Calculate the Range

Projectile Motion Range Calculator

Use this calculator to determine the horizontal range, time of flight, and maximum height of a projectile launched at a specific initial velocity and angle, from a given height.

function calculateProjectileRange() { var initialVelocity = parseFloat(document.getElementById('initialVelocity').value); var launchAngleDegrees = parseFloat(document.getElementById('launchAngle').value); var launchHeight = parseFloat(document.getElementById('launchHeight').value); var gravity = parseFloat(document.getElementById('gravity').value); if (isNaN(initialVelocity) || isNaN(launchAngleDegrees) || isNaN(launchHeight) || isNaN(gravity) || initialVelocity < 0 || gravity <= 0) { document.getElementById('horizontalRangeResult').textContent = 'Invalid Input'; document.getElementById('timeOfFlightResult').textContent = 'Invalid Input'; document.getElementById('maxHeightResult').textContent = 'Invalid Input'; return; } var launchAngleRadians = launchAngleDegrees * Math.PI / 180; var v0x = initialVelocity * Math.cos(launchAngleRadians); var v0y = initialVelocity * Math.sin(launchAngleRadians); var timeOfFlight; var horizontalRange; var maxHeight; if (initialVelocity === 0) { timeOfFlight = Math.sqrt(2 * launchHeight / gravity); horizontalRange = 0; maxHeight = launchHeight; } else { // Calculate Time of Flight using quadratic formula for vertical motion: y = h0 + v0y*t – 0.5*g*t^2 // We want y=0, so 0.5*g*t^2 – v0y*t – h0 = 0 // t = [-b +/- sqrt(b^2 – 4ac)] / 2a // a = 0.5*g, b = -v0y, c = -h0 var discriminant = (v0y * v0y) – 4 * (0.5 * gravity) * (-launchHeight); if (discriminant = 0 document.getElementById('horizontalRangeResult').textContent = 'Error'; document.getElementById('timeOfFlightResult').textContent = 'Error'; document.getElementById('maxHeightResult').textContent = 'Error'; return; } timeOfFlight = (v0y + Math.sqrt(discriminant)) / gravity; // Take the positive root horizontalRange = v0x * timeOfFlight; // Calculate Maximum Height if (v0y <= 0) { // If launched horizontally or downwards maxHeight = launchHeight; } else { // If launched upwards var timeToPeak = v0y / gravity; maxHeight = launchHeight + (v0y * timeToPeak) – (0.5 * gravity * timeToPeak * timeToPeak); } } document.getElementById('horizontalRangeResult').textContent = horizontalRange.toFixed(2); document.getElementById('timeOfFlightResult').textContent = timeOfFlight.toFixed(2); document.getElementById('maxHeightResult').textContent = maxHeight.toFixed(2); } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 8px; padding: 25px; max-width: 600px; margin: 30px auto; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08); color: #333; } .calculator-container h2 { text-align: center; color: #0056b3; margin-bottom: 20px; font-size: 1.8em; } .calculator-container p { margin-bottom: 15px; line-height: 1.6; color: #555; } .calc-input-group { margin-bottom: 15px; display: flex; flex-direction: column; } .calc-input-group label { margin-bottom: 7px; font-weight: bold; color: #333; font-size: 1em; } .calc-input-group input[type="number"] { padding: 10px; border: 1px solid #ccc; border-radius: 5px; font-size: 1em; width: 100%; box-sizing: border-box; } .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; } .calc-button:hover { background-color: #0056b3; } .calc-results { margin-top: 25px; padding: 15px; background-color: #e9f7ff; border: 1px solid #b3e0ff; border-radius: 8px; } .calc-results h3 { color: #0056b3; margin-top: 0; margin-bottom: 10px; font-size: 1.4em; } .calc-results p { font-size: 1.1em; margin-bottom: 8px; color: #333; } .calc-results span { font-weight: bold; color: #007bff; }

Understanding Projectile Motion and 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. This calculator helps you analyze such motion by determining key parameters: horizontal range, time of flight, and maximum height.

