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Letter to the Editor – Projectiles Approaching or Receding: A Unique Paradox

Darren Pais
Saint Louis University

Letters to the Editor are accepted from any reader, and may address any topic dealing with science or undergraduate issues. They are published at the editor’s discretion. To submit a Letter to the Editor, please write toeic@jyi.org.

To the Editor –

This letter examines the use of basic equations of motion for a projectile, assuming constant vertical acceleration. A unique feature of this projectile motion is studied here, which is the distance between the point of projection of the projectile and the position of the projectile at various times during its flight. The variation of this distance is studied using the equations of motion and some basic calculus and very interesting and unexpected results are obtained.

The various results and equations are graphed in order to obtain a better visual understanding of what is actually going on. The question that I will pose now will be answered in this paper – Is a traveling projectile projected at some known angle of projection and with a given velocity approaching us or receding away from us? In a later section, I will show the important properties of the 45° angle of projection and its relation to other results.


I will consider basic projectile motion, with a projectile launched from the surface at a known projection angle, i.e. the angle between the initial velocity vector of the projectile and the horizontal is known. The x direction is defined to be the horizontal direction and the y direction is defined to be the vertical direction. All velocities are broken down into x and y components, and motion in these directions is independently analyzed in order to obtain the parametric equations of motion for the parabolic projectile path. Consider a projectile launched at an angle of θ with the x-direction, with an initial velocity of projection of magnitude v. Breaking this velocity into rectangular components we have initial velocities of v(cos θ) in the x-direction and v(sin θ) in the y-direction.



Analyzing motion in the x-direction, we see that we have a constant velocity in this direction with no acceleration. Thus, distance traveled in the x-direction, X, is given by X = v(cos θ) t (equation 1). The parameter t in equation 1 represents the time elapsed since the time of projection (t = 0) at the instant of observation. Analyzing motion in the y-direction, we see that a constant acceleration exists in this direction, which is the acceleration due to gravity. From the equations of motion, the distance traveled in this direction, Y, is given by Y = v(sin θ) t – gt²/2 (equation 2).

Notice that both equations 1 and 2 are defined using the time parameter t. We now define a new variable t_f, which represents the time of flight for the projectile. In order to obtain this we observe that the time variable t equates to the time of flight variable t_f, when Y = 0. Hence, from equation 2 we obtain equation 3.

From equations 1 and 2 we can eliminate parameter t and obtain an equation for the path of the projectile in terms of variables x and y, giving equation 4.



We graph equation 4 on x- and y- axes to obtain the parabolic path of the projectile motion as shown in Figure 1. Notice also that we intend to study the variation of distances r₁, r₂, r₃… as seen in Figure 1, as the core aim of the paper.

Restrictions

As one will notice, we have placed certain restrictions on the projectile motion for purposes of clarity and simplicity. Firstly, the initial and final points of the projectile motion have the same y coordinate, that is y = 0. This implies we expect to launch our projectile from the ground level and expect it to return to that same ground level altitude. In addition, we restrict ourselves to constant vertical acceleration and drag free motion. We also require that the velocity vector of projection is known, that is, we know the angle of projection and the magnitude of velocity. These are reasonable assumptions made to simplify the math involved and do not take away from the general big picture conclusions of this paper.


In this section, we do the mathematical work necessary to answer the question set out in the introduction: is a traveling projectile approaching us or receding away from us?

In order to do this we must first define a quantity D that is the point-to-point distance between the point of projection of the projectile at t = 0 and the point of observation at some time t (distances r₁, r₂, r₃… in Figure 1). From the x and y distances in equations 1 and 2, using Pythagoras’s theorem we have equation 5.



Now that we have a formula that provides us with the distance of the projectile from the point of projection, we are interested in studying how this distance varies, particularly by finding points of maxima or minima for this distance equation. Since maximizing the function D is similar to maximizing the function D², we define a function F = D². We look at the case where we have a known constant angle of projection θ and a known initial velocity magnitude v, and find critical points for F by setting the derivative of function F with respect to time to be zero. This leads to equation 6.



Now, for this equality to hold true, either t = 0, in which case no projection takes place, or the conditions in equations 7 must be met.

Solving the quadratic equation 7 for t_max/min gives equation 8, from which it is clear that for real values of t_max/min to exist, (9) must be the case.

But since the values of projection angle θ are restricted to being between 0° and 90°, the sine expression in equation 9 can take on only positive values (equation 10). Therefore, we notice that for critical points to exist on a distance from projection vs. time of flight graph, the angle of projection must be above this particular angle of 70.5287° obtained in equation 10. We will call this angle the critical angle. It is also important to notice that the critical angle obtained is independent of projection velocity or vertical acceleration magnitudes.

We have equation 3 for the total time of flight for the projectile t_f and equation 8 for time of critical point t_max/min. The ratio t_max/min/ t_f is a fraction expressing the x-direction component of maximum/minimum distance point with respect to the total horizontal (x-direction) flight distance or range. This is because, as seen in equation 1, the x-direction distance X is directly proportional to time for a given angle of projection. We will call this fraction the position ratio, and it is displayed in equation 11.


