Transcript I Believe I can fly

What's a real world problem?
“How Does a Airplane Fly?”
Separating
point
Lower side
End point
 In these three pictures, it demonstrates how the wind molecules flow
as it comes in contact with the wing blade.
 The closer the lines are, the closer the air molecules are, which forms
greater pressure. (Lower side has closer lines)
 Note: Pressure difference creates net force (Pressure=Force/Area)
 Net force at the bottom of the plane cancels and the gravitational
force and lifts up the plane.
Once we understood the concept, we can start to do a
real world question!
How to calculate the minimum pressure difference that ‘s needed
to lift up a plane?
Application of Physics and
Calculus
 In order to precisely find out the minimum pressure difference (the minimum net force ) to lift up the plane,
we must incorporate:
 Bernoulli Equation

- Used to establish the relationship between velocity of air molecules and air pressure.
 Integration & Arc Length Formula

Used to calculate the total distance that the air molecules traveled (Both the molecules on
the upper side and the molecules on the lower side)
 Mean Value Theorem(Average Value of Y)
 Used to evaluate the average velocity of the flowing air molecules.
 Trigonometry
 Calculation needed
 Related rate of change

Used to find out the time it takes for air molecules to travel from the separating point to the
ending point respects to different angles of attack

Angle of attack- the angle between the lifting body's reference line and the oncoming flow.
 Fluid Mechanics
 Finding average pressure that the body of the plane applies on the air below it.
General Forms of Needed
Formulas
 Bernoulli Equation
𝟏
𝟏
 𝑷𝟏 + 𝟐 𝝆𝒗𝟏 𝟐 + (𝝆𝒈𝒉𝟏 ) = 𝑷𝟐 + 𝟐 𝝆𝒗𝟐 𝟐 + (𝝆𝒈𝒉𝟐 )
 We put ( ) on 𝝆𝒈𝒉 because height difference will have negligible impact
on the pressure in atmosphere.
 Integration & Arc Length Formula

𝒃
𝒇
𝒂

𝒃
𝒂
𝒕 𝒅𝒕
𝟏+
𝒅𝒚 𝟐
𝒅𝒙
𝒅𝒙
 Mean Value Theorem(Average Value of Y)

