Make sure you understand the tutorial and the previous integration lab before doing this lab.

In the last lab, you implemented position update integration. As you observed, this was limited to fixed velocities. In this lab, you'll learn how to update the velocity so that we can simulate the fixed force of gravity on Earth.

Velocity-Update

The velocity update equation is:

v_f = v_0 + a * dt

Use this equation and the code from earlier to write a velocity update script. Fill in the question marks with the correct velocity update code.

set_integration(false);

add_body(#{pos: vec(0, 0), accel: vec(0.0, -0.1), radius: 50.0});

let update = |ids, bodies| {
    for ?? {
        let x_0 = id.get_pos();
        let v_0 = id.get_vel();
        let a_0 = id.get_accel();
        let dt = DT();

        let x_f = ??
        let v_f = ??

        id.set_pos(x_f);
        id.set_vel(??);
    }
}

If you've done it correctly, the ball should fall, accelerating down.

To check that this behavior is exactly what we expect, let's run an experiment. Fill in the ellipses below using your previous code to complete the script.

set_integration(false);

add_body(...);

let t = 0.0;
let update = |ids, bodies| {
    if (is_paused()) {
        return;
    }
    if (t >= 300.0) {
        set_paused(true);
    }

    for ... {
        ...
    }

    t += DT();
}

Don't worry if you don't understand the added code. Its purpose it to keep track of the time and pause the simulation after about 5 seconds.

Now, calculate the expected position and velocity of the ball by hand. Note that the simulation's time does not match up with the real world.

• Delta Time: 300 seconds

• Start Position: 0.0 meters

• Start Velocity: 0.0 m/s

• Start Acceleration: -0.1 m/s²

• End position: ??

• End Velocity: ??

Click the body in the simulation. Does its velocity and position match what you calculated?

The velocity should match up exactly, but the position will be different. This is because of computational integration error; we are essentially taking a Riemann sum with non-zero width rectangles. Numerical integration is an advanced topic that won't be covered in this class. If you're interested, read about it on Wikipedia.