6.055 / Art of approximation
42
the contact force from the track. In choosing the shape of the track, you affect the contact
force on the roller coaster, and thereby its acceleration, velocity, and position. There are an
inﬁnity of possible tracks, and we do not want to analyze each one to ﬁnd the forces and
acceleration. An invariant, energy, simpliﬁes the analysis. No matter what tricks the track
does, the kinetic plus potential energy
1
2
mv
+
mgh
2
is constant. The roller coaster starts with
v
=
0
and height
h
start
; it can never rise above
that height without violating the constancy of the energy. The invariant – the conserved
quantity – solves the problem in one step, avoiding an endless analysis of an inﬁnity of
possible paths.
The moral of this section is:
When there is change, look for what does not change.
6.2 Flight
How far can birds and planes ﬂy? The theory of ﬂight is difﬁcult and involves vortices,
Bernoulli’s principle, streamlines, and much else. This section offers an alternative ap-
proach: use conservation estimate the energy required to generate lift, then minimize the
lift and drag contributions to the energy to ﬁnd the minimum-energy way to make a trip.
6.2.1 Lift
Instead of wading into the swamp of vortices, study what does not change. In this case, the
vertical component of the plane’s momentum does not change while it cruises at constant
altitude.
Because of momentum conservation, a plane must deﬂect air downward. If it did not, grav-
ity would pull the plane into the ground. By deﬂecting air downwards – which generates
lift – the plane gets a compensating, upward recoil. Finding the necessary recoil leads to
ﬁnding the energy required to produce it.
Imagine a journey of distance
s
. I calculate the energy to produce lift in three steps:
1. How much air is deﬂected downward?
2. How fast must that mass be deﬂected downward in order to give the plane the needed
recoil?
3. How much kinetic energy is imparted to that air?
The plane is moving forward at speed
v
, and it deﬂects air over an area
L
2
where
L
is the
wingspan. Why this area
L
2
, rather than the cross-sectional area, is subtle. The reason is that
the wings disturb the ﬂow over a distance comparable to their span (the longest length). So
when the plane travels a distance
s
, it deﬂects a mass of air
m
air
∼
ρ
L
2
s
.