As an exercise in ballistics, we use the example of a soccer ball approximated as a point object (i.e., not subject to rotations, and the "effects" of the ball related to it). We also neglect all dissipative sources (friction, deformation, etc.). A player kicks a free kick from a distance \(L=19\) \(\mathrm{m}\) from the opponent's goal (whose regulation height is \(h=2.44 \mathrm{~m}\) ). The speed, estimated with the Video Assistant Referee (VAR) systems with which the ball is kicked is \(v_{0}=18 \mathrm{~m} / \mathrm{s}\), at an angle \(\alpha=27^{\circ}\) with the plane of the playing field. Determine:
1. the expression of the maximum height \(z_{\max }\) reached by the balloon as a function of \(v_{0}\) and \(\alpha\). Determine the numerical value in the case presented.
2. The maximum range \(x_{\max }\) (again as a function of \(v_{0}\) and \(\alpha\) ), that is, how many meters away the ball would touch the plane of the soccer field again (neglecting the presence of any stands).
3. Determine whether the ball arrives at the goal position above or below the crossbar.
4. At what velocity \(v_{1}\) would the ball have to be kicked for it to arrive under the crossbar, keeping the angle \(\alpha\) unchanged.
5. In that case, would it pass the opponents' barrier (approximately \(2 \mathrm{~m}\) high) placed \(9.15 \mathrm{~m}\) away?