Final answer:
The greatest horizontal displacement for a projectile launched and landing at the same height is achieved with an initial launch angle of 45 degrees, which equally balances the horizontal and vertical components of initial velocity.
Step-by-step explanation:
Understanding Projectile Motion for Maximum Horizontal Displacement
When a projectile is launched and lands at the same height, the horizontal displacement is maximized when the projectile is launched at an angle of 45 degrees to the horizontal. This is because the horizontal and vertical motions are independent of each other in projectile motion. The greatest horizontal displacement, or range, occurs when these two components of the initial velocity are equal, which happens at 45 degrees. At this angle, the initial velocity's horizontal and vertical components are balanced, allowing the projectile to stay in the air for the maximum amount of time while still moving forward at the greatest possible rate.
Therefore, to achieve the greatest horizontal displacement, the initial velocity should be directed at a 45-degree angle to the horizontal. This ensures that the projectile spends the longes