Lesson 10 · Explicit Euler with acceleration

Explicit Euler with acceleration

Gravity is an acceleration - it changes velocity, which changes position. Today you write the naive integrator, explicit Euler, which moves the body first and then updates its velocity, so you can see next lesson why that order is a problem.

The goal

Add an acceleration to the body and integrate it with explicit Euler, moving position before updating velocity.

Start here - the target

TO DO

Scenario: One explicit Euler step under acceleration

Givena body at {0, 0} with velocity {0, 0} and acceleration {0, -10}

WhenIntegrateExplicit is called with dt = 1

Thenits position is {0, 0} (moved by the OLD velocity) and its velocity is {0, -10}

Anda body at {0, 0} with velocity {0, -5} and acceleration {0, -10} becomes position {0, -5}, velocity {0, -15} after one step

Background

Add an Acceleration field to Body. Acceleration is how forces enter the simulation - for now, gravity is a constant {0, -10} pulling downward. To integrate, you now update two quantities each step: velocity changes by acceleration * dt, and position changes by velocity * dt. The question is the order, and explicit (forward) Euler picks the simple one: move the position using the velocity you currently have, then update the velocity.

That ordering is the catch. Starting from rest, the position does not move at all on the first step (the old velocity was zero), even though the body is clearly accelerating - the motion always lags a step behind the velocity. It is the obvious way to write an integrator and it is what a naive implementation does, which is exactly why the next lesson contrasts it with a one-line change that behaves far better. Keep this method named IntegrateExplicit; you will keep it around as the counterexample.

Make it work

// add an Acceleration Vec2 field to Body first
func (b *Body) IntegrateExplicit(dt float64) {
  // explicit: move by the CURRENT velocity, THEN change velocity
  b.Position = Add(b.Position, Scale(b.Velocity, dt))
  b.Velocity = Add(b.Velocity, Scale(b.Acceleration, dt))
}

CheckpointDONE

A body accelerates using explicit Euler, updating position from the old velocity. Commit and stop here.