Library / Ballistics / BAL-01
Ballistics
Projectile range (no drag)
Level-ground range of a projectile launched at a given speed and angle, ignoring air resistance.
- Ref
- BAL-01
- Rev
- v1
- Field
- Ballistics
- Result
- R [m]
with g = 9.80665 m/s² and θ measured from the horizontal.
Launch speed splits into a horizontal part that carries the projectile forward and a vertical part that sets how long it stays in the air. The product is largest at 45°, and any two angles that add up to 90° land at the same spot.
Calculator
- Maximum range at 45°9,177 m
- Time of flight35.09 s
- Apex height1,510 m
- Other angle, same range55 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Range grows with the square of launch speed. Speed is the strongest lever by far.
- Near 45° the curve is flat, so a degree of error costs little; away from 45° each degree matters more.
- θ and 90° − θ give the same range. These are the low-angle and high-angle solutions used in indirect fire.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| v0 Launch (muzzle) speed | 100 to 1,000 m/s | 958.2 | 95,820 | 94,860 | +10% | +21 % |
| θ Launch angle | 5 to 85 ° | 1,594 | 9,177 | 7,584 | +10% | +3.69 % |
Sweep of Launch angle
| Launch angle (°) | Horizontal range (m) |
|---|---|
| 5 | 1,594 |
| 15 | 4,589 |
| 25 | 7,030 |
| 35 | 8,624 |
| 45 | 9,177 |
| 55 | 8,624 |
| 65 | 7,030 |
| 75 | 4,589 |
| 85 | 1,594 |
Sweep one variable
Variables and how each one contributes
Horizontal range
What the equation gives you. Everything else feeds it.
Launch (muzzle) speed
Speed of the projectile as it leaves the launcher. Range scales as v0². Doubling launch speed quadruples range, and a 1% speed error becomes a 2% range error.
Launch angle
Elevation above the horizontal. Range follows sin 2θ: a maximum at 45°. Angles equally far either side of 45° (30° and 60°, say) land at the same point.
Worked example with the values above
- Double the angle: 2θ = 2 × 35° = 70°. Then sin(70°) = 0.9397 (calculator in degree mode).
- Square the speed: v0² = 300² = 90,000 m²/s².
- R = v0² sin 2θ / g = 90,000 × 0.9397 / 9.80665 = 8,624 m.
Assumptions and limits
- Vacuum: no drag, wind, spin or Coriolis effect.
- Flat level ground, with launch and impact at the same height.
- Constant g = 9.80665 m/s².
- The projectile is a point mass.
Common mistakes
- Calculator in radian mode while the angle is in degrees (or the other way round). This is the most common slip.
- Using sin θ instead of sin 2θ.
- Assuming 45° is best in air. With drag the best angle is lower than 45°.
- Quoting this as a real range. Drag usually cuts range substantially, so treat it as an upper bound.
Statistics and uncertainty
Real impact points scatter even when the formula is right. To first order a relative speed error δv/v gives about 2δv/v in range, and an angle error δθ (in radians) gives δR/R = 2 δθ cot 2θ. Dispersion measures such as circular error probable (CEP) describe the resulting scatter on target.
Where it is used
- First-pass range estimates and sanity checks for ballistic software.
- Comparing low-angle and high-angle fire for the same target.
- Checking a drag model: with drag switched off it must collapse to this result.
Maths you need
- Trigonometry: sin 2θ, and the double-angle identity 2 sin θ cos θ
- Degrees and radians: check the calculator mode before every trig call
- Squares: range grows as v2