Parabolic Sandwiches for Functions on a Compact Interval and an Application to Projectile Motion

Abstract

About a century ago, the French artillery commandant Charbonnier envisioned an intriguing result on the trajectory of a projectile that is moving under the forces of gravity and air resistance. In 2000, Groetsch discovered a significant gap in Charbonnier’s work and provided a valid argument for a certain special case. The goal of the present article is to establish a rigorous new approach to the full result. For this, we develop a theory of those functions which can be sandwiched, in a natural way, by a pair of quadratic polynomials. It turns out that the convexity or concavity of the derivative plays a decisive role in this context.

The English Galileo and His Vision of Projectile Motion under Air Resistance

“…and thus holds by the strict concavity of f ′ , as shown in the proof of assertion (i). Note that the condition f(u) � f(v) is not essential here and that the preceding estimate for arbitrary distinct points u, v in [a, b] actually characterizes the strict concavity of f ′ on [a, b], see Corollary 1 of [7]. (v) To establish the first inequality, we recall from the proof of part (ii) that…”

“…First, it turns out that the function x is strictly increasing on [0, ∞) and that its range is an interval of the form [0, x ∞ ) where 0 < x ∞ ≤ ∞. Furthermore, by eorem 12 of [7], we have Theorem 2. Suppose that W is a strictly positive admissible drag function, and let (x(t), y(t)) for t ≥ 0 denote the solution of the corresponding initial value problem (30).…”

“…One of these forces is gravity in the direction of the negative y-axis which results in acceleration of a given magnitude g > 0. e other force is air resistance whose direction is opposite to the velocity vector r ′ (t) � 〈x ′ (t), y ′ (t)〉 of the projectile and whose magnitude is proportional to the mass m of the projectile. Specifically, as in [7], the retarding force is supposed to be represented by…”

“…As in [7], we assume that the drag function W is admissible in the sense that, for every choice of s > 0 and −π/2 < θ < π/2, the initial value problem (30) has a unique solution (x(t), y(t)) that is defined for all t ≥ 0. In practice, admissibility is guaranteed by existence and uniqueness results of the Picard-Lindelöf type under mild Lipschitz conditions on the function W with respect to the last four variables, see, for instance, [9].…”

“…In fact, even in the special case of drag quadratic in speed, no such formulas are known, see [8,10], and [11]. Nevertheless, as shown in Section 4 of [7], it is possible to obtain geometric insight into the shape of the trajectory traced by the projectile. First, it turns out that the function x is strictly increasing on [0, ∞) and that its range is an interval of the form [0, x ∞ ) where 0 < x ∞ ≤ ∞.…”