The Physics of Paintball

Impact for Distance

This page shows how distance affects the impact force and impulse of a paintball hitting a target. The animation below (left) shows how brittle paintballs are and how explosive they are when fired at a solid object. However with the right equipment, such as padding or baggy cloths, the change in velocity can occur over a longer period of time, making the paintball less likely to break on impact. This same concept explains why paintballs bounce off of inflatable bunkers so easily, shown by the animation below (right).

In order to determine the impact force (impulse) of the paintball hiting it's target, the following equations must be used.

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

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

$v=\sqrt{v_{x}^{2}+v_{y}^{2}}$
$F_{ave}=m\frac{\Delta v}{\Delta t} = m\frac{\Delta v^{2}}{\Delta d}$

$Impulse = m \Delta v$

Where $\Delta$ v is equal to v because the collision is considered to be inelastic so the paintball comes to complete stop. Also, $\Delta$ t is equal to the time of transit of the paintball's diameter. Therefore $\Delta$ d is the diameter of the paintball. The impact force can be decreased by increasing the period of time the paintball is decelerating. This can be accomplished by increasing $\Delta$ d by wearing padding/baggy clothes or by being fat.

This should help:

Distance (m)	Force (N)	Impulse (Ns)
0	1549.59	0.2927
10	1506.23	0.2886
20	1464.67	0.2846
30	1424.81	0.2807
40	1383.26	0.2765
50	1343.44	0.2725
60	1308.50	0.2690
70	1269.07	0.2649
80	1234.19	0.2612
90	1203.61	0.2580
100	1166.41	0.2539

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)