Mechanics texts tell us that a particle in a bound orbit under a gravitational central force moves in an ellipse, while introductory physics texts approximate the earth as flat, and tell us that the particle moves in a parabola. The latter is meant to be an approximation to a small segment of the true central-force/elliptical orbit. To look more deeply into this connection, we convert earth-centered polar coordinates to “flat-earth coordinates” by treating radial lines as vertical and by treating lines of constant radial distance as horizontal. We consider questions such as whether gravity is purely vertical in this picture, and whether the central force nature of gravity is important only when the height or range of a ballistic trajectory is comparable to the earth’s radius. Somewhat surprisingly, the answers to both questions are “no,” and therein lie some interesting lessons.

Topics
Central-force, Coordinate system, Educational aids
1.
I. Newton, The Principia, translated by A. Motte (Prometheus Books, New York, 1995), Scholium following Proposition X of Book I, Sec. II. Newton does not give much detail about how the limit should be taken. While taking the center of the ellipse to infinity and letting the eccentricity approach unity (from below), the latus rectum is to be kept fixed. Notice that the center of the earth, at the distant focus, also is taken to infinity. For more detail on a closely related way to take the limit see S. Chandrasekhar, Newton’s Principia for the Common Reader (Oxford U. P., Oxford, 1995).
2.
K. R. Symon, Mechanics (Addison-Wesley, Reading, MA, 1971), pp. 111–112. This intermediate level textbook also discusses the complicated issue of how the changing resistance of air with altitude affects the trajectory, but only discusses the height as a condition for the trajectory to be parabolic in the absence of air resistance.
3.
I. M. Freeman, Physics: Principles and Insights (McGraw–Hill, New York, 1973), p. 107;
R. Wolfson and J. M. Pasachoff, Physics, 3rd ed. (Addison–Wesley, Reading, MA, 1999), Vol. 1, p. 72;
R. A. Serway, Principles of Physics (Saunders, Orlando, FL, 1994), p. 63;
D. E. Roller and R. Blum, Physics (Holden-Day, San Francisco, CA, 1981), Vol. 1, pp. 76 and 346.
4.
D. C. Giancoli, Physics for Scientists and Engineers with Modern Physics, 3rd ed. (Prentice–Hall, Upper Saddle River, NJ, 2000), p. 56;
S. M. Lea and J. R. Burke, Physics: The Nature of Things (Brooks/Cole, St. Paul, MN, 1997), p. 92. The last reference also explicitly addresses the parabolic versus elliptic trajectories.
5.
E. Hecht, Physics: Calculus (Brooks/Cole, Pacific Grove, CA, 1996), p. 94.
6.
Our choice is not the only reasonable way of carrying out this mapping. We could alternatively have chosen x=rφ, for example. By comparison, our choice suffers from having unequal physical distances, for equal increments of x, along x coordinate lines. Our choice, however, leads to simpler dynamical equations, the equations for d2x/dt2 and d2y/dt2 given in Eq. (14). The choice made has no distinguishable effect on the appearance of the results given in the figures.
7.
See, for example, J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems, 4th ed. (Saunders, Fort Worth, TX, 1995);
H. Goldstein, C. P. Poole, and J. L. Safko, Classical Mechanics, 3rd ed. (Pearson, Upper Saddle River, NJ, 2001).
8.
This sort of high velocity motion is not typical of practical ballistic trajectories, in which projectiles are launched at an angle near 45°. For such trajectories α is of order unity, and the parabolic approximation is accurate. For a given range and ε, a larger value of α means a larger value of Vhoriz, or equivalently of the required muzzle velocity. Although the conclusion from Eq. (9) is meant to be a point of principle, not of practical importance, it is interesting that large guns exist that are able to achieve muzzle velocities of several km/s necessary to have peak height h and range of the same order as the values h∼1 km and range ∼100 km in our example.
9.
The radius of curvature at a point in a plane curve is defined as the radius of the “osculating circle,” or the circle of curvature, the circle whose first and second derivatives agree with those of the curve at that point. This curvature is equivalent to the magnitude of the derivative, with respect to arc length, of the unit tangent to the curve. See, for example, M. P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice–Hall, Upper Saddle River, NJ, 1976).
10.
The radius of curvature of the ellipse at its apogee is R=hα(1+ε)2/[1−αε(1+ε)]. For αε≪1 (which is equivalent to R/R≪1) and ε≪1, this result coincides with Eq. (10) at the apogee.
11.
The condition that the parabolic trajectory is a good approximation for the true orbit, that is, R/R≪1, is equivalent to l/R≪1, with ℓ the semilatus rectum.
12.
These equations can be derived by substituting r=y+R and φ=x/(R+h) in the standard equations for central force motion. A simpler equiva-lent set of equations follows, of course, by writing the two constants of motion, energy and angular momentum, in terms of x,y coordinates. The form in Eq. (14b), with second derivatives with respect to time, is given to emphasize the velocity dependence of the acceleration.
13.
One could claim that we have not shown that “gravity” fails to be vertical. The velocity-dependent terms in Eq. (14b), after all, do not contain the gravitational constant G. Those terms are actually kinematic terms due to the nature of the coordinates. To avoid this criticism, the fallacious argument at the end of Sec. I was made for “a particle being acted upon only by gravitational forces.”
14.
Notice that the velocity-dependent term on the right-hand side of Eq. (14b) is made negligible with respect to the term not depending on the velocity if (dx/dt)/Vhoriz≪(αε)−1/2, but one cannot neglect the velocity-dependent right-hand side of Eq. (14a). Notice, however, that typically indeed (dx/dt)/Vhoriz≪(αε)−1/2 and the right-hand side of Eq. (14a) is rather small, so that Galileo’s picture of separate horizontal and vertical motions is a fair description of the motion. However, for the values of the parameters we used in our examples of failure of the parabolic approximation, αε is about unity, and it is easy to see that the equations of motion (14) are indeed coupled, and “gravity is not vertical.”
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© 2005 American Association of Physics Teachers.
2005
American Association of Physics Teachers
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