[color=#0000ff][i][color=#0000ff][i][color=#999999]This activity belongs to the GeoGebra book [url=https://www.geogebra.org/m/mes4bgft]The Domain of the Time[/url].[/color][/i][/color][/i][/color][br][br]This animation simulates the thought experiment of the monkey and the hunter in [b]real-time[/b], disregarding air resistance, of a projectile with a given [i]initial velocity[/i] [color=#cc0000][b]v[sub]0[/sub][/b][/color]. The animation [b]does not use formulas[/b] (no equations, trigonometry, or differential calculus); it only makes the necessary variations in the vectors that direct the motion.[br][br]At point A is the hunter and at point B is the monkey. You can move both points. To better visualize the animation (the speed of the ammunition is too high compared to the distance to be covered), we have replaced the bullet or pellets of the hunter's weapon with a rubber ball launched from a [i]slingshot[/i], which clearly benefits the monkey (you’re welcome).[br][br]This thought experiment places the hunter aiming the slingshot [b]directly [/b]at the monkey. At the same instant he fires, the monkey drops from the tree. Bad decision. As we have seen, by breaking down the motion of the ball into two motions (horizontal and vertical), the ball does not follow a straight-line motion because [i]at every instant it falls vertically[/i] just like the monkey. In fact, if the monkey watches the ball at all times, he will see it approaching him directly, in motion (that he observes) completely linear, since they both fall at the same time. So when the necessary time has passed to cover the horizontal distance between the hunter and the monkey, the ball and the monkey will meet.[br][br]Therefore, in reality, to hit the monkey without him leaving his initial position, the hunter must either aim above him or, as is usually the case, adjust the firing angle by using the [i]sight adjustment[/i] of the weapon.

[b]SCRIPT FOR SLIDER anima[/b][br][br][color=#cc0000][color=#cc0000]# Calculate the elapsed seconds dt; add one second if t1(1) < tt[/color][/color][br][color=#999999]SetValue(tt, t1(1))[br]SetValue(t1, First(GetTime(), 3))[br]SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)[/color][br][br][color=#cc0000]# [color=#cc0000]Move M and N[/color][/color][br][color=#999999][color=#999999]SetValue[/color](v, v + dt g)[/color][br][color=#0000ff]SetValue(vN, vN + dt g)[/color][br][color=#999999][color=#999999]SetValue[/color](M, M + dt v)[/color][br][color=#0000ff][color=#0000ff]SetValue[/color](N, N + dt vN)[/color][br][br][color=#cc0000]# Adds the position M to the record for the polyline trace and controls the end[/color][br][color=#999999][color=#999999]SetValue[/color](reg, Append(reg, M))[/color][color=#0000ff][br]StartAnimation(anima, y(M) > 0 ∧ abs(M − N) > abs(A − B)/50)[/color][color=#999999][br][br][br][br][br][color=#999999][color=#999999][color=#0000ff][color=#0000ff][color=#999999][color=#999999]Author of the activity and GeoGebra construction: [/color][/color][/color][color=#0000ff][color=#999999][color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color][/color][/color][/color][/color][/color][/color]