Launch Angle & Projectile Range
Lob a projectile over a wall, plus the arithmetic: 45 degrees is only best on level ground, and your height matters four times more than your aim.
Clear the wall and land on the target. There are usually two angles that reach it — a low arc and a high one — and only one of them gets over the wall. No air resistance, which is the model every "45 degrees" argument is really about. Nothing is uploaded.
45° is right on level ground and nowhere else
"Forty-five degrees for maximum range" is true of a shot that lands at the height it left from, and of nothing else. Launch from h above the landing point and the best angle has a closed form — sin²θ* = 1 / (2 + 2gh/v²) — which is 45° exactly when h is zero and falls away as you climb. At 20 m/s:
| Launch height | Best angle | Range there | Range at 45° | Given up |
|---|---|---|---|---|
| 0 m | 45.0° | 40.8 m | 40.8 m | — |
| 1 m | 44.3° | 41.8 m | 41.8 m | — |
| 2 m | 43.7° | 42.7 m | 42.7 m | 0.100% |
| 5 m | 41.9° | 45.5 m | 45.3 m | 0.491% |
| 10 m | 39.3° | 49.8 m | 49.1 m | 1.402% |
| 20 m | 35.4° | 57.4 m | 55.5 m | 3.336% |
| 50 m | 28.3° | 75.8 m | 69.9 m | 7.700% |
| 100 m | 22.4° | 99.1 m | 87.4 m | 11.773% |
The angle falls at every step as the launch point rises. The formula is derived rather than searched for, and then searched for anyway as a check — the two agree to within a hundredth of a degree at every height in the table.
But being wrong about the angle costs almost nothing
Read the last column of the table again. Firing at 45° from five metres up — the wrong angle by three degrees — gives up half a per cent of the range. From ten metres it is 1.4%. The range curve is flat at its maximum, so the angle is the forgiving part of the problem:
| Launch height | Within 1% of the best range | Within 5% |
|---|---|---|
| 0 m | ±4.1° | ±9.1° |
| 5 m | ±4.5° | ±10.3° |
| 20 m | ±5.5° | ±12.6° |
| 100 m | ±7.1° | ±16.4° |
On level ground you can be four degrees out and keep 99% of your distance, or nine degrees out and keep 95%. And it gets more forgiving the higher you launch from, not less.
The correction only becomes a real mistake a long way up: at 100 m, firing at 45° throws away 11.8% of the range.
The thing that actually matters is the height
At 20 m/s, launching from ten metres up instead of level adds 22.1% to the range. Being ten degrees off the best angle on level ground costs 6.0%. The height is worth about four times the aim — and it is the one nobody argues about.
So the popular question, what angle, has a forgiving answer; the unpopular one, what are you standing on, has a decisive one. All of this is the vacuum model — with drag the best angle drops further still, which only strengthens the point rather than weakening it.
How to use
- Set the launch angle with the slider.
- Press Fire to send the shot over the wall.
- Two angles usually reach the target — only one clears the wall.
- Press Show the best angle if you get stuck.
- Press New shot for a fresh target, height and speed.
Frequently asked questions
Is 45 degrees really the best launch angle?
Only when the shot lands at the height it left from. That is the case everybody is taught and the only one where it holds. Launch from higher up and the best angle is lower — at 20 metres per second it is 39.3 degrees from ten metres up and 22.4 degrees from a hundred.
What is the formula for the best angle?
The angle that carries furthest satisfies sin squared theta = 1 / (2 + 2gh/v squared), where h is the launch height above the landing point. At h = 0 that gives exactly 45 degrees. The page derives it and then searches for it numerically as a check; the two agree to within a hundredth of a degree.
How much does getting the angle wrong cost?
Far less than you would think, because the range curve is flat at its maximum. On level ground you can be 4.1 degrees out and keep 99 per cent of your distance, or 9.1 degrees out and keep 95 per cent. Firing at 45 degrees from five metres up gives away half a per cent.
When does the correction actually matter?
A long way up. From five metres the mistake is worth 0.5 per cent and from ten metres 1.4 per cent, but from a hundred metres firing at 45 degrees throws away 11.8 per cent of the range. Below ten metres it is not worth thinking about.
What matters more than the angle?
The height you launch from. At 20 metres per second, going from level ground to ten metres up adds 22.1 per cent to the range, while being ten degrees off the best angle costs 6.0 per cent. Height is worth about four times aim, and it is the one nobody argues about.
Why are there two angles that hit the same target?
A low arc and a high one, and both land on exactly the same spot. They collapse into a single angle when the target sits at the very limit of your range, and beyond that there is no solution at all. In this game only one of the two usually clears the wall.
Does this account for air resistance?
No — this is the vacuum model, which is what every "45 degrees" argument is really about. Drag lowers the best angle further still, so it strengthens the conclusion rather than weakening it.
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