Once upon a time I was playing catch with a small child. The child was just learning to catch a ball, and so I drew closer to her because she dropped the last one I threw. However, the dear child was habitually teased by an older sibling for being a “baby” and she was therefore keen to appear grown up. When I took a step towards her, she took a step back to avoid ridicule from her sibling, keeping the distance between us constant.
What to do?
And thus was born a calculus/physics problem. For a fixed range, determine the initial velocity of a projectile such that the target is hit and yet the final speed is minimized. (You want to minimize the final speed to make the ball as easy as possible to catch.)
Note that there are two adjustable parameters in the projectile’s initial velocity; you can take them to be the two components of the velocity, or you can take them to be the initial speed and the initial angle of inclination.
(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)