Aside from the path of a projectile there are several more examples of using quadratic equations to solve problems here's a few I've used in the past:
Areas of shapes with variable length sides. Example a frame with 3" border would have an area of A=(x+6)(y+6) and if you related x is say twice y then you'd have a quadratic equation for the area A=(x+6)(x/2 +6)
Approximating any curve based on three points.
Trianglular numbers 1, 3, 6, 10, 15 ... governed by (n)(n+1)/2
My favorite example from structural engineering is Moment in a beam. Example a simple concrete beam with steel at the bottom has the bending capacity of M = As fy (d - As fy/(0.85 f'c b)) Typically you want to solve for the Area of steel As. Rearranging that equation plugging in values for the Moment (M), yield strength of the steel (fy), depth of the beam (d), width of the beam (b), and concrete compressive strenght (f'c), you would have a nice little quadratic with Area of steel (As) as your unknown. I doubt it could factor be factored though so the student has to resort to the quadratic formula ;)
Also, anytime you need the area under a line, calculus tells us that it's a quadratic so that opens up almost any topic that you can think up. I guess you could go the other way and say the instantaneous slope of a cubic is a quadratic too but that seems a little far a field from high school algebra :)