 Above is a Hohmann transfer orbit from Earth to Mars. At perihelion (the orbit's closest point to the sun), the Hohmann orbit differ's from earth's orbit by about 3 kilometers/second. At aphelion (the orbit's farthest point from the sun), the Hohmann orbit differs from Mars orbit by about 2.5 kilometers/second.
- We zoom in on this Hohmann orbit.
- At a different scale, the path is well approximated by a hyperbola with regard to earth. The 3 km/s difference between the sun centered Hohmann ellipse and earth's orbit is the hyperbolic excess velocity for this hyperbola. Hyperbolic excess velocity is also known as Vinfinity.

-
What is the speed of an object in a hyperbolic orbit?
-
A naive approach is to add earth's escape velocity to Vinfinity.
This incorrect approach I call the Dr. Murphy Method.
-
The speed of an object in earth orbit can be determined by the
vis-viva equation:
v2 = Gm(2/r - 1/a)
-
Where
v is velocity
G is the gravitational constant
m is earth's mass
r is the object's distance from earth's center.
a is the semi major axis of the orbit.
-
(The vis-viva equation works for elliptical orbits, as well as hyperbolic orbits).
-
The vis-viva equation can tell us the hyperbola's speed
in terms of escape velocity (vesc) and hyperbolic excess velocity (vinf)
-
v2 = Gm(2/r - 1/a)
v2 = 2Gm/r - Gm/a
Now vesc = sqrt(2Gm/r)
And vinf = sqrt(-Gm/a)
so
v2 =  vesc2 + vinf2
-
Now the above looks a lot like the Pythagorean Theorem.
A memory device I use is to think of a hyperbola's velocity as
the hypotenuse of a right triangle with vesc and vinf as the legs: -
Earth's escape velocity near the surface is about 11 km/s.
The vinf for an insertion to a Mars Hohmann orbit is 3 km/s.
-
The Dr. Murphy Method would tell the needed delta V is (11 + 3) km/s or 14 km/s.
It is actually sqrt(112 + 32) km/s.
sqrt(112 + 32) km/s = sqrt(121 + 9) km/s = sqrt(130) km/s which is about 11.5 km/s.