Ballistic trajectory calculations are fundamental to understanding how projectiles, such as bullets, fly through the air. This involves predicting the path (trajectory) of a projectile from the moment it leaves the barrel of a firearm or is launched, to the point it impacts a target. Accurate calculations are crucial for marksmen, military personnel, and sports shooters to achieve precision at various distances.
The Physics Behind the Flight
The flight of a projectile is governed by several key physical principles:
Initial Velocity: The speed and direction at which the projectile leaves the muzzle. This is the primary driver of the projectile's forward momentum.
Gravity: The constant downward acceleration due to Earth's gravitational pull. This causes the projectile to fall towards the ground.
Aerodynamic Drag: The resistance force exerted by the air on the projectile. This force opposes the projectile's motion and depends on its speed, shape, size, and the density of the air.
Wind: External forces, primarily wind, can push the projectile off its intended path.
Spin Drift: The rifling in a barrel imparts spin to a bullet, which can cause a slight drift to one side (usually right in right-hand twist barrels) due to the Magnus effect.
Coriolis Effect: On very long-range shots, the Earth's rotation can also have a minor effect.
Key Input Parameters for Ballistic Calculations:
Our calculator utilizes the following parameters:
Muzzle Velocity (m/s): The speed of the projectile as it exits the firearm.
Bullet Weight (grams): The mass of the projectile.
Ballistic Coefficient (BC): A measure of a bullet's ability to overcome air resistance. It's a ratio comparing the projectile to a standard projectile (often a 1-inch diameter flat-based bullet) with a BC of 1. Higher BC means better aerodynamic efficiency. Common types are G1 and G7.
Launch Angle (degrees): The angle above the horizontal at which the projectile is fired. For most applications, this is close to 0 degrees, but it's included for angled shots.
Target Distance (meters): The horizontal range to the target.
Wind Speed (m/s) and Wind Angle (degrees): Wind is a significant factor. The angle is typically measured relative to the line of fire (0° is directly behind, 180° is directly in front, 90° is a direct crosswind from the left or right).
Air Density (kg/m³): This is influenced by temperature, pressure, and altitude. Denser air creates more drag.
Temperature (°C) and Barometric Pressure (hPa): These values are used to calculate the actual air density if not provided directly.
Gravity (m/s²): The acceleration due to gravity, typically around 9.81 m/s².
How the Calculator Works (Simplified):
This calculator uses a simplified ballistic model. For precise long-range shooting, sophisticated algorithms (often iterative numerical methods like Runge-Kutta) are employed to solve the differential equations of motion, taking into account air resistance and other factors. This calculator provides an approximation by:
Calculating Air Density: If temperature and pressure are provided, it estimates air density.
Estimating Bullet Drop: It calculates the effect of gravity and drag over the given distance to determine how much the bullet will fall from the line of sight.
Estimating Windage Correction: It models how the wind will push the bullet laterally based on wind speed, direction, and the projectile's BC.
Time of Flight: The duration it takes for the bullet to reach the target.
Note: For professional use, it's always recommended to verify results with specialized ballistic software or by live fire testing, as environmental conditions and firearm/ammunition variations can influence actual performance.
function calculateTrajectory() {
var muzzleVelocity = parseFloat(document.getElementById("muzzleVelocity").value);
var bulletWeight = parseFloat(document.getElementById("bulletWeight").value); // in grams
var ballisticCoefficient = parseFloat(document.getElementById("ballisticCoefficient").value);
var launchAngle = parseFloat(document.getElementById("launchAngle").value); // in degrees
var distance = parseFloat(document.getElementById("distance").value); // in meters
var windSpeed = parseFloat(document.getElementById("windSpeed").value); // in m/s
var windAngle = parseFloat(document.getElementById("windAngle").value); // in degrees
var airDensity = parseFloat(document.getElementById("airDensity").value); // in kg/m³
var temperature = parseFloat(document.getElementById("temperature").value); // in Celsius
var pressure = parseFloat(document.getElementById("pressure").value); // in hPa
var gravity = parseFloat(document.getElementById("gravity").value); // in m/s²
// — Input Validation —
if (isNaN(muzzleVelocity) || muzzleVelocity <= 0 ||
isNaN(bulletWeight) || bulletWeight <= 0 ||
isNaN(ballisticCoefficient) || ballisticCoefficient <= 0 ||
isNaN(launchAngle) ||
isNaN(distance) || distance <= 0 ||
isNaN(windSpeed) ||
isNaN(windAngle) ||
isNaN(airDensity) || airDensity <= 0 ||
isNaN(temperature) ||
isNaN(pressure) || pressure <= 0 ||
isNaN(gravity) || gravity Cd * A = m / BC
// So, F_drag = 0.5 * rho * v^2 * (m / BC)
// 4. Simplified Time of Flight Estimation
// A highly simplified approach would be to assume constant horizontal velocity,
// but drag significantly reduces it. We need an iterative approach for better accuracy.
// For this example, we'll use an approximation based on range and average velocity.
// A better method uses numerical integration (solving ODEs).
// Let's implement a basic iterative approach to approximate trajectory.
