Trajectories, Part 1

By Donna Cline
Copyright © 2003

Gravitational Forces | Vacuum Trajectory | Envelope
Flat-Fire Approximation | Uphill/downhill | Example

will not go into the proofs and deriving each formula from one another. If that is what you are looking for there are a number of books that will give you that type of information. "Mathematics for Exterior Ballistics" by Gilbert Ames Bliss, 1944, "Exterior Ballistics" by McShane, Kelley, and Reno, University of Denver Press, 1953, "Sierra Reloading Manual 4th Edition" by Sierra Bullets, L.P., 1995, "Modern Exterior Ballistics" by Robert L. McCoy, Schiffer Publishing Ltd., 1999, and "A Ballistics Handbook" by Geoffery Kolbe, Pisces Press, 2000, just to name a few.

It was Sir Isaac Newton (1642-1727) that, while standing on the shoulders of giants, built his great theory of motion.

Newton's First Law of Motion:

Every body continues in its state of rest or of uniform speed in a straight line unless it is compelled to change that state by forces acting on it. The tendency of a body to maintain its state of rest or uniform motion in a straight line is called inertia. As a result, Newton's first law is often called the law of inertia.

Newton's Second Law of Motion:

The Acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the applied net force. A net force exerted on an object may make its speed increase or if it is in a direction opposite to the motion, it will reduce the speed. If the net force acts sideways on a moving object, the direction as well as the magnitude of the velocity changes. Thus, a net force gives rise to acceleration: a = F ÷ m or F = m * a. Acceleration is the velocity that is changing and is equal to velocity divided by time, a = (V2 - V1) ÷ (t2 - t1). This same equation can give you time, t = V ÷ a and velocity, V = a * t. Therefore: F = m * a = m * V ÷ t or F * t = M * V.

Newton's Third Law of Motion:

Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first object. Or to put it another way; for every action there is an equal and opposite reaction. This is what makes jets and rockets fly and it is also what makes a gun recoil and you feel it as a kick.

These three laws are what govern all motion in the universe and we have known about them ever since Sir Isaac Newton first published his great work, the Principia, in 1687.

The curved path of a projectile is called a trajectory. The equations for a vacuum trajectory where gravity is the only force acting on the projectile is quite simple. The trajectory of a projectile in a vacuum would inscribe nearly a parabolic path.

Figure 1-1

The general shape of the trajectory is called a parabolic because on a flat world and in a vacuum, an unresisting medium, absence of an atmosphere, the path of a projectile is actually a parabola, as was first proved by Galileo in 1638. Galileo noticed that, owing to the curvature of the earth, the force of gravity did not cat in parallel lines, but acted instead in lines that converged to the center of the earth. As a consequence, the usual path of a projectile fired on the earth in the absence of air would be a portion of an ellipse. If the shot were fired horizontally from an eminence at any ordinary small arms velocity the elliptical path would intersect the earth’s surface and the projectile would fall to earth striking the surface. But if the velocity were to be 26,000 fps, the projectile would never return to earth, but would be itself a satellite with a circular orbit passing through its original point of firing once every seventeen revolutions. A circle is an ellipse that has both of its foci (plural of focus) at the same point in space and whose equation of the graph is r2 = x2 + y2. If the velocity were greater than (>) 26,000 fps and less than (<) 36,000 fps, the orbit would be elliptical. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points (foci), each called a focus, is a constant and whose equation of the graph is 1 = (x2/a2) + (y2/b2). If the velocity was exactly at 36,000 fps it would tolly-trop into space in a parabola. The geometric definition of a parabola is the set of points in the plane equidistant from a fixed point called the focus and a fixed line called the directrix and whose equation of the graph is y = ax2 + bx + c. And if the velocity were greater than (>) 36,000 fps the trajectory would be a hyperbola. The geometric definition of a Hyperbola is the set of all points in the plane, the difference of whose distances from two fixed points, called the foci, is a constant and whose equation of the graph is 1 = (x2/a2) - (y2/b2). The hyperbola also has lines called asymptotes associated with it. Asymptotes are lines that the hyperbola approaches close to but never touches for ever larger values of x and y. The only time the trajectory would be a straight line is if it was fired straight upward or downward.

For all practical purposes we will never shoot a bullet in a vacuum trajectory. So, why am I even going into showing a vacuum trajectory? Well, atmospheric trajectories display many of the same properties of a vacuum trajectory, but lack the convenience of a simple analytical solution. A vacuum trajectory is the simplest trajectory, only dealing with the force of gravity, which will show the basic fundamentals of trajectories without a bunch of clutter and the mathematical methods presented here will form the framework on which the higher approximations are based on.

Simply put, in a vacuum, the projectile leaves the bore, at the origin, with the barrel at a positive angle from a straight line to the target is called the inclination (slope) angle of departure. As the projectile climbs in height or gains altitude to the summit of its flight this is the ascending branch of its trajectory. The height between the origin and the summit of the projectile's flight path is it's maximum ordinate and the summit lays half way between the origin and the target. From the summit till the point of impact, the projectile's path is on the descending branch of the trajectory. The angle from the projectile's path on the descending branch of the trajectory to the straight line to the origin is the inclination angle of fall. The projectile's inclination angle of departure is the same angle as the inclination angle of fall. And the vertical velocity up ward and horizontal velocity at the point of origin will be the same as the vertical velocity down ward and horizontal velocity at the point of impact.