𝒃
𝟏
𝒇
𝒃−𝒂 𝒂
𝒕 𝒅𝒕
 Related rate of change
 𝒇 𝒕 = 𝒖 =>𝒇′ 𝒕 = 𝒅𝒖/𝒅𝒕
 Fluid Mechanics
𝑭
 P=𝑨
Choice of Plane:
Concorde
General
Information:
Useful load:
245,000 lb
(111,130 kg)
Empty mass:
173,500 lb
(78,700 kg)
Wing area:
3,856 ft2
(358.25 m2)
Length: 202
ft 4 in (61.66
m)
Wingspan:
84 ft 0 in
(25.6 m)
Height: 40 ft
0 in (12.2 m)
The Following Graph Is a Left Side Image
of Concorde
Find the arc length of the upper side
Ending Point
Separating Point
Find the arc length of the lower side
The Following Graph Is a Cross-Section of
the Wind Blade
y
Find the arc length of the upper side
Upper side is represented by a
function U(x)
x
Separating Point
End Point
Angle of
Attack
Find the arc length of the lower side
Lower side is represented by a function D(x)
The Average Velocity of Flowing Air
Molecules
Note: the arc length of the upper side also equals to the total distance that the air
molecules on the upper side traveled.
𝐴𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑢𝑝𝑝𝑒𝑟 𝑠𝑖𝑑𝑒;
𝐿𝑒𝑛𝑔𝑡ℎ
1 + 𝑈 ′ 𝑥 𝑑𝑥
𝑈𝑎𝑟𝑐 =
𝑜
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑖𝑟 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑡𝑟𝑎𝑣𝑒𝑙𝑖𝑛𝑔 𝑎𝑙𝑜𝑛𝑒 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ 𝑈 𝑥 :
1 𝐿𝑒𝑛𝑔𝑡ℎ
𝑣𝑢 =
1 + 𝑈 ′ 𝑥 𝑑𝑥
𝑡 𝑜
𝐴𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑤𝑒𝑟 𝑠𝑖𝑑𝑒;
𝐿𝑒𝑛𝑔𝑡ℎ
1 + 𝐷 ′ 𝑥 𝑑𝑥
𝐷𝑎𝑟𝑐 =
𝑜
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑖𝑟 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑡𝑟𝑎𝑣𝑒𝑙𝑖𝑛𝑔 𝑎𝑙𝑜𝑛𝑒 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ 𝑈 𝑥 :
1 𝐿𝑒𝑛𝑔𝑡ℎ
𝑣𝑑 =
1 + 𝐷 ′ 𝑥 𝑑𝑥
𝑡 𝑜
Note: it takes the same amount time for both upper side air molecules and lower side air
molecules to travel from separating point to end point.
Pressure difference
𝟏
𝟏
𝑷𝒖 + 𝝆𝒗𝒖 𝟐 = 𝑷𝒅 + 𝝆𝒗𝒅 𝟐
𝟐
𝟐
𝟏
𝟏
𝟏
𝚫𝐏 = 𝑷𝒅 − 𝑷𝒖 = 𝝆𝒗𝒖 𝟐 − 𝝆𝒗𝒅 𝟐 = 𝝆(𝒗𝒖 𝟐 − 𝒗𝒅 𝟐 )
𝟐
𝟐
𝟐
Now we plug in the functions of 𝒗𝒖 𝒂𝒏𝒅 𝒗𝒅
𝟏
𝜟𝐏 = 𝝆
𝟐
𝜟𝐏 =
𝟏
𝝆
𝟐
𝟏
𝒕
𝟐
𝑳𝒆𝒏𝒈𝒕𝒉
𝟏+
𝑼′
𝒙 𝒅𝒙
𝒐
𝑳𝒆𝒏𝒈𝒕𝒉
𝒐
𝟏+𝑼′ 𝒙 𝒅𝒙
𝟐
𝟏
−
𝒕
𝑳𝒆𝒏𝒈𝒕𝒉
− 𝒐
𝒕𝟐
𝟐
𝑳𝒆𝒏𝒈𝒕𝒉
𝟏+
𝑫′
𝒙 𝒅𝒙
𝒐
𝟏+𝑫′ 𝒙 𝒅𝒙
𝟐
time t here is a dependent variable, not a independent variable.
-.- it Means that t is a function.
Once you know how to find the pressure difference between the upper and the lower side of a
wind blade, you can apply the same method to evaluate the pressure difference between the upper
and lower side of a plane body.
How Much Time Do We Need?
The times it takes for air molecules to travel from
the separating point to the end point depends on
the angle of attack.
𝑡=
x
𝑥
𝐿𝑐𝑜𝑠𝜃
=
𝑣𝑝𝑙𝑎𝑛𝑒 𝑣𝑝𝑙𝑎𝑛𝑒
Up to this point, we can tell that both
𝜃 𝑎𝑛𝑑 𝑣𝑝𝑙𝑎𝑛𝑒 are independent
variable because we can change
them anytime we want.
Area of the Plane
B(X)
A(X)
b
Body of the plane can be
represented by two different
functions: A(x) and B(x)
Total area of the plane equals to:
𝑎
𝐴=2
𝑏
𝐴 𝑥 𝑑𝑥 +
0
𝐵 𝑥 𝑑𝑥
𝑎
a
Final Battle!
precisely find out the minimum pressure
difference to lift up the plane
 Weight of the plane =𝑊𝑝𝑙𝑎𝑛𝑒 =mg
 Pressure that the plane applies on the air:
𝑃𝑝𝑙𝑎𝑛𝑒 =
𝑊𝑝𝑙𝑎𝑛𝑒
𝐴
=
𝑊𝑝𝑙𝑎𝑛𝑒
𝑎
𝑏
2 0 𝐴 𝑥 𝑑𝑥+ 𝑎 𝐵 𝑥 𝑑𝑥
In order to lift up or balance the plane, the pressure difference
needs to be more than or equal to the pressure that the plane
applies on the air
.’.
𝑊𝑝𝑙𝑎𝑛𝑒
2
𝑎
0 𝐴
𝑥 𝑑𝑥+
𝑏
𝑎 𝐵
𝑥 𝑑𝑥
𝟏
𝟐
≤ 𝝆
𝑳𝒆𝒏𝒈𝒕𝒉
𝒐
𝟏+𝑼′ 𝒙 𝒅𝒙
𝟐
𝑳𝒆𝒏𝒈𝒕𝒉
− 𝒐
𝐿𝑐𝑜𝑠𝜃
𝑣𝑝𝑙𝑎𝑛𝑒
𝟐
𝟏+𝑫′ 𝒙 𝒅𝒙
𝟐
+ 𝜟𝑷𝒑𝒍𝒂𝒏𝒆 𝒃𝒐𝒅𝒚
Conclusion
 With all these knowledge, you can make
yourself fly.
 Didn’t pay attention?!?!
 Feeling regretful?!?!
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Citation
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