The fact: Gliders have ballast tanks that can be filled with water. The addition of ballast increases weight, and this allows the glider to fly at faster airspeeds while maintaining the same glide ratio. This means that if two gliders P1 and P2, being both of the same model, start at some point A, and P2 weights more than P1, they will both end at point B, but P2 will arrive before P1. The movement is rectilinear, and during the flight, each plane is able to maintain constant airspeed and vertical speed, being P2's speed greater than P1's. Angle of attack might be different too. This is a little diagram I made:
To simplify things, let's consider the movement is made inside uniform air, with no density variation with height, and winds are calm. I've read some aviation manuals, and the most frequent explanation provided is that the glider retains the same Lift/Drag (L/D) ratio, but nowhere in the books I've read so far includes the proof. I'm trying with my very limited skills to reach an expression relating speed with weight, but almost always get lost in the process.
This second diagram I made shows the different forces involved. As gliders don't (usually) have an engine, there's no thrust, and the only forces present are Lift (L), Drag (D) and Weight (W). (Sorry, arrow lengths are not proportional to vectors magnitudes):
As the movement is uniform, the acceleration is null. Then, Newton's Second law tell us that:
${\vec F = m·\vec a}$
As the mass cannot be null, then the resulting force F should be null. So we have:
${\vec F = \vec F_x + \vec F_y}$
${\vec F_x = \vec L_x + \vec D_x = \vec 0}$
${\vec F_y = \vec L_y + \vec D_y + \vec W_y= \vec 0}$
The lift formula is:
${L = \frac{C_L(\alpha)V^2 \rho S}{2} }$
Where:
- ${C_L(\alpha)}$ Is the coefficient of lift, which is proportional to the angle of attack below the stall.
- ${V}$ is the speed. Shame on FAA handbook for not detailing which speed it was. According to Wikipedia, it is IAS (Indicated Airspeed) in the direction of movement, which makes sense.
- ${\rho}$ Is the air density
- S is the wing surface area.
Drag formula is:
${D = C_D(\alpha) q S }$
Where:
- ${C_D(\alpha)}$ Is the drag coefficient, which is a function of the angle of attack, roughly quadratic.
- ${q}$ is the dynamic pressure (???).
- S is the wing surface area.
Again, sorry for not including units. I continued transforming the expressions but couldn't get a clear W/V equation, instead I ended up with a messy trigonometric expression. There might be an easier way, perhaps approximating some coefficients? Anyway, my questions:
- Are all assumptions above correct?
- In the real thing, airspeed may be constant, but ground speed is a different thing. Is the movement really uniform?
(Sorry for my awful English, it is not my native language)
Thanks in advance.
EDIT:
Here's my attempt to express speed as function of the mass:
${\vec F_y = \vec L_y + \vec D_y + \vec W_y= \vec 0}$
${|\vec L|\cos(\varphi) = -|\vec D|\sin(\varphi) -gm}$
${\frac{C_L(\alpha)V^2 \rho S}{2}\cos(\varphi) = -C_D(\alpha) q S \sin(\varphi) -gm}$
As lift and drag coefficients are constant for a given angle of attack, ${\rho}$ and q are constants because the air is uniform, and S is constant because the glider flies without modifying the wing area (no flaps added), we can group the constants and this is what we get:
${k V^2 = -g m + k'}$