Here are my four picks for this week
The first is an old but very good problem called "Rolling Down the Stairs". Here we have a cylinder rolling down a staircase which of course at first accelerates. Unlike the inclined plane, the staircase imparts some retarding momentum to the cylinder, and at some point this causes the cylinder to enter an oscillation about some mean terminal velocity. Finding this balance point requires some careful thinking, and in the end it is quite a rewarding problem.
The next is "Safe Three Pointer". Here we ask about the smartest angle at which a basketball player can launch a three point shot. He can launch at whatever angle and velocity he picks, but some are going to be less stable choices than others. Finding the optimal angle requires us to balance constraints. The final step can be approached in several ways, either a clever analytic approach or a numerical intersection.
The third choice is "Out for a Hike". Here we examine the tradeoff between packing a backpack full of food and the decreased performance of our hiking. If we pack a small amount of food, we won't be able to hike very far, but we'll be able to do that hiking very efficiently. If we pack a lot of food, we'll be able to hike longer distances, but will be quite inefficient for much of the hike. How far can we go on a given weight of food?
Finally, we end with the classic scenario of a rotating fluid. Whether it be a rotating bucket or stirred coffee, spinning liquids form a stable equilibrium surface. Can you find the right one?