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

5. Acceleration of Mass 2 due to the Gravitational Force

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. 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) Where should Mars be at launch?

12. Solving Equations

13. Second Attempt

14. Getting Home (Euler’s Method)

15. Getting Home (Short Stay)

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

17. There and Back Again

18. Trip Length Breakdown Outward Journey181.5 days Time On Mars533.5 days Return Trip183.25 days 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?