Projectile Motion Calculator
A launched projectile's range, maximum height and total flight time all come from the same launch speed and angle — the classic 'cannonball' problem, and the reason 45° maximises range for a given speed.
Inputs
Result
Range 40.775 m
Max height 10.194 m · Flight time 2.883 s
How the projectile motion calculator works
Range = v² × sin(2θ) ÷ g.
Maximum height = (v × sin θ)² ÷ (2g).
Flight time = 2v × sin θ ÷ g.
All three assume launch and landing at the same height, with no air resistance.
Worked example: 20 m/s launch speed at 45°
- Range = 20² × sin(90°) ÷ 9.81 = 400 × 1 ÷ 9.81 ≈ 40.77 m.
- Max height = (20 × sin 45°)² ÷ (2×9.81) = (14.14)² ÷ 19.62 ≈ 10.19 m.
- Flight time = 2×20×sin(45°) ÷ 9.81 ≈ 2.88 s.
Common mistakes to avoid
Assuming range increases the higher the launch angle goes
Range peaks at exactly 45° for a given speed and then decreases again — a 70° launch, for instance, gives a lower range than 45° despite being a 'higher' angle, because it trades distance for height.
Applying this formula when launch and landing heights differ
This formula assumes the projectile lands at the same height it launched from. A projectile launched from a cliff or thrown from shoulder height landing on the ground needs a more complete equation accounting for that height difference.
Frequently asked questions
Why does 45° give the maximum range?
Range depends on sin(2θ), which peaks at 2θ = 90°, i.e. θ = 45° — any angle above or below that reduces sin(2θ) and therefore range, for the same launch speed.
Do two different angles ever give the same range?
Yes — angles that add up to 90° (like 30° and 60°) produce the same range for the same launch speed, since sin(2×30°) = sin(60°) = sin(120°) = sin(2×60°).
Does this account for air resistance?
No — this is the idealised, vacuum-style projectile motion model. Real projectiles (especially fast or light ones) experience drag that reduces range and alters the trajectory shape.
How would this change on the Moon or another planet?
Simply change the gravity value — lower gravity increases both range and flight time for the same launch speed and angle, which is exactly why golf balls travel further on the Moon.
Learn more
Why 45 Degrees Isn't Always the 'Best' Launch Angle
Physics says 45° maximises range — but only under an assumption that almost never holds in the real world.
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