At what angle should a projectile be launched to achieve maximum range in a vacuum?

45 degrees. To achieve the maximum range for a projectile launched in a vacuum, the optimal angle is 45 degrees relative to the ground. This angle provides the ideal balance between the horizontal and vertical components of the initial velocity, allowing the projectile to travel the farthest distance before returning to the same height from which it was launched. At angles greater or less than 45 degrees, either the horizontal or vertical component of velocity will be reduced, resulting in a shorter range. This principle can be mathematically demonstrated using the equations of projectile motion, which show that the maximum range is achieved when the launch angle is 45 degrees.