Bili Rate of Rise Calculator

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Bili Rate of Rise Calculator

Understanding the Bili Rate of Rise Calculator

The Bili Rate of Rise Calculator is a tool designed to help understand the vertical component of an object's velocity over time during projectile motion. This calculator is particularly useful in physics and engineering contexts where analyzing the trajectory of an object is crucial.

What is Projectile Motion?

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity (neglecting air resistance). When an object is launched at an angle, its motion can be broken down into two independent components: horizontal and vertical.

The Vertical Component and Rate of Rise

The vertical component of the velocity determines how fast the object is moving upwards or downwards. The "rate of rise" specifically refers to the change in this vertical velocity over time. Initially, as an object is launched upwards, its vertical velocity decreases due to gravity. It reaches its peak when the vertical velocity momentarily becomes zero, and then it starts to decrease as the object falls back down.

How the Calculator Works

The Bili Rate of Rise Calculator takes three key inputs:

  • Initial Velocity (m/s): This is the speed at which the object is launched.
  • Launch Angle (degrees): This is the angle relative to the horizontal at which the object is projected.
  • Time Step (s): This is a small increment of time used to calculate the velocity at specific points in the object's trajectory.

The calculator uses the principles of kinematics to determine the vertical velocity at each time step. The core formulas involved are:

  1. Vertical component of initial velocity (v₀y): `v₀y = Initial Velocity * sin(Launch Angle)`
  2. Vertical velocity at time t (vy(t)): `vy(t) = v₀y – g * t` (where 'g' is the acceleration due to gravity, approximately 9.81 m/s²)

The "rate of rise" at any given time step can be thought of as the instantaneous vertical velocity. This calculator will show you the vertical velocity at different points in time, illustrating how it changes.

Example Calculation:

Let's say an object is launched with an Initial Velocity of 20 m/s at a Launch Angle of 30 degrees. We want to see the rate of rise at intervals of 0.5 seconds.

  • Initial Velocity: 20 m/s
  • Launch Angle: 30 degrees
  • Time Step: 0.5 s

First, we calculate the initial vertical velocity:

`v₀y = 20 m/s * sin(30 degrees) = 20 m/s * 0.5 = 10 m/s`

Now, let's calculate the vertical velocity at a few time steps:

  • At t = 0.5 s: `vy(0.5) = 10 m/s – 9.81 m/s² * 0.5 s = 10 – 4.905 = 5.095 m/s`
  • At t = 1.0 s: `vy(1.0) = 10 m/s – 9.81 m/s² * 1.0 s = 10 – 9.81 = 0.19 m/s`
  • At t = 1.1 s: `vy(1.1) = 10 m/s – 9.81 m/s² * 1.1 s = 10 – 10.791 = -0.791 m/s` (The object is now falling)

This calculator will output these values, showing how the vertical velocity (rate of rise) changes and eventually becomes negative as the object descends.

var g = 9.81; // Acceleration due to gravity in m/s^2 function calculateBiliRateOfRise() { var initialVelocity = parseFloat(document.getElementById("initialVelocity").value); var launchAngleDegrees = parseFloat(document.getElementById("launchAngle").value); var timeStep = parseFloat(document.getElementById("timeStep").value); var resultDiv = document.getElementById("result"); resultDiv.innerHTML = ""; // Clear previous results if (isNaN(initialVelocity) || isNaN(launchAngleDegrees) || isNaN(timeStep) || initialVelocity < 0 || timeStep <= 0) { resultDiv.innerHTML = "Please enter valid positive numbers for all fields. Time step must be greater than 0."; return; } if (launchAngleDegrees 90) { resultDiv.innerHTML = "Launch angle must be between 0 and 90 degrees."; return; } // Convert launch angle from degrees to radians var launchAngleRadians = launchAngleDegrees * Math.PI / 180; // Calculate initial vertical velocity var initialVerticalVelocity = initialVelocity * Math.sin(launchAngleRadians); // Calculate and display vertical velocity for a few time steps var outputHtml = "

Vertical Velocity Over Time:

    "; var currentTime = 0; var maxTimeToShow = 10; // Limit to prevent infinite loops or excessive output // Calculate until vertical velocity becomes significantly negative or max time is reached while (currentTime <= maxTimeToShow) { var verticalVelocity = initialVerticalVelocity – g * currentTime; // Stop if the object has clearly started falling significantly and is far past its peak if (verticalVelocity (2 * initialVerticalVelocity / g)) { // Heuristic to stop after peak and descent break; } outputHtml += "
  • At time " + currentTime.toFixed(2) + " s: " + verticalVelocity.toFixed(2) + " m/s
    • "; currentTime += timeStep; // Break if time step is so large that it skips over the peak and goes directly to negative if (currentTime > maxTimeToShow && initialVerticalVelocity > 0) break; } outputHtml += "
"; resultDiv.innerHTML = outputHtml; }

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