// We'll step through time and update velocity, position, and forces.
var dt = 0.01; // time step in seconds
var x = 0; // horizontal position
var y = 0; // vertical position
var vx = v0x; // horizontal velocity
var vy = v0y; // vertical velocity
var t = 0; // time
var totalDistanceAchieved = 0;
var timeToReachTarget = -1; // Initialize as not found
// Find time to reach target distance without considering wind first
var tempX = 0;
var tempVx = v0x;
var tempVy = v0y;
var tempT = 0;
var tempDt = 0.01;
while (tempX < distance && tempT 0) ? -dragMagnitude * (tempVx / currentSpeed) : 0;
var dragFy = (tempVy != 0) ? -dragMagnitude * (tempVy / currentSpeed) : 0;
// Update velocities
var ax = dragFx / bulletWeightKg;
var ay = (dragFy / bulletWeightKg) – gravity;
tempVx += ax * tempDt;
tempVy += ay * tempDt;
// Update positions
tempX += tempVx * tempDt;
tempY = (tempVy < 0 && tempY < 0) ? 0 : tempY + tempVy * tempDt; // Prevent going below ground unless starting angle is negative
tempT += tempDt;
}
timeToReachTarget = tempT; // This is an approximation without wind
// Re-calculate trajectory considering wind with the determined time of flight
x = 0;
y = 0;
vx = v0x;
vy = v0y;
t = 0;
var windFx = 0;
var windFy = 0;
var windDragMagnitude = 0;
var calculatedBulletDrop = 0;
var finalX = 0;
while (x < distance && t < 30) { // Limit loop
var currentSpeed = Math.sqrt(vx * vx + vy * vy);
// Calculate drag force magnitude
var dragCoefficient_Area = bulletWeightKg / ballisticCoefficient;
var dragMagnitude = 0.5 * airDensity * currentSpeed * currentSpeed * dragCoefficient_Area;
// Drag force components (opposite to velocity direction)
var dragFx = (vx != 0) ? -dragMagnitude * (vx / currentSpeed) : 0;
var dragFy = (vy != 0) ? -dragMagnitude * (vy / currentSpeed) : 0;
// Wind force calculation
var windEffectiveSpeed = windSpeed * Math.cos(windAngleRad); // Component of wind pushing with or against bullet's path
var windDragForce = 0.5 * airDensity * (windSpeed * windSpeed) * dragCoefficient_Area; // Simplified: uses wind speed as if it were bullet speed
// A better model would consider the relative velocity between bullet and wind.
// For simplicity, we'll apply a lateral force proportional to wind speed and BC.
// Calculate wind force based on angle
// This is a simplification. A more accurate approach would involve vector addition of velocities.
// We'll approximate windage correction as a lateral force.
var crosswindComponent = windSpeed * Math.sin(windAngleRad); // Perpendicular component
var headTailWindComponent = windSpeed * Math.cos(windAngleRad); // Parallel component (already partially handled by speed)
// Simplified windage correction: apply a lateral force.
// Assume this force is proportional to the crosswind component and BC.
var windageForceMagnitude = 0.5 * airDensity * (crosswindComponent * crosswindComponent) * dragCoefficient_Area;
var windageFx = windageForceMagnitude * Math.sin(windAngleRad); // push in X direction
var windageFy = 0; // Not directly affecting vertical for simplicity here.
// Total forces
var totalFx = dragFx + windageFx;
var totalFy = dragFy – gravity * bulletWeightKg + windageFy; // Added windageFy if modelled
// Acceleration
var ax = totalFx / bulletWeightKg;
var ay = totalFy / bulletWeightKg;
// Update velocity
vx += ax * dt;
vy += ay * dt;
// Update position
x += vx * dt;
y += vy * dt;
// Store drop at target distance if reached
if (Math.abs(x – distance) = distance) {
calculatedBulletDrop = -y; // Bullet drop is the negative of the vertical position
finalX = x;
}
t += dt;
}
// — Windage Correction —
// This is a very rough estimation. Accurate windage correction is complex.
// It depends on the wind's effect on the bullet's trajectory over its entire path.
// For simplicity, we'll calculate it based on the crosswind component and approximate flight time.
var windDragFactor = 0.00001; // A tuning factor to make windage correction reasonable.
var windageCorrection = (windSpeed * Math.sin(windAngleRad)) * timeToReachTarget * windDragFactor * (100 / ballisticCoefficient) ; // Simplified unit: clicks/MOA or mrad. Let's output in meters for simplicity.
// Re-calculate windage more accurately: force balance at target
var finalSpeed = Math.sqrt(vx * vx + vy * vy);
var finalDragMagnitude = 0.5 * airDensity * finalSpeed * finalSpeed * (bulletWeightKg / ballisticCoefficient);
var finalDragFx = (vx != 0) ? -finalDragMagnitude * (vx / finalSpeed) : 0;
var finalDragFy = (vy != 0) ? -finalDragMagnitude * (vy / finalSpeed) : 0;
var crosswindComponentMetersPerSecond = windSpeed * Math.sin(windAngleRad);
// Simplified: assume wind pushes bullet laterally for the duration of flight.
// A better model would integrate wind's effect on bullet velocity vector.
var estimatedWindPush = (crosswindComponentMetersPerSecond * 0.5) * timeToReachTarget; // This is a VERY crude estimate
// — Output Results —
document.getElementById("impactPoint").textContent = "Impact Point: " + finalX.toFixed(2) + " m horizontally";
document.getElementById("timeOfFlight").textContent = "Time of Flight: " + timeToReachTarget.toFixed(3) + " s";
document.getElementById("bulletDrop").textContent = "Bullet Drop: " + calculatedBulletDrop.toFixed(2) + " m";
document.getElementById("windageCorrection").textContent = "Estimated Windage: " + estimatedWindPush.toFixed(2) + " m";
// You might want to convert meters to MOA or Mils for actual scope adjustments.
// 1 MOA at 100 yards is approx 1 inch (0.0254 m).
// 1 Mil at 100 meters is 0.1 meters.
// For MOA: (estimatedWindPush / distance) * (180 / PI) * 60 => MOA
// For Mils: (estimatedWindPush / distance) * 1000 => Mils
}