Calculator Range

Projectile Range Calculator

Calculate the range, time of flight, and maximum height of a projectile given its initial velocity, launch angle, and starting height. This calculator uses standard physics formulas for projectile motion, assuming no air resistance and a constant gravitational acceleration of 9.81 m/s².

function calculateProjectileRange() { var initialVelocity = parseFloat(document.getElementById('initialVelocity').value); var launchAngleDegrees = parseFloat(document.getElementById('launchAngle').value); var initialHeight = parseFloat(document.getElementById('initialHeight').value); // Validate inputs if (isNaN(initialVelocity) || initialVelocity < 0) { document.getElementById('result').innerHTML = 'Please enter a valid non-negative initial velocity.'; return; } if (isNaN(launchAngleDegrees) || launchAngleDegrees 90) { document.getElementById('result').innerHTML = 'Please enter a valid launch angle between -90 and 90 degrees.'; return; } if (isNaN(initialHeight) || initialHeight < 0) { document.getElementById('result').innerHTML = 'Please enter a valid non-negative initial height.'; return; } var g = 9.81; // Acceleration due to gravity in m/s^2 var launchAngleRadians = launchAngleDegrees * Math.PI / 180; var v0x = initialVelocity * Math.cos(launchAngleRadians); // Initial horizontal velocity var v0y = initialVelocity * Math.sin(launchAngleRadians); // Initial vertical velocity var timeOfFlight; var range; var max_height; // Calculate Time of Flight using quadratic formula: 0 = initialHeight + v0y*t – 0.5*g*t^2 // Rearranging: (0.5*g)*t^2 – v0y*t – initialHeight = 0 // a = 0.5*g, b = -v0y, c = -initialHeight var a = 0.5 * g; var b = -v0y; var c = -initialHeight; var discriminant = b * b – 4 * a * c; if (discriminant < 0) { // This should theoretically not happen for real-world projectile motion starting above or at ground level document.getElementById('result').innerHTML = 'Error: Discriminant is negative. Check input values.'; return; } // We take the positive root for time timeOfFlight = (-b + Math.sqrt(discriminant)) / (2 * a); // Calculate Range range = v0x * timeOfFlight; // Calculate Maximum Height // Max height occurs when vertical velocity is 0 (vy = v0y – g*t_peak = 0) // t_peak = v0y / g var t_peak = v0y / g; if (t_peak < 0) { // If initial vertical velocity is downwards, max height is initial height max_height = initialHeight; } else { max_height = initialHeight + (v0y * t_peak – 0.5 * g * t_peak * t_peak); } // Ensure max_height is not less than initialHeight if launched downwards max_height = Math.max(initialHeight, max_height); var resultHTML = '

Calculation Results:

'; resultHTML += 'Time of Flight: ' + timeOfFlight.toFixed(2) + ' seconds'; resultHTML += 'Projectile Range: ' + range.toFixed(2) + ' meters'; resultHTML += 'Maximum Height: ' + max_height.toFixed(2) + ' meters'; document.getElementById('result').innerHTML = resultHTML; } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 25px; border-radius: 10px; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1); max-width: 600px; margin: 30px auto; border: 1px solid #e0e0e0; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; font-size: 1.8em; } .calculator-container p { color: #555; line-height: 1.6; margin-bottom: 15px; } .calc-input-group { margin-bottom: 15px; display: flex; flex-direction: column; } .calc-input-group label { margin-bottom: 8px; color: #333; font-weight: bold; font-size: 1em; } .calc-input-group input[type="number"] { padding: 10px 12px; border: 1px solid #ccc; border-radius: 5px; font-size: 1em; width: 100%; box-sizing: border-box; transition: border-color 0.3s ease; } .calc-input-group input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 0 3px rgba(0, 123, 255, 0.25); } .calc-button { background-color: #007bff; color: white; padding: 12px 25px; border: none; border-radius: 5px; cursor: pointer; font-size: 1.1em; font-weight: bold; display: block; width: 100%; margin-top: 20px; transition: background-color 0.3s ease, transform 0.2s ease; } .calc-button:hover { background-color: #0056b3; transform: translateY(-2px); } .calc-button:active { background-color: #004085; transform: translateY(0); } .calc-result { margin-top: 25px; padding: 15px; background-color: #e9f7ff; border: 1px solid #cce5ff; border-radius: 8px; color: #004085; font-size: 1.1em; } .calc-result h3 { color: #0056b3; margin-top: 0; margin-bottom: 10px; font-size: 1.4em; } .calc-result p { margin-bottom: 8px; color: #004085; } .calc-result p strong { color: #002752; } .calc-result .error { color: #dc3545; font-weight: bold; }

Understanding Projectile Range

Projectile motion is a fundamental concept in physics that describes the path an object takes when launched into the air, subject only to the force of gravity. The "range" of a projectile refers to the total horizontal distance it travels from its launch point until it returns to its initial height or hits the ground.

Key Factors Influencing Projectile Range:

• Initial Velocity: The speed at which the object is launched. A higher initial velocity generally leads to a greater range.
• Launch Angle: The angle at which the object is launched relative to the horizontal. For a given initial velocity and flat ground, an angle of 45 degrees typically yields the maximum range. However, if launched from a height, the optimal angle can be less than 45 degrees.
• Initial Height: The vertical position from which the projectile is launched. Launching from a greater height allows the projectile more time in the air, potentially increasing its range.
• Gravity: The acceleration due to gravity (approximately 9.81 m/s² on Earth) constantly pulls the projectile downwards, affecting its vertical motion and ultimately its time of flight and range.
• Air Resistance (ignored in this calculator): In real-world scenarios, air resistance (drag) significantly affects projectile motion, reducing both range and maximum height. This calculator provides an idealized result by ignoring air resistance.

How the Calculator Works:

This calculator uses the principles of kinematics to determine the projectile's trajectory. It breaks down the initial velocity into horizontal (v₀x) and vertical (v₀y) components. The horizontal motion is constant (assuming no air resistance), while the vertical motion is influenced by gravity.

The calculation involves:

1. Converting Angle: The launch angle is converted from degrees to radians for trigonometric calculations.
2. Velocity Components: Initial velocity is resolved into horizontal (v₀ * cos(θ)) and vertical (v₀ * sin(θ)) components.
3. Time of Flight: This is the total time the projectile spends in the air. It's calculated by solving a quadratic equation derived from the vertical motion equation (y = y₀ + v₀y*t - 0.5*g*t²), where y is the final height (usually 0 for ground level).
4. Range: Once the time of flight is known, the horizontal range is simply the horizontal velocity multiplied by the time of flight (Range = v₀x * Time of Flight).
5. Maximum Height: This is the highest point the projectile reaches. It's calculated by determining the time it takes for the vertical velocity to become zero (t_peak = v₀y / g) and then plugging that time into the vertical motion equation.

Examples of Projectile Range:

• Throwing a Ball: If you throw a ball with an initial velocity of 20 m/s at a 45-degree angle from the ground (0 m height), it would travel approximately 40.77 meters.
• Cannonball from a Cliff: A cannonball fired horizontally (0-degree angle) from a 100-meter high cliff with an initial velocity of 100 m/s would have a significantly longer range due to the initial height, even with a 0-degree launch angle.
• Golf Drive: A well-hit golf ball might leave the club at 60 m/s at a 12-degree angle from the ground. Its range would be substantial, though air resistance plays a huge role in real golf.

Use this calculator to explore how different launch parameters affect the trajectory of a projectile!