http://www.ck12.org Chapter 4. Technical Chapters
Plift=kinetic energy of sausage
time(C. 7 )
=
1
t1
2
msausageu^2 (C. 8 )=
1
2 t
ρvtAs(
mg
ρvAs) 2
(C. 9 )
=
1
2
(mg)^2
ρvAs. (C. 10 )
The total power required to keep the plane going is the sum of the drag power and the lift power:
Ptotal=Pdrag+Plift (C. 11 )=1
2
cdρApv^3 +1
2
(mg)^2
ρvAs, (C. 12 )
whereApis the frontal area of the plane andcdis its drag coefficient (as in Chapter Cars II).
The fuel-efficiency of the plane, expressed as the energy per distance travelled, would be
energy
distance∣∣
∣
ideal=
Ptotal
v=
1
2
cdρApv^2 +1
2
(mg)^2
ρv^2 As, (C. 13 )
if the plane turned its fuel’s power into drag power and lift power perfectly efficiently. (Incidentally, another name
for “energy per distance travelled” is “force,” and we can recognize the two terms above as the drag force^12 cdρApv^2
and the lift-related force^12 (mg)
2
ρv^2 As. The sum is the force, or “thrust,” that specifies exactly how hard the engines have
to push.)
Real jet engines have an efficiency of aboutε=^13 , so the energy-per-distance of a plane travelling at speedvis
energy
distance=
1
ε(
1
2
cdρApv^2 +1
2
(mg)^2
ρv^2 As)
. (C. 14 )
This energy-per-distance is fairly complicated; but it simplifies greatly if we assume that the plane isdesignedto
fly at the speed thatminimizesthe energy-per-distance. The energy-per-distance, you see, has got a sweet-spot as a
function ofv(figure C.5). The sum of the two quantities^12 cdρApv^2 and^12 (mg)
2
ρv^2 As is smallest when the two quantities
are equal. This phenomenon is delightfully common in physics and engineering: two things that don’t obviously
haveto be equalareactually equal, or equal within a factor of 2.