The Physics of Paintball

General Projectile Motion With Air Resistance

This page gives a general overview of projectile motion.

These are general projectile motion equations to be manipulated in other sections. These equations take air resistance into account.

$V_{t}=\sqrt{\frac{2 m g}{\rho A C_{d}}}$

$a_{x} = -g\frac{v_{x}}{v_{t}}$
$a_{y} = -g\left ( 1 + \frac{v_{y}}{v_{t}} \right )$

$v_{x} = v_{o}\cos\left ( \theta\right )e^{\frac{-g t}{v_{t}}}$
$v_{y} = v_{o}\sin\left ( \theta\right )e^{\frac{-g t}{v_{t}}}-v_{t}\left (1 - e^{\frac{-g t}{v_{t}}} \right )$
$v=\sqrt{v_{x}^{2}+v_{y}^{2}}$

$x = \frac{v_{o}v_{t}\cos\left (\theta \right )}{g}\left (1-e^{\frac{-g t}{v_{t}}} \right )$
$y = \frac{v_{t}}{g}\left (v_{0}\sin\left (\theta \right ) + v_{t} \right )\left (1-e^{\frac{-g t}{v_{t}}} \right ) - v_{t}t$

$F_{ave}=m\frac{\Delta v}{\Delta t}=m\frac{\Delta v^{2}}{\Delta x}$

$Impulse = m \Delta v$

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)
$V_{o} = 91.44\frac{m}{s}$     (NPPL regulation for marker velocity)
$\empty = 0.017272m$ (diameter of a paintball)