What determines the best glide speed?

Question

I found the best glide speed of cessna 152 is 60 kts. My question is why it is not more or less than 60 kts? What are the factors upon which the best gliding speed is determined?

Answer 1

Basically, the faster you go, the more lift and drag a wing will produce. These two values are not proportional, though. When speeding up, the amount of drag produced is higher than the extra amount of lift - which is why you need additional power to maintain level at higher speeds.

When you go slower, the amount of drag will reduce more than the amount of lift produced - at least for a while. That's why going slow is better in terms of gliding distance. The amount of "drag per lift" is very low. However, slowing down beyond a certain point, the wing will rapidly start producing less lift, because airflow separates from the wing. This is what's known as a stall. The best glide speed is the speed as which the drag is as low as possible while the wing is still producing a relatively large amount of lift.

This is illustrated on a speed polar, like this one:

The black line indicates the rate of sink for a given airspeed. The optimum glide speed is the speed corresponding to the point where the red line touches the black line (Vbg).

The red line is a straight line going from (0,0) and touching the speed polar at exactly one point.

A change in aircraft mass will shift the curve along the vertical axis, which is why a heavier aircraft has a higher best glide speed than a lighter one. The intersecting point between the black and red line would shift to the right as the black line is shifted down, and vice versa.

Answer 2

The most important factors for the best glide speed are the aircraft's wing loading, the air density, the wing's aspect ratio, and the aerodynamic quality of the aircraft.

The aircraft must create lift equal to its own weight. The drag for doing so varies with airspeed, and to find the point where the glide ratio has its maximum, drag must be minimal. To find this speed, we describe drag mathematically as the sum of two components:

1. Parasitic drag, which goes up with the square of airspeed. We express this as the zero-lift drag, a drag component which is independent of lift: $D_0 = \frac{\rho}{2}\cdot v^2\cdot S\cdot c_{D0}$
2. Lift-dependent or induced drag which goes down with the inverse of the square of airspeed: $D_i = \frac{\rho}{2}\cdot v^2\cdot S\cdot \frac{c_L^2}{\pi\cdot AR\cdot\epsilon}$

Now it helps to find the lift coefficient to create the needed lift at a given speed: $$c_L = \frac{m\cdot g}{\frac{\rho}{2}\cdot v^2\cdot S}$$ Which, when inserted into the formula for induced drag, produces $$D_i = \frac{(m\cdot g)^2}{\frac{\rho}{2}\cdot v^2\cdot S\cdot \pi\cdot AR\cdot\epsilon}$$ Now it should be obvious that induced drag is indeed proportional to the inverse of flight speed squared. We can simplify this a little by inserting $AR = \frac{b^2}{S}$ and express the total drag as the sum of both components: $$D = \frac{\rho}{2}\cdot v^2\cdot S\cdot c_{D0} + \frac{(m\cdot g)^2}{\frac{\rho}{2}\cdot v^2\cdot \pi\cdot b^2\cdot\epsilon}$$ Next, we differentiate with respect to speed $v$ and need to set the result to zero to arrive at an equation for the speed of lowest drag: $$\frac{∂ D}{∂ v} = \rho\cdot v\cdot S\cdot c_{D0} - \frac{(2\cdot m\cdot g)^2}{\rho\cdot v^3\cdot \pi\cdot b^2\cdot\epsilon} = 0$$ $$\rho\cdot v^4\cdot S\cdot c_{D0} = \frac{(2\cdot m\cdot g)^2}{\rho\cdot \pi\cdot b^2\cdot\epsilon}$$ $$v = \sqrt[4]{\frac{(2\cdot m\cdot g)^2}{\rho^2\cdot\pi\cdot b^2\cdot\epsilon\cdot S\cdot c_{D0}}}$$ $$v = \sqrt{\frac{2\cdot m\cdot g}{\rho\cdot S\cdot\sqrt{\pi\cdot AR\cdot\epsilon\cdot c_{D0}}}}$$ There you have it: The best glide speed is proportional to the square root of both the wing loading $\frac{m\cdot g}{S}$ and the inverse of air density $\rho$, and the fourth root of the inverse of the aspect ratio $AR$, the Oswald factor $\epsilon$ and the zero-lift drag coefficient $c_{D0}$. The Oswald factor is a measure of the quality of lift production and is close to unity in most cases.

