The Domain of the Time (intuitive kinematics)

Rafael Losada Liste, Oct 14, 2024

[color=#008486]This GeoGebra book is the English translation of the Spanish GeoGebra book "El dominio del tiempo" ([url]https://www.geogebra.org/m/nfjy7ug4[/url]), which contains the material from the lecture "Perpetuum mobile" delivered at the 5th National GeoGebra Day held in Cuenca, Spain, on October 4th and 5th, 2024. This content is also available in its (Spanish) original format used during the lecture at: [url]https://geogebra.es/pm/[/url] In this book, various [b]real-time[/b] kinematic experiments are simulated involving a mass at position [color=#1551b5]M[/color] with velocity [color=#c51414][b]v[/b][/color] subjected to an acceleration [color=#0a971e][b]g[/b][/color]. The animations [b]do not use formulas[/b] (neither equations, trigonometry, nor differential calculus); they only implement the necessary variations in the vectors that direct the motion. These variations essentially reduce to the execution of two instructions: SetValue([color=#c51414][b]v[/b][/color], [color=#c51414][b]v[/b][/color] + [i]dt[/i] [color=#0a971e][b]g[/b][/color])  SetValue([color=#1551b5]M[/color], [color=#1551b5]M[/color] + [i]dt[/i] [color=#c51414][b]v[/b][/color]) where [i]dt[/i] is a very small time interval (a few hundredths of a second). In other words, every short time period, [color=#c51414][b]v[/b][/color] changes by an amount equal to "a little bit of [color==#0a971e][b]g[/b][/color]" and the position [color=#1551b5]M[/color] of the mass shifts by "a little bit of [color=#c51414][b]v[/b][/color]". It is important to note that these sums are not numerical but vectorial, meaning a certain quantity is added in a certain direction. We will see that this idea, presented as variation tables by Richard Feynman in his famous book [i]The Feynman Lectures on Physics[/i] (Volume I, 9-7, [i]Planetary motions[/i]), can be perfectly adapted to animations made with GeoGebra, thanks to the ability to synchronize with the user's computer (or other device) clock. This allows for a fairly accurate simulation of various situations without resorting to higher mathematics. Rafael Losada Liste GeoGebra Institute of Cantabria[/color]

Table of Contents

  1. Introduction
    1. Summary
    2. Introduction
  2. Paths, Traces and Polyline
    1. GeoGebra Paths
    2. Time proportional to length (at constant speed)
    3. Length of a path and Polyline trace
  3. Capturing the Instant
    1. Virtual time (Geometric clock)
    2. Time fractions
    3. Time recording (STUDENT TEMPLATE)
  4. Uniform Motion
    1. Uniform Rectilinear Motion
    2. Doppler Effect and Mach 1
    3. Pursuit curve
    4. Mice problem
    5. Hathaway's pursuit curve
    6. UCM: Uniform Circular Motion
    7. UCM with polar coordinates
  5. Acceleration
    1. Centripetal acceleration in UCM
    2. Gravitational acceleration
    3. Conical Pendulum and UCM
    4. Formula 1
  6. Uniformly Accelerated Motion
    1. Free fall
    2. Apollo 15
    3. Free fall with marks and endpoint
    4. Relative error
    5. Free fall with horizontal v₀
    6. Vertical launch
    7. Parabolic motion
    8. Bounces of a ball
    9. Ballistic trajectory
    10. The Monkey and the Hunter
    11. Inclined plane
    12. Free fall on a sphere
  7. Simple Harmonic Motion
    1. SHM of a horizontal spring
    2. SHM and UCM: isochronism
    3. SHM and UCM: multiple projection
    4. Harmonograph (Lissajous figures)
    5. SHM of a vertical spring
  8. Pendulums
    1. Simple gravity pendulum
    2. Domino Effect
    3. Domino Chain Reaction
    4. Simple pendulum and SHM
    5. Seconds pendulum
    6. Metronome
    7. Double pendulum
    8. Zip Line
  9. The Cycloid
    1. A "cuenco" (bowl) in Cuenca
    2. Fall along a cycloid
    3. Tautochrone ("equal time")
    4. Cycloidal pendulum
    5. Brachistochrone ("minimum time")
    6. A Chronology of the Chronometer
  10. Time Scales
    1. The Fly and Trains problem
    2. Foucault pendulum
    3. Orbiting satellite in UCM
    4. Circular orbits
    5. Geostationary satellites
    6. Elliptical orbit
  11. Bibliography
    1. Bibliography

1.

