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Power
Storyboard 
Power is the energy per time that the object is capable of supplying (airplane / bird) and that limits the conditions of both flight, takeoff and landing.
For a finite power it is observed that there is a minimum takeoff speed and a maximum speed at which the object can be held in the air which limits takeoff and landing. Similarly, there is a maximum speed that can be reached that limits the attack and escape capacity of birds.
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Mechanisms
Iframe
Mechanisms
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Flight power
Description
The following diagram shows the two components of the total flight power. The first component corresponds to the high resistance encountered at low speeds due to the required angle of attack for generating sufficient lift. The second component illustrates how the power required for flight increases quite dramatically at higher speeds:
The sum of both curves represents the total power required as a function of flight velocity.
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Takeoff problem
Description
Both airplanes and birds require reaching a minimum speed to be able to fly. Airplanes achieve this by accelerating on the runway for takeoff, while birds have the ability to run or drop themselves, for example, from a cable or branches where they have perched.
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Landing problem
Description
Both airplanes and birds have a speed limit below which they cannot fly. This means that during the landing process, there will always be a residual horizontal velocity, and it will be necessary to use brakes to come to a complete stop. In the case of higher speeds, a longer runway will be required, and it is crucial to be prepared for possible incidents or setbacks:
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Model
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Parameters
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Calculations
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Target : Equation To be used 
Equations
$\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$
P / P_0 = v_0 / v + v^3 / v_0 ^3
$ P = F_R v $
P = F_R * v
$ P =\displaystyle\frac{1}{2} \rho S_p C_W v ^3 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v }$
P = rho * S_p * C_W * v ^3/2+2* m ^2* g ^2/( c ^2* S_w * rho * v )
$ P_0 =\left(\displaystyle\frac{4 m ^6 g ^6 C_W S_p }{ c ^6 \rho ^2 S_w ^3}\right)^{1/4}$
P_0 =((4* m ^6* g ^6* C_W * S_p )/( c ^6 * rho ^2 * S_w ^3))^(1/4)
$ P_{opt} =\left(3^{1/4}+\displaystyle\frac{1}{3^{3/4}}\right) P_0 $
P_opt =(3^(1/4)+1/3^(3/4))* P_0
$ v_0 =\left(\displaystyle\frac{4 m ^2 g ^2}{ c ^2 \rho ^2 C_W S_w S_p }\right)^{1/4}$
v_0 =((4* m ^2* g ^2)/( c ^2* rho ^2* C_W * S_w * S_p ))^(1/4)
$ v_{max} =\left(\displaystyle\frac{ P_{max} }{ P_0 }\right)^{1/3} v_0 $
v_max = ( P_max / P_0 )^(1/3)* v_0
$ v_{max} =\left(\displaystyle\frac{ 2 }{ \rho S_p C_w }\right)^{1/3} P_{max} ^{1/3}$
v_max = (2/( rho * S_p * C_w))^(1/3)* P_max ^(1/3)
$ v_{min} =\displaystyle\frac{ 2 m ^2 g ^2 }{ \rho S_w c ^2}\displaystyle\frac{1}{ P_{max} }$
v_min = 2* m ^2* g ^2/( rho * S_w * c ^2* P_max )
$ v_{min} =\displaystyle\frac{ P_0 }{ P_{max} } v_0 $
v_min = P_0 * v_0 / P_max
$ v_{opt} =\displaystyle\frac{1}{3^{1/4}} v_0 $
v_opt = v_0 /3^(1/4)
ID:(15185, 0)
Flight power
Equation
Power $P$ is the energy per unit of time that needs to be supplied to sustain a given force $F_R$. Therefore, it can be calculated based on the force by multiplying it by the velocity $v$:
| $ P = F_R v $ |
The power is defined as the energy $\Delta W$ per time $\Delta t$ according to the equation:
| $ P =\displaystyle\frac{ \Delta W }{ \Delta t }$ |
Since energy is equal to force $F$ multiplied by the distance traveled $\Delta s$, we have:
| $ \Delta W = F \Delta s $ |
Thus, we obtain:
$P=\displaystyle\frac{\Delta W}{\Delta t}= F_R \displaystyle\frac{\Delta s}{\Delta t}$
However, since the distance traveled in a time interval is the velocity $v$:
| $ \bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$ |
Finally, we can write the expression for power as:
| $ P = F_R v $ |
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General flight power
Equation
To obtain the power of flight ($P$), you need to multiply the total resistance force ($F_R$) by the speed with respect to the medium ($v$). Since the total resistance force ($F_R$) is a function of the density ($\rho$), the surface that generates lift ($S_w$), the total object profile ($S_p$), the coefficient of resistance ($C_W$), the proportionality constant coefficient sustainability ($c$), the body mass ($m$), and the gravitational Acceleration ($g$), which is equal to
| $ F_R = \displaystyle\frac{1}{2} \rho S_p C_w v ^2 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v ^2}$ |
,
the potential is
| $ P =\displaystyle\frac{1}{2} \rho S_p C_W v ^3 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v }$ |
The total resistance force ($F_R$) is a function of the density ($\rho$), the surface that generates lift ($S_w$), the total object profile ($S_p$), the coefficient of resistance ($C_W$), the proportionality constant coefficient sustainability ($c$), the body mass ($m$), and the gravitational Acceleration ($g$), which is equal to
| $ F_R = \displaystyle\frac{1}{2} \rho S_p C_w v ^2 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v ^2}$ |
,
therefore, using the equation for the power of flight ($P$)
| $ P = F_R v $ |
,
we obtain:
| $ P =\displaystyle\frac{1}{2} \rho S_p C_W v ^3 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v }$ |
.
