Understanding Projectile Motion with the Elimination Method
Projectile motion describes the curved path of an object launched into the air, subject only to the force of gravity. Common examples include a thrown ball, a fired bullet, or a kicked football. Understanding the trajectory, range, and maximum height of a projectile is crucial in fields like sports, ballistics, and engineering.
There are two primary methods to solve for projectile motion parameters: the parametric method (solving for time 't' first) and the elimination method. The elimination method directly relates the horizontal and vertical displacements without explicitly calculating time. This is particularly useful when you need to find the horizontal distance traveled to reach a specific vertical position.
The Physics Behind It
The motion of a projectile can be broken down into independent horizontal (x) and vertical (y) components:
Horizontal Motion: Assumes constant velocity (ignoring air resistance).
Initial horizontal velocity: v₀ₓ = v₀ * cos(θ)
Horizontal position at time t: x(t) = v₀ₓ * t = v₀ * cos(θ) * t
Vertical Motion: Assumes constant acceleration due to gravity (g ≈ 9.81 m/s²).
Initial vertical velocity: v₀ = v₀ * sin(θ)
Vertical position at time t: y(t) = y₀ + v₀ * t - (1/2) * g * t² = y₀ + v₀ * sin(θ) * t - (1/2) * g * t²
Where:
v₀ is the initial velocity magnitude.
θ is the launch angle with respect to the horizontal.
y₀ is the initial height.
g is the acceleration due to gravity (positive value, direction handled by the formula).
The Elimination Method Explained
The elimination method involves eliminating the time variable (t) from the equations of motion. We can express time from the horizontal motion equation:
From x = v₀ * cos(θ) * t, we get: t = x / (v₀ * cos(θ))
Now, substitute this expression for t into the vertical motion equation:
y = y₀ + x * (sin(θ) / cos(θ)) - (1/2) * g * x² / (v₀² * cos²(θ))
Using the trigonometric identity tan(θ) = sin(θ) / cos(θ) and sec²(θ) = 1 / cos²(θ), we can rewrite it as:
y = y₀ + x * tan(θ) - (g * x²) / (2 * v₀² * cos²(θ))
This equation represents the trajectory of the projectile. However, for our calculator, we want to find the horizontal distance (x) when the projectile reaches a specific target height (y). Rearranging the equation to solve for x can be complex as it becomes a quadratic equation in x. A more practical approach for this calculator is to solve for 't' at the target height and then find 'x'.
Calculator Logic: Finding Horizontal Distance (x) at Target Height (y)
The calculator first finds the time(s) 't' at which the projectile reaches the targetHeight (y) using the vertical motion equation, which is a quadratic equation in 't':
(1/2) * g * t² - v₀ * sin(θ) * t + (y - y₀) = 0
This is in the form At² + Bt + C = 0, where:
A = (1/2) * g
B = -v₀ * sin(θ)
C = (y - y₀)
The solutions for 't' are given by the quadratic formula: t = [-B ± sqrt(B² - 4AC)] / (2A)
If there are real solutions for 't' (meaning the projectile reaches the target height), the calculator will use the *largest positive* time value (representing the moment the projectile reaches the target height on its way down, or the later time if it passes through the height twice). This time 't' is then plugged into the horizontal position equation: x = v₀ * cos(θ) * t to find the horizontal distance.
Use Cases
Sports: Estimating where a ball will land on a field or court relative to its launch point, given specific heights.
Ballistics: Calculating the distance a projectile travels to reach a certain elevation.
Engineering: Designing trajectories for launching objects or understanding the path of falling debris.
Note: This calculator ignores air resistance for simplicity. In real-world scenarios, air resistance can significantly affect the actual trajectory.
function calculateProjectileRange() {
// Get input values
var initialVelocity = parseFloat(document.getElementById("initialVelocity").value);
var launchAngleDegrees = parseFloat(document.getElementById("launchAngle").value);
var initialHeight = parseFloat(document.getElementById("initialHeight").value);
var targetHeight = parseFloat(document.getElementById("targetHeight").value);
// Constants
var g = 9.81; // Acceleration due to gravity in m/s^2
// Validate inputs
if (isNaN(initialVelocity) || isNaN(launchAngleDegrees) || isNaN(initialHeight) || isNaN(targetHeight)) {
document.getElementById("calculatedDistance").innerText = "Invalid Input";
return;
}
if (initialVelocity 0″;
return;
}
// Convert launch angle from degrees to radians
var launchAngleRadians = launchAngleDegrees * (Math.PI / 180);
// Calculate initial velocity components
var v0x = initialVelocity * Math.cos(launchAngleRadians);
var v0y = initialVelocity * Math.sin(launchAngleRadians);
// Coefficients for the quadratic equation At² + Bt + C = 0 for vertical motion y(t) = y₀ + v₀y*t – 0.5*g*t²
// Rearranged: 0.5*g*t² – v₀y*t + (y – y₀) = 0
var A = 0.5 * g;
var B = -v0y;
var C = targetHeight – initialHeight;
// Calculate the discriminant (delta) of the quadratic equation
var discriminant = B * B – 4 * A * C;
var timeToTarget = -1; // Initialize time to an invalid value
if (discriminant >= 0) {
// There are real solutions for time
var t1 = (-B + Math.sqrt(discriminant)) / (2 * A);
var t2 = (-B – Math.sqrt(discriminant)) / (2 * A);
// We want the latest positive time the projectile reaches the target height
// If t1 and t2 are both positive, pick the larger one
// If only one is positive, pick that one
// If both are negative or zero, it means the target height is never reached (or only at t=0)
if (t1 > 0 && t2 > 0) {
timeToTarget = Math.max(t1, t2);
} else if (t1 > 0) {
timeToTarget = t1;
} else if (t2 > 0) {
timeToTarget = t2;
} else {
// Target height not reached after launch, or only at t=0
document.getElementById("calculatedDistance").innerText = "Target Height Not Reached";
return;
}
} else {
// Discriminant is negative, meaning the projectile never reaches the target height
document.getElementById("calculatedDistance").innerText = "Target Height Not Reached";
return;
}
// Calculate horizontal distance using the determined time
var horizontalDistance = v0x * timeToTarget;
// Display the result, rounded to 2 decimal places
document.getElementById("calculatedDistance").innerText = horizontalDistance.toFixed(2);
}