External Ballistics of Rifle Bullets

bullet properties

scope setting

muzzle velocity (feet/second)

3000.

scope height (inches)

1.

% muzzle velocity at 100 yards

93.

initial angle (degrees)

0.17

This Demonstration shows the trajectory of a rifle bullet determined as a function of bullet characteristics and settings of the telescopic sight of the rifle.

The equations describing the bullet trajectory are:

dx/dt=

V

x

,

dy/dt=

V

y

,

d

V

x

/dt=-

C

d

V

x

2

V

x

+

2

V

y

,

d

V

y

dt=-

C

d

V

y

2

V

x

+

2

V

y

-g

,

with initial conditions

dx(0)/dt=

V

x

(0)=

V

0

cosθ

,

dy(0)/dt=

V

y

(0)=

V

0

sinθ

,

x(0)=0

,

y(0)=-h

.

Here

t

is time,

x

and

y

are the horizontal and vertical dimensions,

V

x

and

V

y

represent the horizontal and vertical velocities,

θ

is the initial rifle bore angle,

C

d

stands for the overall drag coefficient due to air resistance,

g

is the force of gravity, and

h

is the height of the telescopic sight above the rifle bore.

These equations are solved using the built-in Mathematica function NDSolveValue. The result shows the bullet trajectory and bullet impact in a target at 100 yards for different settings of the telescopic sight. Published bullet velocities at the muzzle and at 100 yards are used to determine the overall drag coefficient

C

d

. The solutions of this simple model agree closely with published ballistic tables obtained with advanced professional ballistic models [1].