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Reference power
Equation
The reference power ($P_0$) is calculated using the body mass ($m$), the gravitational Acceleration ($g$), the coefficient of resistance ($C_W$), the surface that generates lift ($S_w$), the total object profile ($S_p$), the proportionality constant coefficient sustainability ($c$), and the density ($\rho$):
| $ P_0 =\left(\displaystyle\frac{4 m ^6 g ^6 C_W S_p }{ c ^6 \rho ^2 S_w ^3}\right)^{1/4}$ |
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Reference speed
Equation
The speed with respect to the medium ($v$) is calculated using the body mass ($m$), the gravitational Acceleration ($g$), the coefficient of resistance ($C_W$), the surface that generates lift ($S_w$), the total object profile ($S_p$), the proportionality constant coefficient sustainability ($c$), and the density ($\rho$):
| $ v_0 =\left(\displaystyle\frac{4 m ^2 g ^2}{ c ^2 \rho ^2 C_W S_w S_p }\right)^{1/4}$ |
ID:(4550, 0)
Generalized power based on references
Equation
The power of flight ($P$) is expressed as a function of the density ($\rho$), the surface that generates lift ($S_w$), the total object profile ($S_p$), the coefficient of resistance ($C_W$), the proportionality constant coefficient sustainability ($c$), the body mass ($m$), and the gravitational Acceleration ($g$) as:
| $ P =\displaystyle\frac{1}{2} \rho S_p C_W v ^3 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v }$ |
Combining these definitions with those of the reference power ($P_0$) and the reference speed ($v_0$), we obtain:
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
The power of flight ($P$) is expressed as a function of the density ($\rho$), the surface that generates lift ($S_w$), the total object profile ($S_p$), the coefficient of resistance ($C_W$), the proportionality constant coefficient sustainability ($c$), the body mass ($m$), and the gravitational Acceleration ($g$) as:
| $ P =\displaystyle\frac{1}{2} \rho S_p C_W v ^3 + \displaystyle\frac{2 m ^2 g ^2}{ c ^2 S_w \rho }\displaystyle\frac{1}{ v }$ |
It can be rewritten by introducing the reference power ($P_0$) as:
| $ P_0 =\left(\displaystyle\frac{4 m ^6 g ^6 C_W S_p }{ c ^6 \rho ^2 S_w ^3}\right)^{1/4}$ |
and the reference speed ($v_0$) as:
| $ v_0 =\left(\displaystyle\frac{4 m ^2 g ^2}{ c ^2 \rho ^2 C_W S_w S_p }\right)^{1/4}$ |
,
resulting in:
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
ID:(4552, 0)
Optimal flight speed
Equation
The minimum flight power is obtained by taking the derivative of the expression for the power of flight ($P$), which depends on the speed with respect to the medium ($v$), the reference power ($P_0$), and the reference speed ($v_0$),
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
with respect to the speed with respect to the medium ($v$) and setting it equal to zero, resulting in:
| $ v_{opt} =\displaystyle\frac{1}{3^{1/4}} v_0 $ |
The speed for minimal power ($v_{opt}$) is defined as the value at which the power of flight ($P$), which depends on the speed with respect to the medium ($v$), the reference power ($P_0$), and the reference speed ($v_0$),
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
is minimized, meaning the first derivative of this equation is equal to zero (and the second derivative is positive).
