2021-USAPhO_plus-01
The Jet Stream
USAPhO · 2021 · Mekanika
The Jet Stream
The jet stream is an eastward wind current that moves over the continental United States at an
altitude of 23, 000 to 35, 000 feet (the range of typical cruising altitudes of commercial airlines).
This strong current affects flight times significantly: flights traveling eastward fly significantly faster
than flights traveling westward.
This problem consists of two independent parts. In the first part, you will consider a simple
model for airplane flight. In the second part, you will determine the jet stream speed on a fictitious
planet called Orb.
1. The power that a plane expends is used both to combat drag and to generate lift. Throughout
this part of the problem, you may assume that the plane travels with horizontal velocity vrel
relative to the air, the density of air is air , the mass of the plane is m, and the cross-sectional
area of the plane is Acs .
(a) The drag force on an airplane is given by
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Fdrag = cd air Acs |vrel |vrel ,
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where cd is the drag coefficient (which depends on the shape of the plane). Write an expression
for the power expended by the airplane to combat the drag force from the air.
(b) Airplanes generate lift by deflecting air downward.
i. Estimate the air mass per unit time which is deflected by the wings of the plain.
ii. Estimate the power expended by the plane for lift.
(c) Estimate the speed at which an airplane flies relative to the air by minimizing the power
expended by the plane. To get a numeric answer, you may use the following parameters:
mplane ∼ 8 ,
air ∼ ,
cd ∼ 102 ,
Acs ∼ .
Next, we estimate the jet stream speed using flight times. Because the jet stream speed on
Earth varies greatly with location, time of year, and climate effects (such as El Niño and La Niña),
you will instead consider the fictitious planet Orb, where the jet stream is eastward and uniform
in the region of interest. At the end of the problem is a map of the region, whose area is much
smaller than the surface area of Orb (i.e., you can neglect the curvature of Orb).
2. At cruising altitude, we asume all airplanes travel at a fixed speed vrel relative to the air. (This is
not necessarily the same as your answer to 1(c), which was just a rough estimate.) Additionally,
we assume that flights occur in three stages – (1) taxi and takeoff, (2) flight at cruising altitude,
(3) landing and taxi – and that stages (1) and (3) take a fixed total time t0 for every flight.
(a) Suppose a plane, at cruising altitude, is traveling at an angle away from due east relative
to the ground. What is the speed of the plane relative to the ground? Give your answer in
terms of vrel , , and vw , the speed of the jet stream relative to the Earth’s surface.
(b) If the plane travels a distance D, what is the total travel time t, including taxi, takeoff, and
landing?
Copyright ©2021 American Association of Physics Teachers
2021 USAPhO+
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(c) Below, we present some data on airplane flights on Planet Orb. Each of the flight times shown
below has an independent uncertainty of t = 5 min. From the data and the map, determine
vw and vrel , giving your answers in km per hour with uncertainties. Indicate clearly what
two quantities you are plotting against each other on each graph that you plot.
Flight-time data (each time has independent uncertainty min):
| Departure city | Arrival city | Time (min) |
|---|---|---:|
| Noethersville | Rubinstead | 185 |
| Rubinstead | Noethersville | 286 |
| Curieton | Franklinport | 107 |
| Franklinport | Curieton | 244 |
| Planck Town | Maxwellbury | 143 |
| Maxwellbury | Planck Town | 256 |
| Rubinstead | Boltzmannburg | 92 |
| Boltzmannburg | Rubinstead | 190 |
| Einsteinopolis | Maxwellbury | 160 |
| Maxwellbury | Einsteinopolis | 384 |
| Planck Town | Franklinport | 128 |
| Franklinport | Planck Town | 266 |
| Einsteinopolis | Franklinport | 188 |
| Franklinport | Einsteinopolis | 431 |
| Boltzmannburg | Maxwellbury | 135 |
| Maxwellbury | Boltzmannburg | 150 |
| Noethersville | Einsteinopolis | 68 |
| Einsteinopolis | Noethersville | 74 |
| Franklinport | Newton City | 144 |
| Newton City | Franklinport | 129 |
| Curieton | Rubinstead | 186 |
| Rubinstead | Curieton | 175 |
| Planck Town | Curieton | 95 |
| Curieton | Planck Town | 102 |
| Planck Town | Rubinstead | 249 |
| Rubinstead | Planck Town | 250 |
Copyright ©2021 American Association of Physics Teachers
W
Curieton
Planck Town
E
Rubinstead
Noethersville
Einsteinopolis
S
N
Franklinport
Maxwellbury
Newton City
Boltzmannburg
2021 USAPhO+
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Copyright ©2021 American Association of Physics Teachers
2021 USAPhO+
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Question 2
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