Why a bullet drops, and what 'drop' actually measures
A bullet begins falling the instant it leaves the muzzle, and it falls at the same rate as a bullet dropped from your hand. Gravity does not care how fast the projectile is travelling forward. What forward speed buys is time: the faster the bullet covers the distance, the less time gravity has to act, and because drop grows with the square of time, small changes in time of flight produce large changes in drop.
Two different quantities get called drop. Drop from the bore line is the pure physics figure, exactly ½gt², and it is what a bullet does relative to the direction the barrel was pointing. Path relative to the line of sight is what a shooter cares about, and it is the number this calculator reports. It differs from the first by two corrections: your sight sits above the bore, typically 1.5 inches, and the barrel is tilted slightly upward so the bullet crosses your line of sight at the zero distance.
That tilt is why a trajectory rises before it falls. With a 200-yard zero the bullet crosses the sight line on the way up at roughly 25 yards, peaks a couple of inches high near 120 yards, and crosses back down through the sight line at 200. Nothing about the bullet's fall changes; only the reference line does.
Time of flight is therefore the whole story, and time of flight is governed by drag. That is where the ballistic coefficient enters.
The drag model and what a ballistic coefficient really is
Air resistance on a supersonic bullet varies with the square of velocity and with a drag coefficient that changes sharply around the speed of sound. Rather than tabulate every bullet, exterior ballistics uses a standard projectile whose drag has been measured across the whole Mach range, and describes your bullet as a scaled version of it. That scale factor is the ballistic coefficient.
The G1 standard is a flat-based projectile with a short ogive, close in shape to nineteenth-century artillery shells, and it is the model behind almost every BC printed on a box of hunting ammunition. The G7 standard is a long boat-tail form that matches modern low-drag rifle bullets far better. Because the two standards have different drag levels, a single bullet has two very different numbers: a match bullet quoted at 0.475 G1 will be near 0.243 G7. Entering a G1 value with the G7 model selected, or the reverse, produces errors of many inches at distance, which is why the model selector sits next to the BC field.
The equation being integrated is dv/dt = −K · (ρ/ρ0) · Cd(M)/BC · v². K is a constant fixed by the standard projectile's mass and diameter, Cd(M) comes from the tabulated drag function at the current Mach number, and ρ/ρ0 scales for air density. A higher BC divides the retardation down, which is exactly what a high BC means physically: the bullet sheds velocity more slowly than the standard.
Air density comes from the ICAO standard atmosphere. Pressure falls with altitude as (1 − 6.87535×10−6·h)5.2559, and density is that pressure divided by absolute temperature, referenced to 59 °F. Because drag scales linearly with density, shooting at 8,000 feet is arithmetically identical to shooting the same rifle at sea level with a BC about 35% higher.
Solving for the launch angle is a root-find, not a formula. The calculator adjusts the bore elevation until the computed path crosses zero at your stated zero range, then re-runs the full trajectory at that angle. That is why the path output is exactly zero at the zero distance no matter what other inputs you change.
Worked example: checking the solver against pure physics
Set the ballistic coefficient absurdly high — 50 — and drag becomes negligible, which turns the trajectory into a textbook parabola you can compute by hand. Take 3,000 fps, a 100-yard zero, and a 1.5-inch sight height, and find the path at 300 yards.
- Time to the zero. 100 yd is 300 ft, so t = 300 / 3,000 = 0.100 s.
- Drop from the bore line at the zero. ½ × 32.174 × 0.100² = 0.16087 ft = 1.930 in.
- Required bore elevation. The bore must lift the bullet by the sight height plus that drop over 300 ft: tanθ = (1.500 + 1.930) / 3,600 = 3.430 / 3,600 = 0.0009528, which is 3.28 MOA.
- Time to 300 yards. 900 ft / 3,000 fps = 0.300 s.
- Drop from the bore line at 300 yards. ½ × 32.174 × 0.300² = 1.4478 ft = 17.374 in.
- Path relative to the sight line. −1.500 + (10,800 in × 0.0009528) − 17.374 = −1.500 + 10.291 − 17.374 = −8.58 in.
- Convert to holdover. 8.58 / (1.047 × 3) = 2.73 MOA, or 8.58 / (3.6 × 3) = 0.79 mil.
Run those inputs through the calculator and it returns the same figures within a few hundredths of an inch, the residual being the small amount of drag that even a BC of 50 cannot eliminate. Now change the BC to a realistic 0.243 G7 and the answer moves a long way, because the bullet is no longer covering 900 feet in 0.300 seconds. That difference — entirely a time-of-flight effect — is what the drag integration exists to capture.