By taking the derivative of the equation and setting it equal to zero, we obtain
$\displaystyle\frac{1}{P_0}\displaystyle\frac{dP}{dv}=\displaystyle\frac{3v^2}{v_0^3}-\displaystyle\frac{v_0}{v^2}=0$
which implies that the speed for minimal power ($v_{opt}$) is
| $ v_{opt} =\displaystyle\frac{1}{3^{1/4}} v_0 $ |
the speed for minimal power ($v_{opt}$) is approximately 0.76 times the reference speed ($v_0$). For a reference velocity of 17.22 m/s, this is equal to 13.09 m/s.
ID:(4556, 0)
Optimum flight power
Equation
The flight power the optimum speed power ($P_{opt}$) is obtained by evaluating the power of flight ($P$) with the speed with respect to the medium ($v$), the reference power ($P_0$), and the reference speed ($v_0$) as shown in the equation:
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
This evaluation results in the speed for minimal power ($v_{opt}$), leading to:
| $ P_{opt} =\left(3^{1/4}+\displaystyle\frac{1}{3^{3/4}}\right) P_0 $ |
The power of flight ($P$) with the speed with respect to the medium ($v$), the reference power ($P_0$), and the reference speed ($v_0$) is expressed as:
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
which results in the speed for minimal power ($v_{opt}$):
| $ v_{opt} =\displaystyle\frac{1}{3^{1/4}} v_0 $ |
and ultimately, we obtain the optimum speed power ($P_{opt}$):
| $ P_{opt} =\left(3^{1/4}+\displaystyle\frac{1}{3^{3/4}}\right) P_0 $ |
It's important to note that the optimum speed power ($P_{opt}$) is approximately 1.75 times the reference power ($P_0$). For a reference power of 0.36 W, this equates to 0.63 W.
ID:(4557, 0)
Minimum speed estimate
Equation
In the case where the aircraft or bird uses all of the maximum speed power ($P_{max}$) it can produce, we can determine the minimum speed with maximum power ($v_{min}$), the speed at which they can fly, using the equation for the power of flight ($P$) with the reference power ($P_0$), the speed with respect to the medium ($v$), and the reference speed ($v_0$):
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
Since in this case, the speed with respect to the medium ($v$) can be considered much smaller than the reference speed ($v_0$) ($v\ll v_0$), the minimum speed with maximum power ($v_{min}$) in this case is:
| $ v_{min} =\displaystyle\frac{ P_0 }{ P_{max} } v_0 $ |
If the speed with respect to the medium ($v$) is significantly smaller than the reference speed ($v_0$), we can disregard the term $(v/v_0)^3$, simplifying the equation for the power of flight ($P$) with the reference power ($P_0$) to:
$\displaystyle\frac{P}{P_0} = \displaystyle\frac{v_0}{v}$
Therefore, by solving for velocity and evaluating the power of flight ($P$) at the maximum speed power ($P_{max}$), we obtain the minimum speed with maximum power ($v_{min}$):
| $ v_{min} =\displaystyle\frac{ P_0 }{ P_{max} } v_0 $ |
ID:(14516, 0)
Minimum speed based on maximum power
Equation
As the minimum speed with maximum power ($v_{min}$) as a function of the reference power ($P_0$), the reference speed ($v_0$), and the maximum speed power ($P_{max}$) is equal to:
| $ v_{min} =\displaystyle\frac{ P_0 }{ P_{max} } v_0 $ |
We can derive this expression using the equations for the reference speed ($v_0$) and the power of the reference power ($P_0$) with the body mass ($m$), the gravitational Acceleration ($g$), the density ($\rho$), the surface that generates lift ($S_w$), the proportionality constant coefficient sustainability ($c$), and the maximum speed power ($P_{max}$):
| $ v_{min} =\displaystyle\frac{ 2 m ^2 g ^2 }{ \rho S_w c ^2}\displaystyle\frac{1}{ P_{max} }$ |
As the minimum speed with maximum power ($v_{min}$) as a function of the reference power ($P_0$), the reference speed ($v_0$), and the maximum speed power ($P_{max}$) is equal to
