Symbolab Calculator

Projectile Motion Calculator

This calculator determines key parameters for an object launched into projectile motion, such as maximum height, total time of flight, and horizontal range, given its initial velocity, launch angle, and initial height. It's a fundamental concept in physics often solved using tools like Symbolab.

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); if (isNaN(initialVelocity) || isNaN(launchAngleDegrees) || isNaN(initialHeight) || isNaN(gravity) || initialVelocity < 0 || gravity <= 0) { document.getElementById('resultOutput').innerHTML = 'Please enter valid positive numbers for all fields. Gravity must be greater than 0.'; return; } var launchAngleRadians = launchAngleDegrees * Math.PI / 180; var initialVerticalVelocity = initialVelocity * Math.sin(launchAngleRadians); var initialHorizontalVelocity = initialVelocity * Math.cos(launchAngleRadians); // Time to reach maximum height (from launch point) var timeToMaxHeight = initialVerticalVelocity / gravity; // Maximum height above launch point var maxHeightRelative = (initialVerticalVelocity * initialVerticalVelocity) / (2 * gravity); // Total maximum height above ground var totalMaxHeight = initialHeight + maxHeightRelative; // Total Time of Flight (solving quadratic equation: 0.5*g*t^2 – initialVerticalVelocity*t – initialHeight = 0) // Using quadratic formula: t = [-b ± sqrt(b^2 – 4ac)] / 2a // a = 0.5 * gravity // b = -initialVerticalVelocity // c = -initialHeight var discriminant = (initialVerticalVelocity * initialVerticalVelocity) + (2 * gravity * initialHeight); var totalTimeOfFlight = (initialVerticalVelocity + Math.sqrt(discriminant)) / gravity; // Horizontal Range var horizontalRange = initialHorizontalVelocity * totalTimeOfFlight; var resultsHtml = '

Results:

'; resultsHtml += 'Time to Max Height (from launch): ' + timeToMaxHeight.toFixed(2) + ' s'; resultsHtml += 'Maximum Height (above ground): ' + totalMaxHeight.toFixed(2) + ' m'; resultsHtml += 'Total Time of Flight: ' + totalTimeOfFlight.toFixed(2) + ' s'; resultsHtml += 'Horizontal Range: ' + horizontalRange.toFixed(2) + ' m'; document.getElementById('resultOutput').innerHTML = resultsHtml; }

Understanding Projectile Motion

Projectile motion is a form of motion experienced by an object or particle (a projectile) that is thrown near the Earth's surface and moves along a curved path under the action of gravity only. The only force of significance that acts on the object is gravity, which acts downwards, causing the object to accelerate vertically. Air resistance is typically ignored in basic calculations, simplifying the analysis.

Key Concepts:

• Initial Velocity (v₀): The speed and direction at which the projectile is launched. It's often broken down into horizontal (vₓ₀) and vertical (vᵧ₀) components.
• Launch Angle (θ): The angle at which the projectile is launched with respect to the horizontal ground.
• Initial Height (h₀): The vertical position from which the projectile begins its motion. If launched from the ground, this is 0.
• Acceleration due to Gravity (g): The constant acceleration experienced by the projectile in the vertical direction. On Earth, this is approximately 9.81 m/s².

Components of Motion:

Projectile motion is analyzed by separating it into two independent components: horizontal and vertical motion.

• Horizontal Motion: In the absence of 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 called the range.
• 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 and increasing as it falls.

Calculations Explained:

The calculator uses fundamental kinematic equations to determine the following:

• Time to Max Height: This is the time it takes for the vertical velocity to become zero, indicating the peak of the trajectory. It's calculated as vᵧ₀ / g.
• Maximum Height: The highest vertical position reached by the projectile. This is the initial height plus the vertical displacement from the launch point to the peak, calculated using h₀ + (vᵧ₀² / 2g).
• Total Time of Flight: The total duration the projectile spends in the air until it hits the ground (or returns to its initial height). This is derived by solving the quadratic equation for vertical displacement, considering when the height becomes zero.
• Horizontal Range: The total horizontal distance covered by the projectile from its launch point to where it lands. Since horizontal velocity is constant, it's simply vₓ₀ × Total Time of Flight.

Example Scenario:

Imagine a cannonball fired with an initial velocity of 50 m/s at a 30-degree angle from a cliff 10 meters high. Using the calculator with Earth's gravity (9.81 m/s²):

• Initial Velocity: 50 m/s
• Launch Angle: 30 degrees
• Initial Height: 10 m
• Gravity: 9.81 m/s²

The calculator would then output:

• Time to Max Height: Approximately 2.55 s
• Maximum Height: Approximately 42.62 m
• Total Time of Flight: Approximately 5.70 s
• Horizontal Range: Approximately 246.80 m

These calculations are crucial in fields like sports, engineering, and military applications to predict the trajectory of objects.