
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 V_{infinity}. 

What is the speed of an object in a hyperbolic orbit?

A naive approach is to add earth's escape velocity to V
_{infinity}.
This incorrect approach I call the Dr. Murphy Method.

The speed of an object in earth orbit can be determined by the
visviva equation:
v
^{2} = 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 visviva equation works for elliptical orbits, as well as hyperbolic orbits).

The visviva equation can tell us the hyperbola's speed
in terms of escape velocity (v
_{esc}) and hyperbolic excess velocity (v
_{inf})

v
^{2} = Gm(2/r  1/a)
v
^{2} = 2Gm/r  Gm/a
Now v
_{esc} = sqrt(2Gm/r)
And v
_{inf} = sqrt(Gm/a)
so
v
^{2} =
v
_{esc}^{2} + v
_{inf}^{2}

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 v
_{esc} and v
_{inf} as the legs:

Earth's escape velocity near the surface is about 11 km/s.
The v
_{inf }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(11
^{2} + 3
^{2}) km/s.
sqrt(11
^{2} + 3
^{2}) km/s = sqrt(121 + 9) km/s = sqrt(130) km/s which is about 11.5 km/s.