### Exercise 1

Using initial positions on Aug 29, 2015, the first opposition occurs after 276 days. The second opposition is at Day 1107. The difference is 831 days, which is approximately 27.7 months.

### Exercise 2

The mass of Earth is 6×10 24 6×10 24 kg. Its radius is 6.4×10 6 6.4×10 6 m. The rocket is initially close to Earth so r 1=≈R E r 1=≈R E. Use the Energy Principle to find the speed of the rocket at r 2=≈5.6 R E r 2=≈5.6 R E. Define the system to be Earth and rocket. Assume there is no work or heat transferred to the system. Also, assume that the rocket’s interactions with the Sun and Mars are negligible during this change in altitude from Earth.

E 1=E 2(9)(9)E 1=E 2
U g r a v,1+K 1=U g r a v,2+K 2(10)(10)U g r a v,1+K 1=U g r a v,2+K 2
−G m M r 1+1 2 m v 2 1=−G m M r 2+1 2 m v 2 2(11)(11)−G m M r 1+1 2 m v 1 2=−G m M r 2+1 2 m v 2 2
−2 G M r 1+v 2 1=−2 G M r 2+v 2 2(12)(12)−2 G M r 1+v 1 2=−2 G M r 2+v 2 2
v 2 2=2 G M r 2+−2 G M r 1+v 2 1(13)(13)v 2 2=2 G M r 2+−2 G M r 1+v 1 2
v 2 2=2 G M(1 r 2−1 r 1)+v 2 1(14)(14)v 2 2=2 G M(1 r 2−1 r 1)+v 1 2
v 2=2 G M(1 r 2−1 r 1)+v 2 1−−−−−−−−−−−−−−−−−−√(15)(15)v 2=2 G M(1 r 2−1 r 1)+v 1 2
v 2≈5400 m/s(16)(16)v 2≈5400 m/s

### Exercise 3

The completed code uses initial positions on Aug 29, 2015. In this simulation, a possible launch is at Day 125, which is Jan. 01, 2016. Here is the starting point of the simulation.

The rocket arrives on June 25, 2016. This gives a travel time of 176 days which is approximately 6 months. The arrival is shown below (although “arrival” is defined as being within 200 Mars diameters.)

It is worth comparing this result to actual missions. Space.com has an article that gives the following travel time for Mars missions.

Mission | Travel Time
--- | ---
Mariner 4, the first spacecraft to go to Mars (1964 flyby) | 228 days
Mariner 6 (1969 flyby) | 155 days
Mariner 7 (1969 flyby) | 128 days
Mariner 9, the first spacecraft to orbit Mars (1971) | 168 days
Viking 1, the first U.S. craft to land on Mars (1975) | 304 days
Viking 2 Orbiter/Lander (1975) | 333 days
Mars Global Surveyor (1996) | 308 days
Mars Pathfinder (1996) | 212 days
Mars Odyssey (2001) | 200 days
Mars Express Orbiter (2003) | 201 days
Mars Reconnaissance Orbiter (2005) | 210 days
Mars Science Laboratory (2011) | 254 days

There are many solutions to this problem and it’s hard! Defining a large region of success for “reaching” Mars (like a radius of 500 Mars Diameters) is helpful.

### Exercise 4

It is best to collect data from the class on all of their successful orbits.