Projectile motion without calculus

Abstract

Projectile motion is a constant theme in introductory-physics courses. It is often used to illustrate the application of differential and integral calculus. While most of the problems used for this purpose, such as maximizing the range, are kept at a fairly elementary level, some, such as determining the safe domain, involve not so elementary techniques, which can hardly be assumed of the targeted audience. In the literature, several attempts have been undertaken to avoid calculus altogether and keep the exposition entirely within the realm of algebra and/or geometry. In this paper, we propose yet another non-calculus approach which uses the projectile's travel times to shed new light on these problems and provide instructors with an alternate method to address them with their students.

On the elliptic locus of a family of projectiles

“…It follows that, in the laboratory frame, the circle C 1 is squashed into the ellipse (3). To algebraically confirm this result, one replaces z by 2y in (7) and multiply through by 4 to obtain (3). It is worth noting that this compression can be traced back to the well-known fact (see e.g.…”

“…the same axis followed by a unit translation in the positive vertical direction. It is readily seen that the image of C 2 under this transformation is again ellipse (3). Algebraically, this means that ellipse (3) is obtained from (9) too by substituting for z from y = −z/2 + 1 (and multiplying through by 4).…”

“…The launch angle α corresponding to this projectile as well as the angle β at which the point of tangency hangs above the horizontal in the laboratory frame are the special angles we have in mind. From figure 7, we see that cos α = sin(π/2 − α) = 1/3, so Physically, projectiles launched at angle α or shallower will never approach the launch point, whereas those launched at a steeper angle approach the launch point on their way down while inside the ellipse (3). Note that the time spent inside the ellipse is given by |ρ 2 − ρ 1 | = 9 sin 2 θ − 8, which vanishes for θ = α.…”

“…The second property has to do with two special angles associated with the 'singular' trajectory that touches ellipse (3). In general, a projectile, on its way down, either cuts the ellipse(3) twice or not at all; one only projectile has a trajectory touching that ellipse at one (doubly critical) point.…”

“…A well-known property of this family is the so called synchronous circular locus [1], which consists in simultaneously launched projectiles of the family forming a circle at any one instant of time. Another genuinely geometric, independent of time, feature of this family is the parabolic envelope that divides the vertical plane into an accessible zone and a safe domain [2,3].…”