Calculating G-Forces in Matter.js Loops

Calculating the g-forces experienced by a body moving through a loop in Matter.js involves measuring the instantaneous acceleration acting on the body and comparing it to the simulation's standard gravitational acceleration. This article covers the physical principles of centripetal acceleration, details how to extract motion data from the Matter.js physics engine, and provides a clear code implementation to calculate both positive and lateral g-forces in real time.

Understanding the Physics of a Loop

G-force is not an actual force, but a measurement of acceleration relative to free fall, expressed in units of \(g\) (\(1\text{ g} \approx 9.81\text{ m/s}^2\) in real-world physics). When an object travels through a circular loop, it experiences two primary accelerations:

Centripetal Acceleration (\(a_c\)): Directed toward the center of the curve, calculated as: \[a_c = \frac{v^2}{r}\] where \(v\) is the tangential speed and \(r\) is the radius of curvature.
Gravitational Acceleration (\(g\)): Directed downward according to the physics engine settings.

The total felt g-force (apparent weight divided by mass) depends on the position within the loop:

Bottom of the loop: The track must support the object against gravity and push it inward to turn. The accelerations add up: \[G = \frac{\frac{v^2}{r} + g}{g} = \frac{v^2}{r \cdot g} + 1\]
Top of the loop (inside): Gravity acts in the same direction as the centripetal acceleration: \[G = \frac{\frac{v^2}{r} - g}{g} = \frac{v^2}{r \cdot g} - 1\]

Step 1: Extract Velocity and Track Parameters in Matter.js

Matter.js operates using its own internal units (pixels per engine tick). To calculate g-forces accurately, identify:

Speed (\(v\)): Obtainable directly via body.speed or the magnitude of body.velocity.
Radius (\(r\)): The radius of your track's circular section in pixels.
Gravity (\(g\)): Matter.js defines gravity using engine.gravity.y scaled by engine.gravity.scale. The default effective downward acceleration is typically engine.gravity.y * engine.gravity.scale per tick squared.

Step 2: Calculate G-Force for a Fixed-Radius Loop

If the loop has a known center point \((x_c, y_c)\) and a fixed radius \(r\), the normal vector from the body to the center can be derived to find the orientation of the centripetal force.

const Matter = require('matter-js');

// Given: engine, body, loopCenter = { x: 400, y: 300 }, loopRadius = 150

Matter.Events.on(engine, 'afterUpdate', () => {
    const speed = body.speed; // pixels per tick
    const radius = 150;        // pixels
    
    // Centripetal acceleration (pixels / tick^2)
    const a_c = (speed * speed) / radius;

    // Matter.js default gravity acceleration magnitude
    const g = engine.gravity.y * engine.gravity.scale;

    // Angle of the body relative to the loop center (0 = bottom, PI = top)
    const dx = body.position.x - loopCenter.x;
    const dy = body.position.y - loopCenter.y;
    const angleFromBottom = Math.atan2(dx, -dy); // Inverted Y-axis in canvas

    // Component of gravity acting perpendicular to the track (inward/outward)
    const gravityComponent = g * Math.cos(angleFromBottom);

    // Normal acceleration felt by the body
    const normalAcceleration = a_c + gravityComponent;

    // Convert to G-units
    const gForce = normalAcceleration / g;

    console.log(`Current G-Force: ${gForce.toFixed(2)} G`);
});

Step 3: Calculate G-Force for Arbitrary or Dynamic Curves

If the track is an irregular spline or composite body rather than a perfect circle, you cannot rely on a fixed center point. Instead, calculate the total acceleration vector directly from the rate of change of velocity across frames:

Calculate Linear Acceleration (\(\vec{a}\)): \[\vec{a} = \frac{\vec{v}_{\text{current}} - \vec{v}_{\text{previous}}}{\Delta t}\]
Subtract Gravity (\(\vec{g}\)): Because an accelerometer measures proper acceleration (contact forces, excluding free-fall gravity), subtract the gravity vector: \[\vec{a}_{\text{felt}} = \vec{a} - \vec{g}\]
Normalize by \(g\): Divide the magnitude of \(\vec{a}_{\text{felt}}\) by the baseline gravity magnitude:

let previousVelocity = { x: 0, y: 0 };

Matter.Events.on(engine, 'afterUpdate', (event) => {
    const delta = event.source.timing.lastDelta / 1000; // Time step in seconds
    if (delta <= 0) return;

    // Change in velocity
    const dvX = (body.velocity.x - previousVelocity.x) / delta;
    const dvY = (body.velocity.y - previousVelocity.y) / delta;

    // Matter.js gravity vector in pixels/s^2 equivalent
    const gY = engine.gravity.y * engine.gravity.scale * (1000 / event.source.timing.lastDelta);

    // Subtract gravity from vertical acceleration to get felt acceleration
    const feltAx = dvX;
    const feltAy = dvY - gY;

    // Calculate magnitude of felt acceleration
    const feltAcceleration = Math.hypot(feltAx, feltAy);

    // Express relative to baseline gravity
    const gForce = feltAcceleration / Math.abs(gY);

    // Store current velocity for the next tick
    previousVelocity = { x: body.velocity.x, y: body.velocity.y };
});

Using this vector-based approach allows you to evaluate g-forces on clothoids, teardrop loops, banked curves, and freeform roller coaster tracks created with Matter.js constraint chains or vertex-based static bodies.