Calculate the trajectory and impact point of your projectile.
Results:
Maximum Height: – m
Horizontal Distance (Range): – m
Time of Flight: – s
Understanding the Kingshot Calculator
The Kingshot Calculator is a tool designed to simulate projectile motion, commonly known as a 'kingshot' in various contexts like sports, games, or physics demonstrations. It helps determine key aspects of a projectile's flight path based on its initial conditions and the gravitational force acting upon it.
The Physics Behind the Calculation
This calculator uses fundamental kinematic equations derived from classical mechanics. The core principle is breaking down the projectile's motion into independent horizontal (x) and vertical (y) components. Air resistance is typically ignored in these basic models for simplicity.
Inputs Explained:
Initial Velocity (v₀): The speed at which the projectile is launched. Measured in meters per second (m/s).
Launch Angle (θ): The angle relative to the horizontal at which the projectile is released. Measured in degrees.
Gravitational Acceleration (g): The acceleration due to gravity. On Earth, this is approximately 9.81 m/s², but can be adjusted for different celestial bodies or scenarios.
Initial Height (y₀): The starting vertical position of the projectile relative to a reference point (usually the ground). Measured in meters (m).
Formulas Used:
The calculator employs the following standard physics formulas:
1. Initial Velocity Components:
Horizontal velocity (v₀ₓ) = v₀ * cos(θ)
Vertical velocity (v₀y) = v₀ * sin(θ)
Note: The launch angle is converted from degrees to radians for trigonometric functions in the JavaScript.
2. Time of Flight (T):
This is the total time the projectile spends in the air. It's calculated by solving the vertical displacement equation for time when the final height is zero (or the ground level). The quadratic formula is used here:
y(t) = y₀ + v₀y*t - (1/2)*g*t²
Setting y(t) = 0 and solving for t yields:
T = (v₀y + sqrt(v₀y² + 2*g*y₀)) / g
(This formula assumes the projectile lands at or below its initial height. If landing above, a different quadratic solution may apply.)
3. Horizontal Distance (Range, R):
The total horizontal distance traveled by the projectile. Since horizontal velocity is constant (ignoring air resistance):
R = v₀ₓ * T
4. Maximum Height (H):
The highest vertical point the projectile reaches. This occurs when the vertical velocity becomes zero. Using the kinematic equation v² = u² + 2as:
0² = v₀y² - 2*g*H_up
H_up = v₀y² / (2*g)
The total maximum height from the ground is then:
H = H_up + y₀
(This calculation assumes the peak height is reached after launch and before landing.)
Use Cases:
This calculator is useful for:
Understanding basic projectile physics.
Simulating scenarios in video games or simulations.
Estimating trajectories in sports like golf, baseball, or archery.
Educational demonstrations of physics principles.
function calculateKingshot() {
var v0 = parseFloat(document.getElementById("initialVelocity").value);
var angleDegrees = parseFloat(document.getElementById("launchAngle").value);
var g = parseFloat(document.getElementById("gravity").value);
var y0 = parseFloat(document.getElementById("initialHeight").value);
var resultDiv = document.getElementById("result");
var maxHeightResultSpan = document.getElementById("maxHeightResult");
var rangeResultSpan = document.getElementById("rangeResult");
var timeOfFlightResultSpan = document.getElementById("timeOfFlightResult");
// Clear previous results
maxHeightResultSpan.textContent = "-";
rangeResultSpan.textContent = "-";
timeOfFlightResultSpan.textContent = "-";
// Input validation
if (isNaN(v0) || v0 < 0 ||
isNaN(angleDegrees) || angleDegrees 180 ||
isNaN(g) || g <= 0 ||
isNaN(y0) || y0 = 0) {
var t1 = (v0y + Math.sqrt(discriminant)) / g;
var t2 = (v0y – Math.sqrt(discriminant)) / g;
// We need the positive time value
timeOfFlight = Math.max(t1, t2);
if (timeOfFlight = 0 && timeToPeak 0 && timeOfFlight > 0 && timeToPeak > timeOfFlight) {
// If peak time is beyond flight time, calculate height at landing
var landingHeight = y0 + v0y * timeOfFlight – 0.5 * g * Math.pow(timeOfFlight, 2);
maxHeight = Math.max(y0, landingHeight); // Max is initial or landing height
} else {
// If launched downwards or horizontally, max height is initial height
maxHeight = y0;
}
// Display results, formatted to 2 decimal places
maxHeightResultSpan.textContent = maxHeight.toFixed(2);
rangeResultSpan.textContent = range.toFixed(2);
timeOfFlightResultSpan.textContent = timeOfFlight.toFixed(2);
}