Projectile Motion Calculator
Calculate the range, maximum height and flight time of a projectile from its launch speed and angle.
How to use this calculator
- 1Enter the launch speed and the angle above the horizontal.
- 2Set a launch height if the projectile does not start at the level it lands on.
- 3Change gravity to model another body — the Moon is 1.62 m/s², Mars 3.72.
How the calculation works
vₓ = v·cos θ v_y = v·sin θ t = (v_y + √(v_y² + 2gh₀)) ÷ g range = vₓ·t apex = h₀ + v_y² ÷ 2g- v, θ
- Launch speed and angle above the horizontal
- vₓ, v_y
- Horizontal and vertical velocity components
- h₀
- Launch height above the landing surface
- g
- Gravitational acceleration
The horizontal and vertical motions are independent. Horizontally there is no acceleration, so the projectile covers ground at a constant rate; vertically it is in free fall the whole time. Splitting the problem this way is what makes it solvable at all.
From ground level (h₀ = 0) the range simplifies to v²·sin(2θ)/g, which is maximised at θ = 45° because sin(2θ) peaks at 90°. It also explains why complementary angles match: sin(2 × 30°) and sin(2 × 60°) are both sin(60°).
A raised launch breaks that symmetry. Extra height already buys hang time, so the optimal angle drops below 45° — and the greater the height relative to the speed, the lower it goes.
Worked example
25 m/s at 45° from ground level
- 1.Components: vₓ = 25·cos45° = 17.678 m/s, v_y = 25·sin45° = 17.678 m/s.
- 2.Time to apex: 17.678 ÷ 9.80665 = 1.803 s. From ground level total flight is twice that: 3.605 s.
- 3.Apex height: 17.678² ÷ (2 × 9.80665) = 312.5 ÷ 19.6133 = 15.934 m.
- 4.Range: 17.678 × 3.605 = 63.73 m.
- 5.Equivalently, v²·sin(2θ)/g = 625 × sin90° ÷ 9.80665 = 63.73 m.
Result: 63.73 m range, 15.93 m peak, 3.61 s airborne
Two independent motions
The central insight of projectile motion is that the horizontal and vertical components do not interact. Gravity pulls only downward, so it changes the vertical velocity and leaves the horizontal one untouched. A bullet fired horizontally and a bullet dropped from the same height hit the ground at the same moment — the fired one simply travels much further while doing so.
Every projectile formula follows from decomposing the launch velocity into those two components, treating the horizontal as constant-speed motion and the vertical as free fall, then recombining them through the shared variable of time.
Why 45° — and when it is not
From ground level the range works out to v²·sin(2θ)/g, and since sin peaks at 90°, the range peaks when 2θ = 90°, meaning θ = 45°. The same expression explains why complementary angles land together: 30° and 60° both produce sin(60°), so both give identical range with very different trajectories — one flat and fast, one high and slow.
The 45° result depends entirely on launching and landing at the same height. Throw from a cliff, or a shot-putter releasing from shoulder height, and the optimum drops below 45°, because the launch height already supplies hang time that a steeper angle would otherwise have to buy. Competitive shot put releases at closer to 35–40° for exactly this reason.
The air-resistance caveat
These equations describe motion in a vacuum, and the discrepancy with reality is not small for most real projectiles. Drag scales with the square of speed and acts opposite the direction of travel, which shortens range, lowers the apex and makes the descent steeper than the ascent — real trajectories are asymmetric, unlike the perfect parabola here.
The error depends heavily on the object. A dense compact mass at modest speed tracks the ideal curve reasonably well. A ball, a shuttlecock, or anything at sporting speeds can fall well short of the predicted range. Spin adds another layer entirely: the Magnus effect can make a spinning ball curve or carry noticeably further than any drag-only model predicts.
What this assumes, and where it stops
Assumptions
- No air resistance. The projectile follows an ideal parabola.
- Gravity is constant over the flight, which holds for any everyday trajectory.
- The projectile is a point mass with no spin, lift or Magnus effect.
- The landing surface is level and horizontal.
Limitations
- Ignores drag, which for light or fast projectiles substantially overstates range and makes the modelled trajectory more symmetric than a real one.
- Does not model spin, wind, or lift — all of which matter in sport and ballistics.
- Assumes flat ground. Landing on a slope changes both range and flight time.
Common questions
Why is 45 degrees the best launch angle?
Because from ground level the range is proportional to sin(2θ), which is largest when 2θ = 90°. It only holds when launch and landing heights are equal — launching from a height moves the optimum below 45°, since the extra height already provides flight time that a steeper angle would otherwise have to generate.
Do 30° and 60° really give the same range?
Yes, from level ground — sin(60°) and sin(120°) are equal, so both angles produce the same range. The trajectories look completely different: 30° is flat and fast with a short flight time, 60° is high and slow. Only the landing point matches.
How far off is this without air resistance?
It depends entirely on the projectile. A dense compact object at modest speed is close to the ideal. A ball or anything light and fast can fall well short of the predicted range, and the real descent will be steeper than the ascent rather than symmetric. Treat this as an upper bound on range.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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