[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 vertical launch of an object upwards in [b]real time,[/b] neglecting air resistance, with a given [i]initial velocity[/i] [color=#cc0000][b]v[sub]y[/sub][/b][/color]​. The animation [b]does not use formulas[/b] (neither equations nor trigonometry nor differential calculus) and only performs the necessary variations in the vectors that direct the motion.[br][br]Remember that, after using the clock to establish small time intervals ([i]dt[/i]), the animation, at each step, [b]essentially uses these two instructions[/b]:[br][br] SetValue([color=#cc0000][b]v[/b][/color], [color=#cc0000][b]v[/b][/color] + [i]dt[/i] [b][color=#6aa84f]g[/color][/b]) [br] SetValue([color=#0000ff]M[/color], [color=#0000ff]M[/color] + [i]dt[/i] [b][color=#cc0000][b]v[/b][/color][/b])[br][br]This means that each time the value of the "anima" slider is updated, the velocity [color=#cc0000][b]v[/b][/color] increases slightly (by dtdtdt) in the direction and sense of [b][color=#6aa84f]g[/color][/b], and [color=#0000ff]M[/color] moves slightly in the direction and sense of [color=#cc0000][b]v[/b][/color]. As you can observe, the result fits quite well with reality.

[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=#0000ff]SetValue(M, If(y(M + dt v) > 0, M + dt v, (x(M), 0))[/color][color=#999999][br]StartAnimation(anima, y(M) > 0)[br][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]