[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 parabolic motion of an object (like the launch of a projectile or the "flat" throw of a ball) in [b]real time[/b], neglecting air resistance, with a given [i]initial velocity[/i] [color=#cc0000][b]v[sub]0[/sub][/b][/color]. The animation [b]does not use formulas[/b] (neither equations, trigonometry, nor differential calculus) and only performs the necessary variations in the vectors that direct the motion.[br][list][*][color=#999999]Note: Strictly speaking, this motion is not parabolic but elliptical. For it to be truly parabolic, either the gravitational acceleration [b][/b][color=#6aa84f][b]g[/b][/color] must be exactly constant, or the initial velocity [b][color=#cc0000][b]v[sub]0[/sub][/b][/color][sub][/sub][/b] must be equal to the [i]escape velocity[/i] of the Earth (about 40,280 km/h). However, near the Earth's surface and for small velocities, we can assume (as we have been doing) that the magnitude of [color=#6aa84f][b]g[/b][/color][b][/b] remains constant (approximately 9.81 m/s²), so the elliptical arc is practically identical to the parabolic arc.[/color][/*][/list]We can consider parabolic motion as a combination of horizontal [b]Uniform Rectilinear Motio[/b]n and [b]vertical Launch Motion[/b], as each does not influence the other. This is the [color=#cc0000][i]principle of compound motion[/i], established by Galileo[/color] in 1638, which he used to demonstrate the parabolic shape of projectile motion: the horizontal and vertical components of a projectile's velocity are independent of each other. You can activate the "Show theoretical arc" checkbox to display the corresponding parabolic graph.[br][br]Also, note that if there is no friction, the horizontal component of the velocity vector remains constant at all times, equal to the initial horizontal velocity. As a result, the x-coordinate that the mass will reach when it hits the x-axis will be the same as if it had continued moving uniformly in a straight line, that is, it will be equal to the initial x-coordinate plus [color=#cc0000][b][b]v[sub]x[/sub][/b][/b][/color] [i]T[/i], where [i]T[/i] is the total time of travel.

[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=#999999]SetValue[/color](M, If(y(M + dt v) > 0, M + dt v, Intersect(Line(M, M + v), xAxis)))[br]StartAnimation(anima, y(M) > 0)[br][br][/color][color=#cc0000]# Registers M for the polyline trace[/color][color=#999999][br]SetValue(reg, Append(reg, M))[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]