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A Trip to Mars

A Trip to Mars. Douglas Marks NCSSM. The Problem. Find a flight path from the Earth to Mars. Approaches. Define an Archimedean spiral (Pre cal version) Use Kepler’s laws (Physics/Pre cal version) Define the forces due to gravity on the rocket (Calculus version). Gravitational Force.

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A Trip to Mars

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  1. A Trip to Mars Douglas Marks NCSSM

  2. The Problem • Find a flight path from the Earth to Mars.

  3. Approaches • Define an Archimedean spiral (Pre cal version) • Use Kepler’s laws (Physics/Pre cal version) • Define the forces due to gravity on the rocket (Calculus version)

  4. Gravitational Force where

  5. Acceleration of Mass 2 due to the Gravitational Force where

  6. Differential Equations

  7. Euler’s Method (linear)

  8. Euler’s Method (quadratic)

  9. Modeling the Planets’ Orbit • Model the orbits as circles. • Earth’s orbit has a radius of 1 AU and a period of 365 days. • Mar’s orbit has a radius of 1.52 AU and a period of 687 days.

  10. What Happens?

  11. Where should Mars be at launch? • How long does it take to get to Mars? 181.5 days • What is the space ships location when it intersects the Mars Orbit? (-1.1317, 1.0145)

  12. Solving Equations

  13. Second Attempt

  14. Getting Home (Euler’s Method)

  15. Getting Home (Short Stay)

  16. When to launch? • How long is the return trip? 183.25 days • Solve the equations. • Leave Mars after 715 days after launch

  17. There and Back Again

  18. Trip Length Breakdown Neil Degrasse Tyson talking about a trip to Mars.

  19. Other Questions • What is the space ship’s relative velocity to Mars when they meet? • How does including the Gravitational Force from the Earth and Mars affect the path? • How much shorter can you make the trip with better rockets?

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