[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]In this construction, you can observe the uniform circular motion of three artificial satellites (M1, M2, and M3) around the Earth. You can select the altitude of each satellite within certain ranges. The first (M1, blue) is located in a [i]low orbit[/i] (at an altitude between 350 and 2000 km from Earth's surface). The second (M2, red) and the third (M3, yellow) are located in [i]medium orbits[/i]. You can also adjust the angle of each satellite's orbit (when the angle is 90° or 270°, the orbit will be [i]polar[/i]).[br][list][*][color=#999999]Note: Medium orbits range from 2000 km to up to the geostationary orbit, at over 35,000 km, but for better visualization, we've set a maximum of 4000 km. Every time you modify the altitude or angle of a satellite, the animation will restart.[/color][/*][/list]In the animation, the Earth takes 23.93'' to complete one full rotation, the same number of seconds as hours in reality. Therefore, it rotates 3600 times faster than in reality. To maintain the proportion of this 23.93'' period with the satellite periods, we have calculated the real period (using real distances and the Earth's real mass) for each satellite and divided it by 3600. Thus, if the panel shows a satellite with a 1.5-hour period (about 16 orbits per day), in the animation that satellite will complete one full orbit every 1.5 seconds.[br][br]Remember that [b][color=#cc0000]the radius of each orbit is scaled proportionally to the Earth's radius, and the velocity of the corresponding satellite is determined by that radius, as they must remain in uniform circular motion.[/color][/b][br][br]Note that the period increases (i.e., the angular velocity decreases) as the satellite's altitude increases.[br][list][*][color=#999999]Note: Specifically, the real period of each satellite is given by the formula [math]T=2\pi\sqrt{\frac{d^3}{G\cdot m_T}}[/math] seconds, where [i]d [/i]is the distance to Earth's center (in meters), [i]G[/i] is the universal gravitational constant, and [i]m[sub]T[/sub][/i] is Earth's mass (in kilograms).[/color][br][/*][/list][color=#999999][br]For better performance, it is recommended to download the GGB file.[/color]

[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]# Rotate the Earth [color=#cc0000](f radians)[/color] and move M1, M2 and M3[/color] [br][color=#999999][color=#999999]SetValue[/color](f, f + ω dt)[/color][color=#0000ff][br]SetValue(M1, Rotate(M1, ω1 dt, axis1))[br][color=#0000ff]SetValue[/color](M2, [color=#0000ff]Rotate[/color](M2, ω2 dt, axis2))[br][color=#0000ff]SetValue[/color](M3, [color=#0000ff]Rotate[/color](M3, ω3 dt, axis3))[/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]