Calculate key metrics for projectile motion based on initial conditions.
Enter values to see results
Understanding Projectile Motion
Projectile motion is a fundamental concept in physics that describes the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. This type of motion is characterized by a curved path, known as a trajectory. Examples include a thrown baseball, a fired cannonball, or even water from a fountain.
The motion can be analyzed by considering its horizontal (x) and vertical (y) components independently. In the absence of air resistance, the horizontal component of velocity remains constant, while the vertical component is affected by gravity.
The Math Behind the Calculator
This calculator uses standard kinematic equations to determine several key metrics of projectile motion:
Time of Flight: The total time the projectile spends in the air.
Maximum Height: The highest vertical position reached by the projectile.
Horizontal Range: The total horizontal distance covered by the projectile.
Formulas Used:
Let:
\(v_0\) be the initial velocity (m/s)
\(\theta\) be the launch angle (degrees)
\(g\) be the acceleration due to gravity (m/s²)
\(v_{0x} = v_0 \cos(\theta)\) be the initial horizontal velocity
\(v_{0y} = v_0 \sin(\theta)\) be the initial vertical velocity
1. Time to Reach Maximum Height (\(t_{peak}\)): At the maximum height, the vertical velocity is zero. Using \(v_y = v_{0y} – gt\):
\(0 = v_0 \sin(\theta) – gt_{peak}\)
\(t_{peak} = \frac{v_0 \sin(\theta)}{g}\)
2. Maximum Height (\(h_{max}\)): Using the equation \(y = v_{0y}t – \frac{1}{2}gt^2\) with \(t = t_{peak}\):
3. Total Time of Flight (\(T\)): Assuming the launch and landing heights are the same, the total time of flight is twice the time to reach maximum height:
Amateur athletes (e.g., golfers, baseball players) for understanding trajectory basics.
Enthusiasts in fields like archery or even game development for simulating object paths.
Note: This calculator assumes no air resistance for simplicity. In real-world scenarios, air resistance can significantly affect the trajectory, especially for lighter or faster-moving objects.
function calculateProjectileMotion() {
var initialVelocity = parseFloat(document.getElementById("initialVelocity").value);
var launchAngleDegrees = parseFloat(document.getElementById("launchAngle").value);
var gravity = parseFloat(document.getElementById("gravity").value);
var resultDiv = document.getElementById("result");
// Clear previous results and errors
resultDiv.innerHTML = "Enter values to see results";
resultDiv.style.backgroundColor = "#28a745"; // Reset to default green
// Validate inputs
if (isNaN(initialVelocity) || initialVelocity <= 0) {
resultDiv.innerHTML = "Please enter a valid initial velocity (positive number).";
resultDiv.style.backgroundColor = "#dc3545"; // Error red
return;
}
if (isNaN(launchAngleDegrees) || launchAngleDegrees 90) {
resultDiv.innerHTML = "Please enter a valid launch angle between 0 and 90 degrees.";
resultDiv.style.backgroundColor = "#dc3545"; // Error red
return;
}
if (isNaN(gravity) || gravity <= 0) {
resultDiv.innerHTML = "Please enter a valid acceleration due to gravity (positive number).";
resultDiv.style.backgroundColor = "#dc3545"; // Error red
return;
}
// Convert angle to radians for trigonometric functions
var launchAngleRadians = launchAngleDegrees * (Math.PI / 180);
// Calculate components
var initialVelocityX = initialVelocity * Math.cos(launchAngleRadians);
var initialVelocityY = initialVelocity * Math.sin(launchAngleRadians);
// Calculate metrics
// Time of flight (T) = 2 * (v0y / g)
var timeOfFlight = (2 * initialVelocityY) / gravity;
// Maximum height (h_max) = (v0y^2) / (2*g)
var maxHeigh = Math.pow(initialVelocityY, 2) / (2 * gravity);
// Horizontal range (R) = v0x * T
var horizontalRange = initialVelocityX * timeOfFlight;
// Display results
var resultHTML = "Results:" +
"Time of Flight: " + timeOfFlight.toFixed(2) + " s" +
"Maximum Height: " + maxHeigh.toFixed(2) + " m" +
"Horizontal Range: " + horizontalRange.toFixed(2) + " m";
resultDiv.innerHTML = resultHTML;
resultDiv.style.backgroundColor = "#28a745"; // Success green
}