Final answer:
To show that a shell can strike any target within a certain surface, we analyze the equations for projectile motion and demonstrate that by adjusting the angle of projection θ, any target within the surface described by g²r² = Vo⁴ - 2gV² z can be reached.
Step-by-step explanation:
To show that a shell can strike any target within the surface given by g²r² = Vo⁴ - 2gV² z where z is the height of the target and r is its horizontal distance from the gun, let's consider the equations of motion for a projectile.
The horizontal range r of a projectile is given by r = (V₀² × sin(2θ)) / g, where V₀ is the initial velocity, θ is the angle of projection with respect to the horizontal, and g is the acceleration due to gravity.
The maximum height z reached by the projectile is given by z = (V₀² × sin²(θ)) / (2g).
By eliminating the angle θ and manipulating these equations, we can derive the equation g²r² = Vo⁴ - 2gV² z, demonstrating that any target within this surface is reachable by adjusting the angle of projection θ.