How friction changes as you push

Push a $3.2\ \text{kg}$ box harder and harder. Friction grows with your push, then gives way, and drops as the box starts to slide. You will make the program draw the whole story and find the push that gives a chosen acceleration.

The idea. Below the limit $\mu_s mg$, static friction equals the push and the box stays put. Above it, kinetic friction $\mu_k mg$ acts, and

Your tasks.

1. Work out the limit of static friction.
2. Work out the friction for every push in P.
3. Find the push that gives the sliding box an acceleration of $2.0\ \text{m/s}^2$.