This one’s from Young and Freedman, and I pick it out because it’s both from the chapter I’m teaching and it’s a great conceptual problem as well. (I’ve modified it slightly.)
A shotgun fires a large number of pellets upward, with some pellets traveling vertically and some as much a 1 degree from the vertical. Ignore air resistance and assume the pellets leave the gun at 150 m/s. Within what radius from the point of firing will the pellets land? Will air resistance tend to increase or decrease this number?
The book gives the range equation directly even though it can be derived easily, so since we’re still in the “mostly conceptual” week here at BoF, I’m going to just give the equation. Wikipedia follows the derivation I’d use, for the curious.
It’s a short and sweet equation, but make sure to keep track of units and the distinction between degrees and radians.
The interesting part is of course the second part. For instance, air resistance will tend to slow the horizontal motion which should shrink the circle. But it will also limit the downward speed to some comparatively small terminal velocity, giving the pellets more time to come apart – which would tend to increase the circle. Think about this stuff long enough and the answer is clear – so fire away! Mathematically, anyway.