4.3. Planes II http://www.ck12.org
Figure C.4:Air flow behind a plane. Photo by NASA Langley Research Center.
The sausage’s mass is
msausage=density×volume=ρvtAs. (C. 3 )Let’s say the whole sausage is moving down with speedu, and figure out whatuneeds to be in order for the plane to
experience a lift force equal to its weight mg. The downward momentum of the sausage created in timetis
mass×velocity=msausageu=ρvtAsu. (C. 4 )And by Newton’s laws this must equal the momentum delivered by the plane’s weight in timet, namely,
mgt. (C. 5 )Rearranging this equation,
ρvtAsu=mgt, (C. 6 )we can solve for the required downward sausage speed,
u=
mg
ρvAs.
Interesting! The sausage speed is inversely related to the plane’s speedv. A slow-moving plane has to throw down
air harder than a fast-moving plane, because it encounters less air per unit time. That’s why landing planes, travelling
slowly, have to extend their flaps: so as to create a larger and steeper wing that deflects air more.
What’s the energetic cost of pushing the sausage down at the required speedu? The power required is