Muzzle Velocity Calculator

Background.

Ballistic tables, air-gun specifications and arrow-tuning charts often quote a projectile's energy rather than its speed, and the two are linked by one of the oldest equations in mechanics. Kinetic energy is half the mass times the speed squared, E = ½mv², so knowing any two of energy, mass and speed fixes the third. Solved for speed, it becomes v = √(2E/m).

The square root is where the practical lessons sit. Energy grows with the square of speed, so speed grows only with the square root of energy: doubling the energy at the same mass raises the speed by about 41%, not 100%. Mass pulls the other way. At equal energy, a projectile twice as heavy moves about 29% slower, and one four times as heavy moves at half the speed.

The worked example, 1,000 J carried by a 20 g projectile, comes out at about 316 m/s, a little under the speed of sound in room-temperature air, roughly 343 m/s.

The question usually arrives in one of a few forms. An air-gun owner knows the power rating in foot-pounds and the pellet weight in grains and wants the speed in feet per second. An archer wants to know how much speed a heavier arrow gives up at the same energy. A physics student has a spring or a falling mass and wants the launch speed. All of them reduce to the same two numbers once the units are in joules and kilograms.

The number that comes out is the speed implied by the energy you enter, and nothing more. Whether a particular barrel, load or bow really delivers that energy depends on pressure, propellant, barrel length and much else that only measurement settles. A chronograph reading is the true muzzle velocity, and published energy figures are usually calculated from such a reading in the first place. Drag, trajectory and safety lie outside the calculation.

Both inputs are in SI units, joules and kilograms. Sources in foot-pounds and grains need converting first, or the answer will be off by a large factor.

What is muzzle velocity calculator?

Muzzle velocity is the speed of a projectile at the moment it leaves the barrel, launcher or bowstring, usually quoted in metres or feet per second. What is computed here is the energy-equivalent speed: the speed at which a body of the given mass carries the given kinetic energy, from E = ½mv². When the energy you enter is the projectile's muzzle energy, that speed is its muzzle velocity.

Muzzle energy and muzzle velocity are two descriptions of the same moment, tied together by the projectile's mass. In practice the velocity is the one that gets measured: ammunition makers working to the North American SAAMI or European CIP frameworks fire loads through standardized test barrels over a chronograph, and the energy printed in a catalogue is then worked out from that speed and the bullet's weight. Running the conversion backwards, as here, recovers the speed the energy figure was built from, provided you use the same projectile mass.

Speed is the number that shapes everything after the muzzle, because it sets how long gravity has to act before the projectile reaches its target. Even with no air at all, the 20 g example at 316.2 m/s would take 0.316 s to cover 100 m and drop about 0.49 m in that time; real drag slows it and adds to the drop.

How to use this calculator.

1. 01Enter the projectile's kinetic energy in joules. A figure in foot-pounds converts by multiplying by 1.3558, so 500 ft·lbf is about 678 J.
2. 02Enter the mass of the projectile alone in kilograms, leaving out the cartridge case, wad or sabot. Divide grams by 1,000, or multiply grains by 0.0000648, so 20 g is 0.02 kg and 150 grains is about 0.00972 kg.
3. 03Read the speed in metres per second, and multiply by 3.281 if you want feet per second.
4. 04For the speed at a distance rather than at the muzzle, enter the remaining energy a ballistic table gives for that range; the same relation holds at any point in flight.
5. 05To check a specification sheet, enter its quoted energy and projectile weight and compare the speed you get with the speed it quotes. A large mismatch usually means the energy was stated for a different projectile weight.

The formula.

Kinetic energy is E_k = ½mv², with E_k in joules, m in kilograms and v in metres per second. Solving for speed gives v = √(2E_k/m): double the energy, divide by the mass, take the square root. The units work out directly, because a joule is a kilogram square metre per second squared, so joules per kilogram are square metres per second squared and their root is metres per second.

The ½mv² itself comes from the work-energy theorem: the net work done on a body equals the kinetic energy it gains. Pushing a mass from rest with a steady force F over a distance d gives it energy Fd, and the speed follows from setting Fd equal to ½mv². Read the other way, the energy tells you the average push behind the projectile. Delivering 1,000 J over a 0.5 m barrel takes an average net force of 2,000 N; on a 20 g projectile that is an average acceleration of 100,000 m/s², about 10,200 times gravity. Real pressure curves peak early in the barrel, so the peak force is higher than that average.

Speed rises with the square root of energy and falls with the square root of mass, so a fourfold change in either one changes the speed only twofold. The same square root halves measurement errors: an energy figure 2% too high, or a mass 2% too low, moves the speed by only about 1%. For pellet or bullet choice the trade runs both ways: at the same energy, a projectile 10% heavier leaves about 4.7% slower, and to keep the same speed it needs 10% more energy. The equation also rearranges for the third quantity. If a source gives speed and energy but not mass, m = 2E/v², so 1,000 J at 316.2278 m/s implies 0.02 kg.

The relation ignores the energy stored in a projectile's spin, and it is the non-relativistic form, which is accurate far beyond any measurement at the speeds real projectiles reach. Zero or negative values for either input are refused, since neither describes a moving projectile. The speed is shown to four decimal places, which is more precision than any energy figure supports.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, University Physics Volume 1, section 7.2, Kinetic Energy.
2. [2]SAAMI, Glossary of Industry Terms.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.
4. [4]NIST Special Publication 811, Guide for the Use of the International System of Units, 2008 edition.
5. [5]CIP, Decisions, texts and tables: standardized ammunition test framework.