Weapon Systems- Firearms

Modelling weapon damage accurately is often too difficult without a large budget for the equipment necessary to run predictive algorithms or real time CFD.

Damage can be roughly equated to the energy of a projectile that will be transferred to the target. For firearms, this comes in the form of the kinetic energy of the solid bullets. Following this train of through, the toughness of the target’s armor, or it’s ability to absorb energy without failing, is the counterpart to the weapon’s damage.

Another factor to consider is the pressure the bullet will apply to the target, and the target’s compressive strength and thickness.

Damage, D, will be used to refer to the relevant energy of the projectiles.
Pressure, P, will be used to refer to the projectile’s ability to penetrate the target’s armour (for firearms, stress).

HP will refer to the toughness of the target’s armour,

Armor, A, will refer to the compressive strength of the target.


As a basis, the drag equation can be used to determine the velocity, and therefore the energy of a solid bullet in flight.

Take the net force on an object in one direction:

Eq. 1 ∑F_x=m·a_x

where F_x is the net force in the x direction, m is the mass of the object, and a_x is the acceleration of the object

Eq. 2 a_x=v_x d(v_x)/d(x)

where v_x is the velocity of the object in the x direction and x is the distance travelled. Here, d(v_x)/d(x) is the derivative of v_x with respect to x.

substituting equation 2 into equation 1,

∑F_x=m·( v_x d(v_x)/d(x) )

Eq. 3 ∑F_x=m·v_x ·d(v_x)/d(x)


If we ignore gravity, which is in the y direction but will cause the bullet to drop and change the angle and trajectory, the coriolis effect for long distances, the Magnus effect for a spinning body, gyroscopic drift, and the Poisson effect, then the only force acting on the bullet in the x direction is aerodynamic drag.

Given these assumptions, the basic formula for aerodynamic drag is

Eq. 4 F_D=.5*density*v_x^2*C_D*A

where density is the density of air, C_D is the drag coefficient of the object, and A is the frontal surface area of the bullet.

For this analysis, only form drag will be considered.

Combining equations 3 and 4

∑F_x=-F_D

m·v_x ·d(v_x)/d(x)=-.5*density*v_x^2*C_D*A

dividing both sides by v_x

m·d(v_x)/d(x)=-.5*density*v_x*C_D*A

separating variables

d(v_x)/v_x=-.5*density*C_D*A*d(x) /m


 

This leads to an equation that can be integrated on both sides:

Left Hand Side: ∫d(v_x)/v_x from v_o to v_1

=ln(v_x) from v_o to v_1

=ln(v_1)-ln(v_0)

Right Hand Side: ∫d(x)*-.5*density*C_D*A /m from x_o to x_1

=-x*.5*density*C_D*A /m from x_o to x_1

=-(x_1*.5*density*C_D*A /m – x_0*.5*density*C_D*A /m)

since x_0=0,

=-x_1*.5*density*C_D*A /m

Combining the left and right side we get the equation of velocity as a function of distance

ln(v_1)-ln(v_0)=-x_1*.5*density*C_D*A /m

We can remove the 1 subscript for simplicity

ln(v)-ln(v_0)=-x*.5*density*C_D*A /m

moving the ln(v_o) term to the right since it is a known value,

ln(v)=-x*.5*density*C_D*A /m+ln(v_0)

we can now taker the exponent of both sides

e^ln(v)=e^(-x*.5*density*C_D*A /m+ln(v_0))

v=e^(-x*.5*density*C_D*A /m)*e^(ln(v_0))

This yields the final form of the equation of the velocity of a solid bullet as a function of distance travelled assuming only form drag force.

Eq. 5 v=e^(-x*.5*density*C_D*A /m) * v_0


 

Alternatively the ratio of velocities can be used in order to determine the change in damage as a function of distance

Eq. 6 v/v_0 =e^(-x*.5*density*C_D*A /m)

Assuming that the damage consists entirely of kinetic energy, then using the equation for kinetic energy,

Eq.7 KE=.5*m*v_x^2

It is possible to simply substitute the known velocity at the time of impact into equation 7 to find the damage done. alternatively, if the velocity is not known the initial damage can be used to find the final damage by using the ratio of KE_1 to KE_2

KE_2/KE_1=(.5*m*v_x2^2)/(.5*m*v_x1^2)

This simplifies to

KE_2/KE_1=(v_x2^2)/(v_x1^2)

Eq. 8 KE_2/KE_1=(v_x2/v_x1)^2

Substituting equation 6 into equation 8

KE_2/KE_1=(e^(-x*.5*density*C_D*A /m))^2

KE_2/KE_1=e^(-2*x*.5*density*C_D*A /m)

Eq. 9 KE_2/KE_1=e^(-x*density*C_D*A /m)

This can also be simplified using

Eq. 10 A=density*C_D*A /m

Substituting equation 10 into equation 9

KE_2/KE_1=e^(-A*x)

KE_2/KE_1=e^(A)*e^(-x)

KE_2/KE_1=e^A*e^-x

Here we can define a less meaningful variable A2 as

Eq. 11 A2=e^A

Substituting equation 11

Eq. 12 KE_2/KE_1=A2*e^(-x)

This gives the ratio of the damage at a distance to the initial damage of a solid bullet firearm.

 

Written by slizer88

Works Cited:

1. http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node17.html

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