Among the scenarios provided, the one that typically involves a quadratic equation to solve for height or trajectory is:
"A coach models the trajectory of a ball to find how high the player is kicking the ball."
Trajectory problems often involve quadratic functions since the motion of the ball can be modeled using the physics of projectile motion, which follows a parabolic path due to the influence of gravity. The other scenarios involve either exponential growth (savings account doubling) or linear growth (adding quarters to a piggy bank or the constant rise of a balloon).