Launching a Projectile on a Sloped Surface

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This Demonstration shows the ideal launch angle for a projectile to travel the farthest distance down a sloped surface. How close can you get to the maximum range?

Contributed by: Daniel Tokarz and Payton Kim (June 2016)
Open content licensed under CC BY-NC-SA



The time elapsed for a trajectory is:


where 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


[1] T. Nakayama. “Range of Projectile Motion.” (Jun 22, 2016)

[2] OpenStax CNX. S. K. Singh. "Projectile Motion on an Incline." (Nov 12, 2008)

Special thanks to the University of Illinois NetMath program and the mathematics department at William Fremd High School.

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