Physics C Mechanics Calculator

Physics C Mechanics: Projectile Motion Calculator

This calculator helps you analyze the trajectory of a projectile launched at an angle, considering its initial velocity, launch angle, and initial height. It computes the total time the projectile spends in the air (Time of Flight), its maximum vertical displacement from the ground (Maximum Height), and the total horizontal distance it covers (Horizontal Range).

function calculateProjectileMotion() { var initialVelocity = parseFloat(document.getElementById('initialVelocity').value); var launchAngleDegrees = parseFloat(document.getElementById('launchAngle').value); var initialHeight = parseFloat(document.getElementById('initialHeight').value); var gravity = parseFloat(document.getElementById('gravity').value); // Input validation if (isNaN(initialVelocity) || initialVelocity < 0) { alert('Please enter a valid non-negative initial velocity.'); return; } if (isNaN(launchAngleDegrees) || launchAngleDegrees 90) { alert('Please enter a valid launch angle between 0 and 90 degrees.'); return; } if (isNaN(initialHeight) || initialHeight < 0) { alert('Please enter a valid non-negative initial height.'); return; } if (isNaN(gravity) || gravity <= 0) { alert('Please enter a valid positive value for acceleration due to gravity.'); return; } var launchAngleRadians = launchAngleDegrees * (Math.PI / 180); var Vx = initialVelocity * Math.cos(launchAngleRadians); var Vy0 = initialVelocity * Math.sin(launchAngleRadians); // Calculate Time of Flight (using quadratic formula for vertical motion: y = y0 + Vy0*t – 0.5*g*t^2) // 0 = initialHeight + Vy0*t – 0.5*g*t^2 // 0.5*g*t^2 – Vy0*t – initialHeight = 0 // t = [ -b +/- sqrt(b^2 – 4ac) ] / 2a // a = 0.5*g, b = -Vy0, c = -initialHeight var discriminant = (Vy0 * Vy0) – 4 * (0.5 * gravity) * (-initialHeight); if (discriminant < 0) { // This should not happen with valid inputs for projectile motion above ground alert('Error in calculation: Discriminant is negative. Check inputs.'); return; } var timeOfFlight = (Vy0 + Math.sqrt(discriminant)) / (gravity); // Only positive root for time // Calculate Maximum Height var timeToPeakFromLaunch = Vy0 / gravity; var heightAboveLaunch = (Vy0 * timeToPeakFromLaunch) – (0.5 * gravity * timeToPeakFromLaunch * timeToPeakFromLaunch); var maxHeight = initialHeight + heightAboveLaunch; // If launched purely horizontally (angle = 0) or vertically downwards (not covered by 0-90 range), // the peak height is just the initial height. if (launchAngleDegrees === 0 || Vy0 <= 0) { // Vy0 <= 0 handles cases where initial velocity is 0 or angle is 0 maxHeight = initialHeight; } // Calculate Horizontal Range var horizontalRange = Vx * timeOfFlight; // Display results document.getElementById('timeOfFlight').innerText = timeOfFlight.toFixed(3) + ' s'; document.getElementById('maxHeight').innerText = maxHeight.toFixed(3) + ' m'; document.getElementById('horizontalRange').innerText = horizontalRange.toFixed(3) + ' m'; }

Understanding Projectile Motion

Projectile motion is a fundamental concept in classical mechanics, describing the path an object takes when thrown or launched into the air, subject only to the force of gravity. In Physics C Mechanics, we typically analyze this motion under ideal conditions, neglecting air resistance and assuming a constant gravitational acceleration.

Key Concepts:

• Initial Velocity (V₀): The speed and direction at which the projectile begins its motion. It's often broken down into horizontal (Vₓ) and vertical (Vᵧ₀) components.
• Launch Angle (θ): The angle above the horizontal at which the projectile is launched. This angle significantly influences the trajectory.
• Initial Height (y₀): The vertical position from which the projectile is launched. A non-zero initial height affects the total time of flight and maximum height.
• Acceleration due to Gravity (g): The constant downward acceleration experienced by the projectile. On Earth, its approximate value is 9.81 m/s².

The Trajectory:

The motion of a projectile can be analyzed by separating it into independent horizontal and vertical components:

• Horizontal Motion: Assuming no air resistance, there is no horizontal force acting on the projectile. Therefore, the horizontal velocity (Vₓ) remains constant throughout the flight. The horizontal distance covered is simply Vₓ multiplied by the total time of flight.
• Vertical Motion: The vertical motion is influenced by gravity, causing a constant downward acceleration. The vertical velocity (Vᵧ) changes over time, decreasing as the projectile rises, becoming zero at the peak of its trajectory, and then increasing in the downward direction as it falls.

Calculator Outputs Explained:

• Time of Flight: This is the total duration the projectile remains in the air, from launch until it hits the ground (or returns to its initial height, depending on the problem context). Our calculator determines the time until it reaches ground level (y=0).
• Maximum Height: This is the highest vertical position the projectile reaches above the ground during its flight. It occurs when the vertical component of its velocity momentarily becomes zero.
• Horizontal Range: This is the total horizontal distance covered by the projectile from its launch point to where it lands.

Example Calculation:

Let's consider a scenario:

• Initial Velocity: 25 m/s
• Launch Angle: 45 degrees
• Initial Height: 0 m
• Acceleration due to Gravity: 9.81 m/s²

Using the calculator with these values, you would find:

• Time of Flight: Approximately 3.60 seconds
• Maximum Height: Approximately 15.94 meters
• Horizontal Range: Approximately 63.71 meters

This calculator provides a quick way to explore how changing initial conditions affects the flight path of a projectile, a crucial skill for Physics C Mechanics.