Projectile Motion
Range, max height and flight time at a launch angle
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Projectile Motion is built for range, max height and flight time at a launch angle — fast, free, and private. You supply Launch Velocity (m/s), Launch Angle (°) and Gravity (m/s²) to Projectile Motion; the answer follows at once. Projectile Motion puts the headline number first. Anyone comparing options in Science can use Projectile Motion to compare scenarios. Nothing is uploaded while Projectile Motion works, and the figures stay put. Looking for projectile motion calculator range height time launch angle or free online projectile motion calculator? Projectile Motion covers both. Keep Projectile Motion bookmarked for next time. Try Projectile Motion now and keep it handy.
What the defaults give
20 m/s at 45° under standard gravity flies 40.77 m, reaching 10.19 m at the top after 2.88 s in the air.
Range and flight time are symmetric around 45°: 30° and 60° land in the same place, but 60° hangs longer.
The three formulas
Range R = v² sin(2θ) ÷ g
Max height h = v² sin²θ ÷ (2g)
Flight time t = 2v sinθ ÷ g
All three come from splitting the launch velocity into an upward part and a forward part, then treating each independently. Gravity only pulls on the upward part, so it decides how long the object stays up; the forward part just keeps going at a steady speed for that long. Height uses sin²θ because the upward component is squared on the way up and again on the way down.
Worked examples
20 m/s at 45° → 40.77 m range, 10.19 m peak, 2.88 s
20 m/s at 22.5° → 28.83 m, 2.99 m, 1.56 s
20 m/s at 90° → 0 m horizontal, 20.39 m peak, 4.08 s
10 m/s at 45° → 10.19 m, 2.55 m, 1.44 s
40 m/s at 45° → 163.10 m, 40.77 m, 5.77 s
Straight up is the useful edge case. The range collapses to zero and the whole velocity goes into height, which is exactly why the peak at 90° is twice the peak at 45° for the same speed.
Why 45 degrees wins
Range is proportional to sin(2θ), which peaks at 90° — that is, at θ = 45°. Below 45° the object lands sooner than it could; above 45° it spends its speed climbing rather than travelling. Real sports do not sit at 45° because the launch and landing heights differ: a shot put leaves the hand at head height and lands lower, so the optimum shifts below 45°, and a golf drive on a raised tee shifts it slightly up.
What the model ignores
Air resistance is the big one. It cuts range substantially for a light, slow object (a badminton shuttle loses most of its range) and barely touches a dense, fast one over short distances. Drag also breaks the symmetry between 30° and 60°: the flatter shot stays at speed for less time, so it usually beats the steeper one in real air.
Mass never appears in these equations, which is why a cannonball and a pebble leave the same barrel at the same speed on the same path in a vacuum. Spin, wind and the curvature of the Earth are separate corrections that matter only at long range.
Frequently Asked Questions
Why is 45° the best launch angle?
Range is proportional to sin(2θ), and sin(2θ) is largest when 2θ is 90°, so θ is 45°. At any other angle part of the speed is spent on a shorter flight.
Does the launch angle change the flight time?
Yes. Time is 2v sinθ ÷ g, so 90° keeps the object up longest. At 20 m/s, 90° gives 4.08 s against 2.88 s at 45°.
Do 30° and 60° land in the same spot?
They do, because sin(60°) equals sin(120°). Their peak heights differ by a factor of three, so the steeper shot is in the air far longer.
Why does mass not appear in the equations?
Gravity accelerates every mass equally, so the path depends on speed, angle and gravity alone. Only air resistance, which depends on shape and density, separates a pebble from a balloon.
How do I use a different gravity?
Enter it directly. At 4.905 m/s² (about Mars) the same 20 m/s at 45° travels 81.55 m instead of 40.77 m, because range is inversely proportional to g.
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