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→An Approximate Method of Solution: The Analytical Polar Curve
\[ \frac{D}{\frac{1}{2} \rho V^2 S} = C_{D0} + \frac{k}{\pi A} \frac{L^2}{\frac{1}{4}\rho^2 V^4 S^2} \]
Both sides of the equation above need to be multiplied by \( \frac{1}{4} \rho V^4 S^2 \) (notice that this is the square of \( \frac{1}{2} \rho V^2 S\)), then divided by \( \frac{1}{2} \rho V S \), giving:
\[ DV = \frac{1}{2} \rho V^3 S C_{D0} + \frac{2kL^2}{\pi A \rho V S} \]
To proceed, the conservation of energy must be invoked. We realise that the kinetic energy of the glider is not changing because the glider is flying unaccelerated. Therefore, the release of gravitational potential, the rate of which equals to the power of the gravitational force, must balance the rate at which the mechanical energy of the glider is being dissipated by aerodynamic drag, which is the power of the drag force.
By definition, the power of a force is given by:
\[ P = F \times V \]
In words: the power is the product of the magnitude of the force and the speed of the subject in the direction of the force. By using this relationship, we realise that: the power of the gravitational force is given by \( W \times V_S \) (weight times the sink rate), and the power of the drag force is given by \( D \times V \) (drag times the airspeed). This relationship can also be obtained by a geometrical argument using basic trigonometry.
It should be noticed that the above argument is not watertight: this is because \( V \) is the indicated airspeed which generally differs from the true airspeed, and it is the latter that must be used to calculate the drag power. There are two ways to think around this:
# You can think of this as an approximation that is being made: true airspeed is being approximated with indicated airspeed. As a result, some systematic error will be introduced into the results.
# If you can understand the relationship between \( W \times V_S \) and \( D \times V \) from a geometrical perspective, you can think the following: because we are working in the indicated system where \( V \) is the indicated airspeed, the corresponding \( V_S \) obtained geometrically is the indicated sink rate. It needs to be converted to the true sink rate via the compound quantity \( \frac{1}{2} \rho V^2 \).
By using \( D \times V = W \times V_S \), the last equation becomes:
\[ WV_S = \frac{1}{2} \rho V^3 S C_{D0} + \frac{2kL^2}{\pi A \rho V S} \]
This expression should be examined in detail. The following quantities are known (either set, from design, or can be measured):
# \( W \), weight of the glider, depends on the design, cockpit loading, and amount of water carried, but can be known and usually does not change midway in flight.
# \( \rho \), density of air, because we work in the indicated system, this becomes the air density value used in the ASI, which is a fixed number.
# \( S \), wing area of the glider, known and stays constant (we shall not consider the effects of deploying flaps, etc. on the performance).
# \( C_{D0} \), this depends on the aerodynamic design of the glider.
# \( k \), this depends on the wing design of the glider, a highly complex series expansion to obtain a numeric value exists, but for all practical purposes this is a constant.
# \( \pi \), 3.1415926...
# \( A \), aspect ratio of the wing, depends on the glider design and a known constant.
Therefore, there are three changing quantities in this equation:
# \( V_S \), this is the quantity we are interested in, the y.
# \( V \), this is the quantity we can control, the x.
# \( L \), lift on the glider, what is it?
You should realise that, the existence of \( L \) in the equation above prevents us from obtaining a deterministic relationship between \( V_S \) and \( V \) which is the polar equation we desire. \( L \) can be related to \( W \) by using \( V_S \) and \( V \) and geometrical arguments, but this will complicate the equation and prevent us from arriving at an '''explicit''' relationship. In other words, doing so is the equivalence of reverting to the method in the last section.
Instead, we shall introduce the following '''approximation''': '''the weight of the glider equals to the lift force acting on the glider'''. This sounds intuitively true, but there is an error associated with it, whose magnitude is given by \( 1-\cos{\theta} \). Fortunately, this error is gracefully small at typical glide angles. If the glide ratio is 30:1, the error is 0.056%, which becomes even smaller if the glide ratio is higher.
We have shown that this is a good approximation. Therefore, we can replace the \( L \) in the existing equation with \( W \), and the '''analytical polar curve''' is arrived at:
\[ V_S = \frac{1}{2W} \rho S C_{D0} V^3 + \frac{2kW}{\pi A \rho V S} \]
