Case 1
- Launch speed
- 20m/s
- Launch angle
- 45°
40.789m
Open with these valuesProjectile Range Calculator
40.789m
Result: 40.789 mOn level ground the horizontal range is v² × sin(2θ) ÷ g, so 20 m/s at 45° carries 40.789 m. The 45° launch is the farthest for any given speed, and complementary angles — 30° and 60° — land in exactly the same spot. Speed matters more than angle: it enters squared.
The numbers at a glance
Held fixed: Launch speed 20.00 m/s.
| Launch angle (°) | Result (m) |
|---|---|
| 30.00 | 35.324 |
| 40.00 | 40.169 |
| 45.00Your value | 40.789 |
| 50.00 | 40.169 |
| 60.00 | 35.324 |
Worked examples
Case 2
- Launch speed
- 30m/s
- Launch angle
- 45°
91.774m
Open with these valuesCase 3
- Launch speed
- 50m/s
- Launch angle
- 30°
220.775m
Open with these valuesHow it's calculated
R = v² × sin(2θ) ÷ g
- StepEnter the launch speed in metres per second — divide km/h by 3.6.
- StepEnter the angle above the horizontal, between 0° and 90°.
- ResultRead the distance from launch point to landing point, both at ground level.
What this number means
45° wins only if you launch and land level
The formula assumes the landing point sits at exactly the launch height, and the 45° rule rests on that assumption. Throw from shoulder height or off a cliff and it no longer holds.
Complementary angles land in the same place
The range follows sin(2θ), and sin(60°) equals sin(120°), so 20 m/s carries 35.324 m at both 30° and 60°. The steeper throw simply climbs higher and stays up longer to cover the same ground.
Speed is the stronger lever, not angle
The angle enters only through sin(2θ), which never exceeds 1, while the speed enters squared. Half again the speed more than doubles the range: 30 m/s at 45° reaches 91.774 m against 40.789 m for 20 m/s.
No air resistance is modelled here
Real drag shortens every range in the table and pushes the best angle below 45°, most of all for light or fast objects. Read the result as the ceiling a throw could reach, not the distance it will.
Commonly misread
45° is the best angle for every throw.
Only for launch and landing at the same height, which is what this calculator assumes. Once the two heights differ, 45° stops being the answer.
Doubling the launch speed doubles the range.
It quadruples it, because the speed enters squared: 10 m/s at 45° reaches 10.197 m and 20 m/s reaches 40.789 m.
A steeper throw always carries farther than a flat one.
Past 45° the range falls away again: 12 m/s at 75° manages only 7.342 m, less than a much slower 10 m/s at 45° covers with 10.197 m.
Reference table
| Speed, angle | Launch | Range (m) |
|---|---|---|
| 20, 0 | Flat, no flight | 0.000 |
| 12, 75 | Steep, short | 7.342 |
| 10, 45 | Best angle, slow | 10.197 |
| 15, 50 | Just past the best angle | 22.595 |
| 20, 60 | Same as 20 at 30° | 35.324 |
| 20, 45 | Best angle | 40.789 |
| 30, 45 | Half again the speed | 91.774 |
| 50, 30 | Fast and flat | 220.775 |
Questions
How do I calculate the range of a projectile?
Use R = v² × sin(2θ) / g, where v is the launch speed, θ is the launch angle, and g = 9.80665 m/s². With speed in metres per second and angle in degrees, the range comes out in metres. A 20 m/s launch at 45° gives 20² × sin(90°) / 9.80665 = 40.79 m.
What launch angle gives the maximum range?
On level ground with no air resistance, the maximum range always occurs at 45°. The range depends on sin(2θ), which reaches its peak of 1 when 2θ = 90°, so any steeper or shallower angle covers less distance for the same launch speed.
Why do 30° and 60° give the same range?
Because the range depends on sin(2θ), and sin(60°) equals sin(120°). Complementary angles that add up to 90° produce the same sin(2θ) value, so they land the same distance away. The steeper launch simply rises higher and stays in the air longer to cover the equal distance.
How much does the launch speed change the range?
A lot — the range grows with the square of the launch speed. Doubling the speed multiplies the range by four, and tripling it by nine. Speed is a stronger lever than angle, which is why a fast, lower-angle throw can out-distance a perfect 45° throw at a slower speed.
Does this calculator account for air resistance?
No. It gives the idealised range for level ground with no air resistance and equal launch and landing heights. Real drag shortens the range and pushes the best angle below 45°, especially for light or fast objects.
Sources and last check
Information, not professional advice.