The Magic Angles of Projectile Motion

A projectile thrown from the ground in a vacuum with an initial speed at an angle above the horizontal has a trajectory of length and encompasses an area [1].

The length can be rewritten using , an inverse Gudermannian function. In addition, a projectile has range

These equations share a peculiar property: the kinematics of the problem, the length , is a factor separate from the parts that depend on the angle. This separation comes from the parametric representation of the projectile in Cartesian coordinates,

Setting the kinematic factor to 1 allows us to study the global features of the purely geometric aspects of projectile motion.

The given length, area, and range for are

Here is the trajectory's perimeter.

Figure 1. The perimeter (solid line) and the area (dashed line) versus the initial angle .

Contrary to our intuition, the longest perimeter does not encompass the largest area; is the angle that maximizes the area.

Similarly, to determine the angle (in degrees) that maximizes the perimeter, we set the slope of the perimeter to zero and solve the transcendental equation.

Figure 2. The perimeter versus the area for projectiles of initial angles .

The point where the curve crosses itself determines two angles at which the corresponding pairs of areas and perimeters are the same. To evaluate these magic angles, we solve the simultaneous equations.

To plot the unique pair of corresponding kinematic-independent parabolic trajectories, we eliminate t between as given in equation (1).

Figure 3. The magic pair of kinematic-independent trajectories. These two parabolas have the same perimeters and the same areas.