What is Projectile Motion?

A projectile is any object upon which the only force acting is gravity. This means air resistance is typically ignored in basic projectile motion calculations, or assumed to be negligible. The path followed by a projectile is called its trajectory, which is usually a parabolic curve.

The motion of a projectile can be broken down into two independent components:

• Horizontal Motion: Assumed to be constant velocity (no horizontal forces like air resistance).
• Vertical Motion: Subject to constant downward acceleration due to gravity.

Key Factors Affecting Projectile Range

Several factors influence how far, high, and long a projectile travels:

1. Initial Velocity (m/s): The speed at which the object is launched. A higher initial velocity generally leads to a greater range and height.
2. Launch Angle (degrees): The angle relative to the horizontal at which the object is launched. For ground-to-ground launches, an angle of 45 degrees typically yields the maximum horizontal range. However, if launched from a height, the optimal angle can be different.
3. Launch Height (m): The initial vertical position from which the projectile is launched. Launching from a greater height increases both the time of flight and the horizontal range.
4. Acceleration due to Gravity (m/s²): The constant acceleration pulling the object downwards. On Earth, this is approximately 9.81 m/s². Different celestial bodies would have different gravitational accelerations, affecting the projectile's path significantly.

How the Calculator Works (The Physics Behind It)

The calculator uses standard kinematic equations to determine the projectile's characteristics:

• Time of Flight: This is the total time the projectile spends in the air until it hits the ground. It's calculated by solving the vertical motion equation, considering the initial vertical velocity, launch height, and gravity. The formula used is derived from the quadratic equation for displacement: y = h₀ + v₀y*t - ½gt², where y=0 at impact.
• Horizontal Range: This is the total horizontal distance covered by the projectile. Since horizontal velocity is constant (v₀x), the range is simply the product of the horizontal velocity component and the total time of flight: Range = v₀x * Time of Flight.
• Maximum Height: This is the highest vertical point the projectile reaches during its trajectory. It's calculated by considering the point where the vertical velocity becomes zero. If the projectile is launched upwards, the formula is Max Height = h₀ + (v₀y² / 2g). If launched horizontally or downwards, the maximum height is simply the launch height.

Examples of Projectile Motion

• Ground-to-Ground Launch (e.g., a golf ball hit from a tee):
  • Initial Velocity: 60 m/s
  • Launch Angle: 45 degrees
  • Launch Height: 0 m
  • Gravity: 9.81 m/s²
  • Results: Horizontal Range ≈ 367.00 m, Time of Flight ≈ 8.66 s, Maximum Height ≈ 91.74 m
• Launch from a Height (e.g., a cannonball fired from a cliff):
  • Initial Velocity: 100 m/s
  • Launch Angle: 30 degrees
  • Launch Height: 50 m
  • Gravity: 9.81 m/s²
  • Results: Horizontal Range ≈ 970.00 m, Time of Flight ≈ 11.20 s, Maximum Height ≈ 177.47 m
• Horizontal Launch (e.g., an object pushed off a table):
  • Initial Velocity: 5 m/s
  • Launch Angle: 0 degrees
  • Launch Height: 1.5 m
  • Gravity: 9.81 m/s²
  • Results: Horizontal Range ≈ 2.77 m, Time of Flight ≈ 0.55 s, Maximum Height ≈ 1.50 m

How to Use This Calculator

Simply input the required values into the respective fields:

1. Enter the Initial Velocity in meters per second (m/s).
2. Enter the Launch Angle in degrees (0-90 for typical upward launches, but can be extended for downward angles).
3. Enter the Launch Height in meters (m).
4. Adjust the Acceleration due to Gravity if you're not on Earth or need a specific value (default is 9.81 m/s²).
5. Click the "Calculate Range" button to see the Horizontal Range, Time of Flight, and Maximum Height.

This tool is invaluable for students, engineers, and anyone interested in understanding the mechanics of objects in flight.