Final answer:
At the top of its path, a projectile has zero vertical velocity, reaching its maximum height, where it briefly stops before falling back down. The maximum height achieved is influenced solely by the projectile's initial vertical velocity component, independent of air resistance.
Step-by-step explanation:
A projectile launched up into the air has zero vertical velocity at the top of its path. At this highest point, known as the maximum height, the projectile's vertical component of velocity is zero because gravity has slowed it down from its initial upwards speed. When the projectile reaches this point, it will momentarily stop moving upwards before it starts to descend back to the ground.
Considering a fireworks display, where a shell is shot at an angle, the vertical component of initial velocity dictates the maximum height. If a projectile has a 67.6 m/s initial vertical velocity, for example, it would reach a maximum height of 233 m in the absence of air resistance. For a fireworks shell launched at 70.0 m/s at 75.0 degrees above the horizontal, the maximum height, time of flight, and horizontal displacement would need to be calculated using kinematic equations and trigonometric functions based on the components of the initial velocity.