The time elapsed for a trajectory is:
is the launch angle and
is the angle of depression.
The range of the projectile is:
Differentiate with respect to
and set the derivative to zero to find the ideal launch angle to maximize range:
This formula holds true as long as air resistance is negligible. Neither gravity nor initial velocity affects the ideal launch angle.
Units are in MKS and angles are in radians. The trajectory is always a segment of a parabola.
Snapshot 1: this graphic shows the project in "game mode" with the ideal information hidden, so the user must find the ideal launch angle
Snapshot 2: on Mars, the acceleration due to gravity is low, so the projectile goes further; the ideal launch angle is not affected by gravity
Snapshot 3 : with an angle of depression of zero, the ideal launch angle is
, confirming a commonly known fact
Snapshot 4 : this user has gotten close to the ideal launch angle and has been rewarded with an encouraging message
Special thanks to the University of Illinois NetMath program and the mathematics department at William Fremd High School.