6.1 **Differential Equations of Bullet Motion **

In the three-degree-of-freedom analytical model of bullet ballistics, the bullet is considered as a point mass characterized by its mass **m **and ballistic coefficient **C **. The bullet moves along a trajectory as shown in **Figure 6-1 **under the influence of two forces, the aerodynamic drag force **D**and the gravitational force **mg **(where **g **is the acceleration due to gravity, 32.176 ft/sec 2 in the Northern Hemisphere).

**Figure 6-1 **also shows the coordinate system used to describe the bullet motion. The origin of the coordinates is at the muzzle of the gun. The **x **axis is level at the firing point and points downrange. The **y **axis is vertical at the firing point, so that gravity acts in the negative **y **direction. The **z **axis completes a right-handed cartesian coordinate frame, and the **x-z **plan is horizontal.

The bullet is fired along the extended bore centerline, which is in the **x-y **plane. In the absence of a crosswind, the bullet trajectory then remains in the **x-y **plane. In the absence of a crosswind, the bullet trajectory then remains in the **x-y **plane. The bore line is elevated at an angle X with respect to the **x **axis. As the bullet flies downrange its trajectory curves downward. At any point on the trajectory (such as the point shown in **Figure 6-1 **) the bullet velocity vector **V **is in the **x-y **plane and tangent to the trajectory. The aerodynamic drag force **D **acts directly opposite to **V **, so it is also in the **x-y **plane. The velocity vector **V **is elevated at an angle X relative to a line parallel to the **x **axis, as shown in the figure. The angle X is positive when the trajectory is rising, as depicted in the figure. After the trajectory peaks and begins to descend, X becomes negative.

The mass of the bullet is its weight divided by the acceleration due to gravity:

**(6.1-1)**

where **w **is the bullet weight in grains and the factor 7000 converts grains to pounds (7000 grains = 1.0 lb). Equation (6.1-1) expresses the bullet mass in slugs, and **mg **is bullet weight in pounds.

The differential equations of bullet motion result from applying Newton’s second law to the bullet and resolving the motions and forces along the coordinate axes:

**(6.1-2) (6.1-3) (6.1-4)**

In these equations the position coordinates **x, y, **and **z **and the angle X are functions of time **t **. The drag force **D **also varies with time, but the mass **m **and the gravitational acceleration **g **are constants.

The velocity components are defined by

**(6.1-5)**

The angle X is the slope angle of the trajectory and is defined by:

**(6.1-6)**

The bullet leaves the muzzle at t = 0, and the initial conditions of the motion are then:

**x(0) = y(0) = z(0) = 0 **

** v ****x ****(0) = v ****m ****cos X **

** v ****y ****(0) = v ****m ****sin X (6.1-7) **

** v ****z ****(0) = 0 **

where **v ****m **is the muzzle velocity of the bullet and X is the bore elevation angle defined earlier.

With the initial conditions **z(0) = 0 **and **v ****z ****(0) = 0 **, equation (6.1-4) has the solution:

**z(t) = 0; v**

**z**

**(t) = 0**

for all **t **. This corresponds, of course, to bullet motion only in the **x-y **plane. Therefore, the following discussions will be concerned only with the **x **and **y **components of the bullet motion.