How long does it take for an unpowered spacecraft to move from
Earth's orbit to Mars' orbit? There are many orbits possible. The orbit will
be a standard Keplerian elliptical orbit around the Sun. Use the
spacecraft orbit that has perihelion at the Earth’s orbit (which is very
close to a circle – assume it is a perfect circle) and aphelion at Mars’
orbit (also assume that this is a circle). (b) How fast does the spacecraft
need to be moving at the beginning of its trip (i.e. when it is at the Earth's
orbit)? The velocity v (at any point in any orbit!) is given by the vis viva

Question

anonymous · Answer

v^2 = GM ( 2/r - 1/a )

where M is the mass of the Sun, r is the distance from the Sun (which
varies continuously except in the special case of a circular orbit), and a is
the semi-major axis of the orbit. Show your calculations and a sketch of
the three orbits (the Earth's, the Mars', and the spacecraft's orbits). This is
referred to as the "least energy" transfer orbit and is the most efficient
way to move a spacecraft from one planet to another.