Final answer:
The range of a projectile is zero when launched at 0° or 90°, and the optimal launch angle for the maximum range on level ground is 45°. Additionally, vectors of tangential velocity and centripetal force form a 90° angle, and the maximum perpendicular weight component on an inclined plane occurs at a 90° incline.
Step-by-step explanation:
Projectile Motion and Optimum Launch Angles:
When considering projectile motion, different angles of elevation lead to different trajectories and characteristics of a projectile's flight. For instance, when a projectile is launched at a 0° elevation, it moves horizontally without any vertical motion component and thus, has a range of zero. On the other hand, if it is launched directly upwards at 90°, it will have maximum height but again a range of zero as it will come back down to the launch point.
The optimum launch angle for a projectile to achieve maximal horizontal distance, assuming no air resistance and that it is launched from and lands on the same level, is 45°. This is because at this angle, the projectile's initial velocity is equally distributed between the horizontal and vertical components, maximizing the range.
Furthermore, when the angle between the vectors of tangential velocity and centripetal force is measured, they are perpendicular to each other, leading to a 90° angle. Additionally, when considering an object on an inclined plane, the perpendicular component of the box's weight is at its maximum when the angle of incline is 90°, which makes the plane vertical.