A ballistic missile trajectory consists of at least three sectors: 1. Powered flight from launch until engine cut-off, the point we shall refer to as the burnout point. 2. Free flight, which covers most of the trajectory. 3. Re-entry, which begins at some point where atmospheric drag forces become a significant factor in determining the subsequent motion of the missile, this sector lasting until impact with the surface of the earth which we shall refer to generally as the ground. During the free flight sector, the trajectory is part of a conic section—invariably a sector of an elliptic orbit which intercepts the ground—the subject of this article. Figure 1 illustrates some of the important parameters in the analysis. We are concerned here with the free flight sector of the trajectory, which can be treated without any knowledge of the details of the powered flight phase.

The parabolic orbit is rarely found in nature although the orbits of some comets have been observed to be very close to parabolic. The parabola is of interest mathematically because it represents the boundary between the open and closed orbit forms. An object moving along a parabolic path is on a oneway trip to infinity never being able to retrace the same orbit again. The velocity of such an object is the escape velocity and its total energy is zero.

Except for the circle, for which the true anomaly v is proportional to the time t, the position of a body in orbit about a central body at a given time is simplest to derive for a parabola. The classical determination of the time of flight on a parabolic trajectory is through the integration of the dynamic equations of motion. (See Appendix.)

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In a recent paper on Horner’s Method [1], which includes a compact method for dividing expressions, we mentioned that some Computer Algebra Systems (CASs) such as DERIVE could be used to make the calculations, but that such programs, even when obtained through educational establishments, are overly expensive especially when most PCs have spreadsheets on them that could equally well do the calculations. Here we describe the use of an Excel spreadsheet to divide one expression by another, first by the method of detached coefficients and second by Horner’s Method of Synthetic Division (or simply synthetic division). A third example uses Horner’s Method to replace x by (x + c) to form a new expression [2], useful in the determination of the roots of a polynomial.

Apart from false position and double false position, another numerical method for calculating roots of equations was known to the Ancient Chinese. Chia Hsien in the eleventh century is reputed to have given an algorithm for calculating roots as well as describing Pascal’s triangle. The algorithm was mentioned again by the twelfth century scholar Liu I. The Chinese used the method to solve quadratics and cubics as early as 100 BC, but it was not until 1247 that Ch’in Kiu-Shao from South China published its extension to higher order polynomials in his work, Mathematics in nine chapters. A year later in the book, Sea-mirror of circle measurements, Li Yeh, who was from north China, took root-finding for granted. The fact that these quite independent writers published similar work suggests that finding the zeros of polynomials was well known by the middle of the thirteenth century. It was left to one of China’s greatest mathematicians, Chu Shi-kie’ (ca. 1280–1303), to give this algorithm its name fan fa, which means the method of the Celestial Element or sometimes the Celestial Unknown [1,2]. Translations and spellings of these older Chinese words do not always give the same result. This algorithm eventually became known as the Ruffini-Horner Method or more simply Horner’s method.