Projectile Motion Calculator
Range, height and flight time, and why 45° is optimal.
Work out Projectile Motion. Range, height and flight time, and why 45° is optimal. Shows the working, not just the answer.
Standard gravity by definition. Local values run 9.78 to 9.83.
Drag is ignored, which matters more here than for most of these. A real thrown ball falls well short of this range, and a bullet or arrow is not remotely described by it — the optimal angle in air is always below 45°, often far below.
Range (m)
40.789
Peak 10.2 m · airborne 2.88 s
45° is the optimal angle on level ground, and this is the maximum range for 20 m/s. Range is v²sin(2θ)/g, and sin(2θ) peaks when 2θ is 90°.
How the Projectile Motion Calculator works
Range is v²sin(2θ)/g, which peaks at 45° because sin(2θ) peaks when 2θ is 90°. The same symmetry means complementary angles give identical range: 30° and 60° land in the same place, one by a flat fast path and the other by a high slow one.
Also known as: projectile range calculator · trajectory calculator · launch angle calculator · projectile motion equations
Why 45 degrees
Range is v²sin(2θ)/g. Since sin peaks at 90°, the range peaks when 2θ is 90° — that is, at θ = 45°. This holds for any launch speed on level ground.
The same expression explains why complementary angles pair up: sin(60°) equals sin(120°), so 30° and 60° give identical range. One arrives fast and flat, the other slow and high.
Where the real world departs
Drag matters more for projectiles than for almost anything else on this site. A real thrown ball falls well short of these figures, and the optimal angle drops to somewhere around 30–35°.
The reason is that drag scales with the square of speed, so it punishes the fast flat trajectory less than the model suggests while stealing more from the long high one. For a bullet or an arrow the idealised equations are not usable at all.
Treat these as the physics-classroom answer, which is exactly what they are.
Frequently asked questions
What angle gives the maximum range?
45 degrees on level ground, always, regardless of launch speed. In air, drag pulls the optimum well below that — often to 30–35° for a thrown ball.
Why do 30° and 60° give the same range?
Because range depends on sin(2θ), and sin(60°) equals sin(120°). Complementary angles always pair up this way, trading height against flight time.
How high does it go?
v²sin²θ/(2g). At 20 m/s and 45° that is 10.2 metres.
Does this account for air resistance?
No, and for projectiles it matters more than almost anywhere else. A real thrown ball falls well short of these figures, and a bullet or arrow is not described by them at all.
Put this calculator on your own site
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