How it works
A projectile launched without air resistance moves at constant horizontal speed while gravity pulls it down at 9.80665 m/s². The path is a parabola, and the horizontal and vertical motions can be solved separately.
From level ground the range is greatest at 45°, and complementary angles such as 30° and 60° land at the same spot. Launching from a height shifts the best angle a little below 45°.
Because air drag is ignored, this is a textbook model. It works well for slow, dense objects over short distances, but not for bullets, golf balls or anything where drag is significant.
Formula
Range (level ground) = v² sin(2θ) ÷ g Max height = h₀ + (v sin θ)² ÷ 2g Time = (v sin θ + √((v sin θ)² + 2gh₀)) ÷ g
Questions people ask
What angle gives the maximum range?
On level ground with no air resistance, 45°. Launching from above the landing point makes the optimum slightly lower; air drag also lowers it, often to 30–40° for sports balls.
Why do 30° and 60° give the same range?
Range depends on sin(2θ), and sin(60°) = sin(120°). The 60° shot goes higher and stays up longer but covers ground more slowly.
Does mass affect the trajectory?
Not without air resistance. All objects fall at the same rate in a vacuum. With air, heavier and denser objects are slowed less.