The Physics of Paintball

Angle for Distance

This page shows the angle at which to hold a paintball marker in order to mark a target at varying distances. As you can see from the animation below, the paintballs arc, like all projectiles affected by gravity and drag, so it is important to understand how to properly fire a paintball marker so that the paintballs reach their intended target.

In order to determine the angle at which to hold the paintball marker, the following equations must be solved for theta.

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

Where x will be replaced with the distance to the projectile's target and y will be set to zero (assuming the target is at the same height as the paintball marker). It is likely that there will be several solutions, this is because there are different angles at which the projectile can be launched to reach the target distance. The projectile can be fired at a lower angle to take a shorter arc path, reach the target faster, and have more impact, or it can be fired at a higher angle to take a longer arc path, reach the targer slower, and have less impact.

This should help:

Distance (m)	Angle ( $\degree$ )
0	0.00
10	0.35
20	0.70
30	1.05
40	1.40
50	1.80
60	2.15
70	2.55
80	2.95
90	3.30
100	3.75

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)