It is important to graphically realize some of the results obtained in the previous section. The first graph we look at comes directly from equation 11. Figure 2(a) shows the positive version of the position ratio plotted against the angle parameter.

From Figure 2(a), we can clearly see that a position for the maximum/minimum distance exists only for an angle above the angle derived in equation 10. To interpret this graph, for an angle of projection of 75°, the max/min point exists when the projectile has traveled about 91% of its horizontal distance, i.e. the point (75, 0.9132) on the graph.



In Figure 2(b), the complete version of equation 9 is plotted with the position ratio on the y-axis and projection angle on the x-axis.

The horizontal lines on this graph are asymptotes representing y = 0.5 and y = 1, respectively. It is interesting to note that at angles above the critical angle, two critical points exist for the distance formula. We can interpret this as two locations where distance of the projectile (from the point of projection) maximizes and minimizes along the path of the projectile, quite an intriguing result. We also see that at a projection of 90°, the point at which we achieve maximum distance occurs at 50% of our flight time. That result conforms to our intuition of vertical projection, where maximum distance exists at zero velocity and half of total flight distance and time.

To understand the information in Figure 2(b) better, we take the following example. If we observe the x = 75° vertical line, we will see that it intersects the curve at two points, i.e., y = 0.58 and y = 0.91. The interpretation is that with an angle of projection of 75° the distance from the point of projection to the projectile maximizes or minimizes at 58% and 91% of total flight time.

Particular Cases

We will now observe distance from origin (quantity D in equation 5) vs. time of flight graphs for projectiles projected at 3 particular angles. The first angle we observe is projection below critical angle. Let us consider θ = 60°, for example, in Figure 3(a)

As we can see, no critical points exist on in the plot in Figure 3(a), and the distance from projection increases with time. A similar graph will be observed for all θ less than 70.5287°. In Figure 3(b), we observe what happens when we chose the projection angle to be equal to the critical angle, i.e., θ = 70.5286°. As clear from Figure 3(b), a single critical point is observed, indicating that at a projection angle equal to the critical angle, inflection takes place on the graph at a single critical point. This corresponds well to our observations in Figure 2(b), where a single position ratio exists on the plot for critical angle of projection. Lastly, in Figure 3(c) we observe the case where the angle of projection is greater than the critical angle, say,θ = 78°.



Figure 3(c) is an interesting plot to observe because, as expected, we see both a maximum and a minimum point on this graph. Hence for projections greater than the critical angle, the projectile retreats from the point of projections, approaches it, and retreats again.

It is also important to note that in Figures 3(a-c), the velocity and acceleration act only as scaling factors for the distance on the y-axis and have no direct effect on the properties of the graph itself such as critical points. This adds further validity to the assumptions initially made.


It is especially important to consider the 45° angle of projection case. It is known that for a projectile to have a maximum horizontal range from the point of projection, the angle of projection must be 45°. This result can be easily obtained by maximizing equation 1 after setting t = t_f. This, in turn, is intuitively the expected critical angle for any variation in behavior for projectile motion, like the distance variation observed in this paper since it conforms to intuition, being halfway between 0° and 90°. We will now compare the variation of the distance from the point of projection with time for various angles.



As we see in Figures 4 and 5, the 45° angle of projection represents no critical change in graph behavior, but the 80° angle (greater than 70.5287°) angle shows significant deviation in behavior displaying the intriguing critical points.
Thus, paradoxically, the critical angle derived is a special case angle for projectile motion that is seemingly random and very much different from the expected 45° whole number angle. This angle is inherent to projectile motion with regards to distance from point of projection, just as much as the 45° angle is for range optimization.



90° angle of projection

The 90° angle of projection is another important special case to look at. We expect that if we construct a graph between distance from the point of projection and time of flight for a projection angle of 90°, the maximum distance should be observed at one-half of flight time, and the final distance must be zero as the projectile returns back to point of launch. This is observed in Figure 6, where we can see a single critical point on the plot.


In this paper, we observe the properties of the distance between the point of projection for a projectile and the position of the projectile at various times during its flight, for the duration of its flight. The following are the key results.



(1) A critical angle of 70.5287° was derived for projectile motion. Only for projection above this critical angle, the projectile reaches a maximum/minimum distance from the point of projection. Otherwise, the projectile continuously retreats from the point of projection. This angle is independent of projection velocity and acceleration due to gravity and is inherent to any projectile motion.

(2) For projections above the critical angle, two critical points are observed. Therefore, the projectile retreats away from us, approaches us for a fixed period, and then retreats again. This phenomenon is quite different from what we would intuitively expect.

(3) The critical angle derived is an example of an interesting paradox when compared to a 45° angle of projection, which is expected to be an angle at which critical changes happen during projectile motion.


Hibbeler, Russell C. (2003) Engineering Mechanics-Dynamics. 37-39, Prentice Hall.

Hughes-Hallett, et al. (2002) Single and Multivariable Calculus. 196-200, Wiley.