Smaller steps, better answers

A rocket sled starts from rest with an acceleration that grows, $a = 2 + 3t$ (SI units). The equations of uniform acceleration cannot be used. You will make the program step through time, as a computer game does, and see how the answer improves as the steps get smaller.

The idea. In a short time $\Delta t$, the velocity grows by $a\,\Delta t$ and the position by $v\,\Delta t$. Calculus gives the exact answer: $x = t^2 + \tfrac12 t^3$, which is $48\ \text{m}$ at $4\ \text{s}$.

Your tasks.

1. Inside the loop, update the velocity and then the position.
2. Work out the exact position at $4\ \text{s}$ from the formula.
3. Work out the error: how far the step-by-step answer is from the exact one.