What is the relationship between launch angles that yield the same range?

When a projectile is launched in a vacuum, its trajectory can be analyzed using the principles of kinematics. The range of the projectile is influenced by the launch angle, initial velocity, and acceleration due to gravity. For a given initial speed, it turns out that two different angles can produce the same horizontal range.

The relationship is that the two angles that yield the same range indeed add up to 90 degrees. This is a fundamental principle derived from the equations of motion for projectiles. Essentially, if one angle is θ, the other angle that will give the same range is 90 degrees - θ.

This occurs because the horizontal velocity component, which determines how far the projectile travels, and the vertical velocity component, which determines how high the projectile goes, can be balanced in such a way that the time in the air (and thus the range) remains constant for these two complementary angles.

For example, if one angle of launch is 30 degrees, the angle that would yield the same range is 60 degrees (90 - 30 = 60). This duality allows for interesting applications in fields such as sports, engineering, and physics, where optimizing the angle of launch for maximum distance is often crucial. Thus, the correct relationship between