LerpIt

Some terminology we need to familirize ourselves first -

• State - the quantities that define the configuration of the system at that point of time. In this case (positions, velocities).
• Integrator - a numerical method that takes in current state, acceleration and time step size and estimates the state at the next time step.

It would look something like this -

$$State(t) -> Integration\ over \Delta t -> State(t+\Delta t)$$

Let's now take a look into the integration step. Let's say we start with state $s_t = (x_t, v_t)$. To calculate the next state $s_{t+1}$, the most intuitive way to do it is just apply newton's equations on state $s_t$ at time $t$.

$$a_t = k \times x_t$$ $$v_{t+1} = a_t * \Delta t$$ $$x_{t+1} = v_t * \Delta t$$

Okay, now onto the code.