Basic Physical Concepts
our purposes, a vector is any physical quantity that has both a
magnitude and a direction. Examples are a force, linear acceleration,
linear velocity, linear momentum, a torque, angular acceleration,
angular velocity, angular momentum, etc. We will discuss changes
in vector quantities, and it is important to realize that a vector
can change in magnitude, in direction, or both.
Translational motions Translational
motions are linear motions of a body (i.e., a bullet). These motions
can occur in three directions, for example, in the downrange, vertical,
and crossrange directions. The translational motions of a bullet
are governed by Newtons Laws, specifically Newtons Second
Law, which states that the rate of
change of the linear momentum of a body with respect to time is
equal to the force applied to the body.
When the mass of the body is constant,
not changing with time, this relationship becomes the familiar mass
times acceleration is equal to the force applied to the body. These
are relationships among the vector quantities of force and momentum
(or acceleration) of the bullet. When these vectors are resolved
into components along the three translational directions of motion
of the bullet, the results are called the equations of linear motion
of the bullet.
Rotational motions Rotational
motions of a rigid body (a bullet) are caused by torques applied
to the body. A force that does not act at or through the center
of mass of a body produces a torque. The rotational motions of a
bullet are governed by an angular vector relationship, which is
the rate of change of the angular
momentum of a body with respect to time is equal to the torque applied
to the body. Normally, the center
of mass of the body is considered the origin, and the principal
axes of the body are used for calculating components of forces,
torques, moments of inertia, and angular momentum. This choice of
origin and axes greatly simplify the angular vector relationship,
and the resulting equations are called Eulers equations of
angular motion of the body. This choice is used to analyze the angular
motions of bullets.
The implications of these laws of motion
are clear. If a bullet changes its linear state of motion in flight
(speeds up, slows down, moves up or down or sideways), the changes
must be caused by forces applied to the bullet such as gravity,
drag, lift, negative lift, or sideforces. If a bullet changes its
angular orientation in flight (pitches downward to keep the velocity
vector tangent to the trajectory, or turns to follow the wind),
the angular changes must be caused by torques applied to the bullet.
Torques are caused by forces. Except for gravity, all the forces
that can act on a bullet are aerodynamic. Aerodynamic forces that
do not act through the center of mass produce the torques that cause
the angular orientation of the bullet to change.
Fortunately, the drag force on a bullet,
which is by far the largest force, acts through the center of mass
when the bullet is well stabilized, and thus creates no torque on
the bullet. If this were not true, the bullet would lose stabilization
and tumble erratically in flight.
Figure 4.1-1 illustrates basic characteristics
of a bullet in flight. When there is no wind, the trajectory path
lies in a vertical plane that contains the bullet velocity vector
and the gravity vector. The bullet has a center of mass and a center
of pressure, both of which are located on the longitudinal axis
of the bullet. The center of mass is a point at which the gravitational
force acts on the bullet, and the
center of pressure can be thought of as a point at which the aerodynamic
forces act on a bullet. As a bullet flies, the velocity vector is
tangent to the trajectory path at all points, and the longitudinal
axis is almost exactly tangent to the trajectory path at all points.
A very tiny yaw angle (nose left or right of the trajectory plane)
or pitch angle (nose up or down but in the trajectory plane) will
exist to cause the bullet to turn.
Figure 4.1-1 will be referred to in the
following subsections to explain the rotational motions of a bullet
in flight. This figure will be redrawn from other points of view
to illustrate the forces, torques, and angular momentum of the bullet.