Final answer:
The optimum angle at which to launch a projectile for maximum range is 45°. For a projectile's range to be zero, the angle should be either 0° or 90°. The perpendicular component of a box's weight on an inclined plane is maximized at an incline of 90°.
Step-by-step explanation:
The subject of the question relates to the optimal angle for projectile motion to achieve maximum range. In physics, it is established that the optimum angle for a projectile to cover the maximum distance is 45°. This is because the range of a projectile, which is the horizontal distance it travels, is dependent on both the vertical and horizontal components of the initial velocity. At 45 degrees, these components are equal, providing the most efficient combination for maximum distance. This principle is derived from kinematic equations and the assumption of no air resistance.
Moreover, for the case of the projectile's range being equal to zero, the angles would have to be either 0° or 90°, since these angles result in no horizontal movement (0°) or no horizontal velocity component (90°).
To answer a related question provided, the angle of incline at which the perpendicular component of a box's weight is maximized is at 90°, where all of the weight would be acting perpendicularly to the inclined plane.