Report

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 1 2 1 ⋅ 2 =⋅ 2 where = 6.67 ⋅ 10−11 3 ⋅ 2 Acceleration of Mass 2 due to the Gravitational Force 1 ⋅ 2 =⋅ 2 where = 6.67 ⋅ 10−11 |2 | = 2 1 |2 | = 2 3 ⋅ 2 Differential Equations 2 1 = − ⋅ 2 ⋅ 2 2 1 = − ⋅ 2 ⋅ sin() 2 2 1 = − ⋅ ⋅ 2 2 1 = − ⋅ 2 + 2 2 1 = − ⋅ 2 ⋅ cos() 2 3 2 2 1 = − ⋅ ⋅ 2 2 1 = − ⋅ 2 + 2 3 2 Euler’s Method (linear) 0 = 1 0 = 0 2 (, ) = − ⋅ 2 0 = 0 1 2 + 3 2 2 2 = −1 + 2 −1 , −1 ⋅ Δ = −1 + −1 ⋅ Δ 0 = 0.0191 2 (, ) = − ⋅ 2 1 2 + 3 2 2 2 = −1 + 2 −1 , −1 ⋅ Δ = −1 + −1 ⋅ Δ Euler’s Method (quadratic) 0 = 1 0 = 0 2 (, ) = − ⋅ 2 0 = 0 1 2 + 3 2 2 2 = −1 + 2 −1 , −1 ⋅ Δ 1 2 = −1 + −1 ⋅ Δ + −1 , −1 ⋅ Δ 2 2 2 0 = 0.0191 2 (, ) = − ⋅ 2 1 2 + 3 2 2 2 = −1 + 2 −1 , −1 ⋅ Δ 1 2 = −1 + −1 ⋅ Δt + −1 , −1 ⋅ Δ 2 2 2 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. 2 () = cos ⋅ 365 2 () = sin ⋅ 365 () = 1.52 ⋅ sin 2 = 1.52 ⋅ cos ⋅ 687 2 ⋅ 687 What Happens? 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) Solving Equations 2 = 1.52 ⋅ cos ⋅ = −1.1317 687 = 1.52 ⋅ sin 2 ⋅ = 1.0145 687 ⇒ 263.578 − 181.5 = 82.078 = 1.52 ⋅ cos 2 ⋅ ( + 82.078) 687 = 1.52 ⋅ sin 2 ⋅ ( + 82.078) 687 = 263.578 Second Attempt Getting Home (Euler’s Method) 0 = ( ) 0 = ( ) 2 (, ) = − ⋅ 2 1 2 + 3 2 2 2 = −1 + 2 −1 , −1 ⋅ Δ 1 2 = −1 + ⋅ Δ + −1 , −1 ⋅ Δ 2 2 2 0 = 0.012 0 1.52 0 = 0.012 2 (, ) = − ⋅ 2 0 1.52 1 2 + 2 3 2 2 = −1 + 2 −1 , −1 ⋅ Δ 1 2 = −1 + ⋅ Δt + −1 , −1 ⋅ Δ 2 2 2 Getting Home (Short Stay) When to launch? • How long is the return trip? 183.25 days • Solve the equations. • Leave Mars after 715 days after launch There and Back Again Trip Length Breakdown Outward Journey 181.5 days Time On Mars 533.5 days Return Trip 183.25 days Neil Degrasse Tyson talking about a trip to Mars. 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?