| $ v_{min} =\displaystyle\frac{ P_0 }{ P_{max} } v_0 $ |
If we substitute the reference speed ($v_0$) with the body mass ($m$), the gravitational Acceleration ($g$), the proportionality constant coefficient sustainability ($c$), the coefficient of resistance ($C_W$), the surface that generates lift ($S_w$), and the aircraft body mass ($m_p$) with
| $ v_0 =\left(\displaystyle\frac{4 m ^2 g ^2}{ c ^2 \rho ^2 C_W S_w S_p }\right)^{1/4}$ |
and the reference power ($P_0$) with
| $ P_0 =\left(\displaystyle\frac{4 m ^6 g ^6 C_W S_p }{ c ^6 \rho ^2 S_w ^3}\right)^{1/4}$ |
we obtain the expression
| $ v_{min} =\displaystyle\frac{ 2 m ^2 g ^2 }{ \rho S_w c ^2}\displaystyle\frac{1}{ P_{max} }$ |
ID:(14518, 0)
Maximum speed estimate
Equation
In the event that the aircraft or bird utilizes all of the maximum speed power ($P_{max}$) they can produce, we can determine the minimum speed with maximum power ($v_{min}$), the maximum speed at which they can fly, using the equation for the power of flight ($P$) with the reference power ($P_0$), the speed with respect to the medium ($v$), and the reference speed ($v_0$):
| $\displaystyle\frac{ P }{ P_0 }=\left(\displaystyle\frac{ v }{ v_0 }\right)^3+\displaystyle\frac{ v_0 }{ v }$ |
Since in this case, the speed with respect to the medium ($v$) can be considered much larger than the reference speed ($v_0$) ($v\gg v_0$), the maximum speed with maximum power ($v_{max}$) in this case is:
| $ v_{max} =\left(\displaystyle\frac{ P_{max} }{ P_0 }\right)^{1/3} v_0 $ |
If the speed with respect to the medium ($v$) is significantly smaller than the reference speed ($v_0$), we can disregard the term $(v/v_0)^3$, simplifying the equation for the power of flight ($P$) with the reference power ($P_0$) to:
$\displaystyle\frac{P}{P_0} = \left(\displaystyle\frac{v}{v_0}\right)^3$
Therefore, by solving for velocity and evaluating the power of flight ($P$) at the maximum speed power ($P_{max}$), we obtain the maximum speed with maximum power ($v_{max}$):
| $ v_{max} =\left(\displaystyle\frac{ P_{max} }{ P_0 }\right)^{1/3} v_0 $ |
ID:(14517, 0)
Maximum speed as a function of maximum power
Equation
As the maximum speed with maximum power ($v_{max}$) as a function of the reference power ($P_0$), the reference speed ($v_0$), and the maximum speed power ($P_{max}$) is equal to:
| $ v_{max} =\left(\displaystyle\frac{ P_{max} }{ P_0 }\right)^{1/3} v_0 $ |
We can obtain this expression using the equations for the reference speed ($v_0$) and the reference power ($P_0$) with the density ($\rho$), the total object profile ($S_p$), the coefficient of resistance ($C_W$), and the maximum speed power ($P_{max}$):
| $ v_{max} =\left(\displaystyle\frac{ 2 }{ \rho S_p C_w }\right)^{1/3} P_{max} ^{1/3}$ |
The expression for the maximum speed with maximum power ($v_{max}$) as a function of the reference power ($P_0$), the reference speed ($v_0$), and the maximum speed power ($P_{max}$) is equal to
| $ v_{max} =\left(\displaystyle\frac{ P_{max} }{ P_0 }\right)^{1/3} v_0 $ |
If we substitute the reference speed ($v_0$) with the body mass ($m$), the gravitational Acceleration ($g$), the proportionality constant coefficient sustainability ($c$), the coefficient of resistance ($C_W$), the surface that generates lift ($S_w$), and the total object profile ($S_p$) with
| $ v_0 =\left(\displaystyle\frac{4 m ^2 g ^2}{ c ^2 \rho ^2 C_W S_w S_p }\right)^{1/4}$ |
and the reference power ($P_0$) with
| $ P_0 =\left(\displaystyle\frac{4 m ^6 g ^6 C_W S_p }{ c ^6 \rho ^2 S_w ^3}\right)^{1/4}$ |
we obtain the expression
| $ v_{max} =\left(\displaystyle\frac{ 2 }{ \rho S_p C_w }\right)^{1/3} P_{max} ^{1/3}$ |
ID:(14519, 0)
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Video
Video: Power