Reaching higher goals by means of a reflecting wall

A student realizes that a point particle that is able to rise at a given point P<inf>0</inf> at height H when launched vertically from the origin O of a Cartesian plane at a fixed initial speed V<inf>0</inf> cannot reach, by means of a direct shot from a small spring cannon, a point P positioned at the same height H and distance d from P<inf>0</inf>, with 0 < d ≤ R, where R is the maximum horizontal distance attainable with an optimum throw at an angle of 45°. However, the student realizes that, in the absence of air resistance, conservation of energy does not prevent this possibility. Therefore, in order to reach point P, the student uses a perfectly reflecting surface, opportunely inclined with respect to the horizontal. The full story of how this is done will be narrated in this article.