Reading the trajectory table and choosing a zero
The path column is what you dial or hold. A path of −8.58 inches means the bullet strikes 8.58 inches below where the crosshair sat, so you either hold 8.58 inches high or dial 2.73 MOA up. Positive values mean the bullet is above the line of sight, which happens between the two crossings of a distant zero.
Retained velocity is the number that decides terminal performance, not energy. Expanding hunting bullets have a minimum impact velocity below which they behave like solids, commonly quoted by makers around 1,800 to 2,000 fps for controlled-expansion designs and lower for soft cup-and-core bullets; check the specific bullet rather than assuming. Match bullets have no expansion requirement but do become less predictable through the transonic region, roughly 1,340 down to 900 fps, where the drag curve is steepest and small differences in bullet shape matter most. The calculator flags the target range when it falls into that band.
Maximum point blank range is the practical answer to "what zero should I use". Choose the vital-zone diameter of your quarry, and MPBR is the farthest range at which the path never falls below the bottom of that zone, so you can aim at the centre and ignore holdover entirely. A flat-shooting cartridge with a 6-inch zone typically gives an MPBR somewhere near 250 to 300 yards, achieved with a zero around 200 to 250 yards; enter your own numbers rather than adopting a rule of thumb, since the answer depends on velocity and BC.
Once you have the holdover, the MOA and mil scope adjustment calculator converts it into turret clicks for your specific scope, and the muzzle energy calculator handles the terminal-energy side of the same shot.
Air density ratio by altitude and temperature
| Altitude (ft) | Pressure ratio | Density at 20 °F | Density at 59 °F | Density at 90 °F |
|---|---|---|---|---|
| 0 | 1.0000 | 1.081 | 1.000 | 0.944 |
| 2,000 | 0.9298 | 1.006 | 0.930 | 0.877 |
| 4,000 | 0.8637 | 0.934 | 0.864 | 0.815 |
| 6,000 | 0.8014 | 0.867 | 0.801 | 0.756 |
| 8,000 | 0.7428 | 0.803 | 0.743 | 0.701 |
| 10,000 | 0.6877 | 0.744 | 0.688 | 0.649 |
A rifle zeroed at sea level and 59 F, then fired at 8,000 ft and 20 F, sees only 80% of the drag it was zeroed in and will shoot high at distance. Humidity changes density by well under 1% and is ignored here.
What this solver does not model, and other ways to be wrong
- Wind. No crosswind is applied. Wind deflection is usually the dominant error at long range and depends on the same time-of-flight figure the table reports.
- Spin drift and Coriolis. Both push the bullet sideways at extended range: spin drift a few inches to the right for a right-hand twist past 600 yards, Coriolis smaller still and dependent on latitude and azimuth. Neither is included.
- Uphill and downhill shots. Only the horizontal component of the range produces drop, so a steeply inclined shot needs less holdover than its slant range suggests. Multiply the range by the cosine of the angle before entering it.
- Box-flap velocity. Advertised muzzle velocities are measured in test barrels that are often longer than yours. A chronograph reading from your own rifle is worth more than any other input on this page.
- The wrong BC for the model. A G1 number entered against the G7 curve roughly doubles the drag. Check which standard the manufacturer published.
- Assuming the solver replaces confirmation. Every prediction should be verified on paper at distance. True the BC until the calculated and measured drops agree, then trust the table between those points.
Where this model sits among ballistic methods
Three levels of exterior ballistics are in common use. The oldest is the Siacci method, a closed-form flat-fire approximation using tabulated space, altitude and inclination functions, which is what Ingalls' tables and every printed drop chart before the computer age relied upon. It is accurate for flat-fire small arms and requires no integration. The middle level, used here, is numerical point-mass integration: treat the bullet as a particle, integrate the drag equation directly, and you gain the ability to handle any launch angle, any atmosphere, and velocity-dependent drag without approximation.
The top level is the six-degree-of-freedom model, which tracks the projectile's orientation as well as its position and can predict spin drift, aerodynamic jump and yaw-induced drag. It requires aerodynamic coefficients that bullet makers rarely publish, so it is confined to laboratory and military work. For everything a rifle shooter does, a well-trued point-mass solution matches observed drops to within the rifle's own dispersion.
Three practical companions round out the picture. Zeroing and correcting a group is a separate arithmetic problem handled by the scope adjustment calculator, which converts a measured offset into turret clicks. Terminal effect is a function of retained velocity and bullet weight, which the muzzle energy and momentum calculator quantifies in ft-lb, joules, momentum and power factor. And the probability of a hit across several shots, given a per-shot hit chance, follows the same 1 − (1 − p)n arithmetic used by the drop rate probability calculator.