Introduction

  • 1. Summary
  • 2. Introduction

1.1.

–

1.2.

–

2.

Paths, Traces and Polyline

  • 1. GeoGebra Paths
  • 2. Time proportional to length (at constant speed)
  • 3. Length of a path and Polyline trace

2.1.

–

2.2.

–

2.3.

–

3.

Capturing the Instant

  • 1. Virtual time (Geometric clock)
  • 2. Time fractions
  • 3. Time recording (STUDENT TEMPLATE)

3.1.

–

3.2.

–

3.3.

–

4.

Uniform Motion

  • 1. Uniform Rectilinear Motion
  • 2. Doppler Effect and Mach 1
  • 3. Pursuit curve
  • 4. Mice problem
  • 5. Hathaway's pursuit curve
  • 6. UCM: Uniform Circular Motion
  • 7. UCM with polar coordinates

4.1.

–

4.2.

–

4.3.

–

4.4.

–

4.5.

–

4.6.

–

4.7.

–

5.

Acceleration

  • 1. Centripetal acceleration in UCM
  • 2. Gravitational acceleration
  • 3. Conical Pendulum and UCM
  • 4. Formula 1

5.1.

–

5.2.

–

5.3.

–

5.4.

–

6.

Uniformly Accelerated Motion

  • 1. Free fall
  • 2. Apollo 15
  • 3. Free fall with marks and endpoint
  • 4. Relative error
  • 5. Free fall with horizontal v₀
  • 6. Vertical launch
  • 7. Parabolic motion
  • 8. Bounces of a ball
  • 9. Ballistic trajectory
  • 10. The Monkey and the Hunter
  • 11. Inclined plane
  • 12. Free fall on a sphere

6.1.

–

6.2.

–

6.3.

–

6.4.

–

6.5.

–

6.6.

–

6.7.

–

6.8.

–

6.9.

–

6.10.

–

6.11.

–

6.12.

–

7.

Simple Harmonic Motion

  • 1. SHM of a horizontal spring
  • 2. SHM and UCM: isochronism
  • 3. SHM and UCM: multiple projection
  • 4. Harmonograph (Lissajous figures)
  • 5. SHM of a vertical spring

7.1.

–

7.2.

–

7.3.

–

7.4.

–

7.5.

–

8.

Pendulums

  • 1. Simple gravity pendulum
  • 2. Domino Effect
  • 3. Domino Chain Reaction
  • 4. Simple pendulum and SHM
  • 5. Seconds pendulum
  • 6. Metronome
  • 7. Double pendulum
  • 8. Zip Line

8.1.

–

8.2.

–

8.3.

–

8.4.

–

8.5.

–

8.6.

–

8.7.

–

8.8.

–

9.

The Cycloid

  • 1. A "cuenco" (bowl) in Cuenca
  • 2. Fall along a cycloid
  • 3. Tautochrone ("equal time")
  • 4. Cycloidal pendulum
  • 5. Brachistochrone ("minimum time")
  • 6. A Chronology of the Chronometer

9.1.

–

9.2.

–

9.3.

–

9.4.

–

9.5.

–

9.6.

–

10.

Time Scales

  • 1. The Fly and Trains problem
  • 2. Foucault pendulum
  • 3. Orbiting satellite in UCM
  • 4. Circular orbits
  • 5. Geostationary satellites
  • 6. Elliptical orbit

10.1.

–

10.2.

–

10.3.

–

10.4.

–

10.5.

–

10.6.

–

11.

Bibliography

  • 1. Bibliography

11.1.

–

Information