Collisions

Background

An elastic collision is one where the total kinetic energy of the system is conserved before and after the collision. In a perfectly elastic collision, objects will bounce off each other without any loss of energy. The equations governing the final velocities v_1f and v_2f of two objects with masses m_1 and m_2 and initial velocities v_1i and v_2i are:

v_1f = (m_1 - m_2)/(m_1 + m_2)*v_1i + 2m_2/(m_1 + m_2) * v_2i
v_2f = 2m_1/(m_1 + m_2)*v_1i + (m_2 - m_1)/(m_1 + m_2) * v_2i

Implementing Elastic Collisions

First, we'll need to disable the engine's built-in collision system by adding the following line at the beginning of your script:

set_collisions(false);

Now, we'll implement the update function that checks for collisions between objects and calculates their new velocities based on the elastic collision equations. To do this, we'll use the pairs helper function, which finds every combination of two bodies in the list.

let ELASTICITY = 1.0;

let update = |ids, bodies| {
    for pair in pairs(ids) {
        let a = pair[0];
        let b = pair[1];
        
        let m1 = a.get_mass();
        let m2 = b.get_mass();
        let mt = m1 + m2;

        let v1 = a.get_vel();
        let v2 = b.get_vel();

        let r1 = a.get_radius();
        let r2 = b.get_radius();
        let r = b.get_pos() - a.get_pos();
        // the radius after the next timestep
        let next_r = (b.get_pos() + v2) - (a.get_pos() + v1);

        let touching = r.length <= r1 + r2;
        let closer = next_r.length < r.length;
        if touching && closer {
            let v1f = ???;
            let v2f = ???;

            a.set_vel(v1f);
            b.set_vel(v2f);
        }
    }
};

fn pairs(ls) {
    let ret = [];
    for i in 0..(ls.length - 1) {
        for j in (i+1)..ls.length {
            ret.push([ls[i], ls[j]]);
        }
    }

    return ret;
}

To complete the implementation, you'll need to fill in the missing parts of the script by calculating the final velocities (v1f and v2f) using the elastic collision equations.

Exercises

After implementing the elastic collisions script, test your implementation with the following exercises:

1. Create two objects with different masses and observe how their final velocities are affected by the mass ratio. How does the mass ratio impact the energy transfer during the collision?

2. You had to update the v1f and v2f lines of code. What do you think each line of that section does? Can you explain what these conditions mean physically?

3. Experiment with varying values of elasticity. How does the elasticity affect the behavior of the collisions and the conservation of kinetic energy in the system?

4. Bonus: What happens when multiple objects collide at the exact same time? Why does this happen? The solution is to use a constraint solver, which is far outside the scope of this class.