even experienced players can significantly improve their paintball performance by learning how paintballs act after being fired from a paintball marker. I suggest paying extra attention to the Time section.

Notation:
$v_{t}$ : the terminal velocity of the projectile
$m$ : the mass of the projectile
$g$ : the gravitational acceleration on the projectile
$\rho$ : the density of the fluid through which the object is moving
$A$ : the projected area of the object
$C_{d}$ : the drag coefficient of the projectile
$a_{x}$ : the projectile acceleration in the x direction
$a_{y}$ : the projectile acceleration in the y direction
$t$ : the time difference from when the projectile is launched
$v_{x}$ : the projectile velocity in the x direction
$v_{y}$ : the projectile velocity in the y direction
$v_{o}$ : the initial velocity at which the projectile is launched
$\theta$ : the angle at which the projectile is launched
$v$ : the magnitude of the sum of the x and y velocity vectors
$x$ : the distance from the origin in the x direction
$y$ : the distance from the origin in the y direction
$F_{ave}$ : the average force on the projectile
$\Delta v$ : the change in velocity of the projectile
$\Delta t$ : the difference in time
$\Delta d$ the difference in position of the projectile
$Impulse$ : the change in momentum of the projectile

Contsants:
$m=3.201*10^{-3}kg$     (average mass of a paintball)
$g=9.81\frac{m}{s^{2}}$     (gravity on earth)
$\rho = 1.164\frac{kg}{m^{3}}$     (air density at 1 atm and 30 degrees celcius)
$A = 2.343*10^{-4}m^{2}$    (cross sectional area of a paintball)
$C_{d} = 0.47$     (drag coefficient for a smooth sphere)
$\empty = 0.017272m$ (diameter of a paintball)