[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 trajectory towards a target 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 or differential calculus); it only makes the necessary variations in the vectors that direct the motion.[br][br]At point A is a cannon and at point B is the target. You can move both points. This animation follows the same steps as the activity "Parabolic motion", except now the initial value [color=#cc0000][b]v[sub]0[/sub][/b][/color]​ of the velocity vector [color=#cc0000][b]v[/b][/color] is introduced in two stages: first its magnitude (which corresponds to the speed of the cannon used) and then its direction (that is, the cannon is aimed to hit the target B).[br][br]Move the green point, trying to estimate the correct direction, and then press the button [img]https://www.geogebra.org/resource/yxbcmb2f/CZJZaLQBirTUHVXU/material-yxbcmb2f.png[/img]. If you do not want to wait for the travel time, activate the [b]Theoretical arc[/b] box to see the trajectory that the projectile will follow.[br][br]Unless point B is positioned exactly at the limit of the cannon's range, there are either two possible firing angles or none. In the first case, you can see both trajectories by activating the [b]Arches for target[/b] box. In the second case, B is outside the cannon's range, and a message will appear to that effect.

[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]# Move M[/color][br][color=#999999][color=#999999]SetValue[/color](v, v + dt g)[br][/color][color=#999999][color=#999999]SetValue[/color](M, If(y(M + dt v)>0, M + dt v, Intersect(Line(M, M + v), xAxis)))[/color][color=#0000ff][br][br][color=#cc0000]# Adds the position M to the record for the polyline trace and controls the end[/color][color=#999999][br][color=#999999][color=#999999]SetValue[/color][/color](reg, Append(reg, M))[br][/color][color=#0000ff]StartAnimation(anima, y(M) > 0 ∧ abs(M − B) > abs(A − B)/100)[/color][br][br][/color][color=#999999][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]