Thankfully, quiet flyer reminds me in the comments that I need to point out that I always talk about true air speed here. If you want to set your speed using the speed indicator, reduce the value by the applicable density correction.

Nomenclature:
$c_{D0} \:$ zero-lift drag coefficient
$c_L \:\:\:$ lift coefficient
$S \:\:\:\:\:$ reference area (wing area in most cases)
$v \:\:\:\:\:$ airspeed
$\rho \:\:\:\:\:$ air density
$\pi \:\:\:\:\:$ 3.14159$\dots$
$AR \:\:$ aspect ratio of the wing
$\epsilon \:\:\:\:\:$ the wing's Oswald factor
$m \:\:\:\:$ the aircraft's mass
$g \:\:\:\:\:$ gravitational acceleration
$b \:\:\:\:\:$ wingspan

Answer 3

(it's simpler than it may looks)

If you are at a certain height, you have a certain amount of potential energy (or height energy). The only thing you can do is to convert it to kinetic energy (or speed, which then creates lift). The problem: drag takes up energy too. So all the energy you loose due to drag means a loss in kinetic energy (=speed) and therefore a loss in lift.

The question actually is: how to reduce drag to a minimum?

It's actually quite simple: there are roughly two different kinds of drag:

• induced drag, induced by the angle of attack of the airplane. The more your nose goes up (so the lower your airspeed is), the higher the induced drag. This is an exponential relation.

• parasitic drag, comes from the air and is the "usual" drag you also feel with a car or bike. It depends exponential on the airspeed.

The total drag consists of the sum of both. The minimum is the best gliding speed.

Answer 4

I've never heard of the term maximum gliding speed, there's no special limitation to how fast you can fly a c152 without an engine as opposed to with it working. I think what you are talking about is best glide speed, also known as Vbg, which is the speed that gives you the farthest horizontal distance travelled per unit of height lost. If I remember correctly 60kts is the best glide with flaps extended, 65kts was best glide with no flaps.

The best glide speed actually varies based on weight, as do most of the V-speeds. A heavier airplane would mean a faster Vbg and a lighter one a slower Vbg. On a c152 the difference is pretty small, maybe 2 kt either way, so giving a 1 speed answer makes sense as it's easy to remember. Best glide speed on a big airplane will vary much more and would need to be calculated based on weight estimate at that point in the flight.

Answer 5

In this answer, we'll confine ourselves to steady linear flight with the wings level.

We can show that the highest glide ratio (i.e. smallest or flattest glide angle) occurs at the highest ratio of L (Lift) to D (Drag). In fact we can show that the glide ratio is exactly the same as the ratio of L to D. We can also show that the highest ratio of L to D occurs exactly at the point of minimum D. (L is nearly constant and nearly equal to Weight; more precisely, L = Weight * cosine glide angle.)

I found the best glide speed of cessna 152 is 60 kts. My question is why it is not more or less than 60 kts? What are the factors upon which the best gliding speed is determined?

Because that is where D (Drag) is at a minimum. Nothing else matters.

Keep in mind that Drag is proportional to the product of the drag coefficient-- which increases as we increase the angle-of-attack-- and the square of the airspeed.

Other answers give additional insight as to why the point of minimum Drag occurs where it does-- it is the point where Induced Drag and Parasite Drag are exactly equal.

By the way, the ratio of L/D is exactly equal to the ratio of Cl / Cd, where Cl and Cd are the coefficients of Lift and Drag. Unlike Lift, Cl is not nearly constant across a wide range of angles-of-attack or airspeeds. It is always much greater at high angles-of-attack (low airspeeds), at least until we get very close to the stall angle-of-attack. Cd is also always much greater at a high angle-of-attack (low airspeed) than at a low angle-of-attack (high airspeed). So we can also say that the maximum glide ratio occurs at the maximum ratio of Cl to Cd. Lest there be any confusion, we should also observe that this is the point of minimum Drag, but not the point of minimum drag coefficient. If we were flying at the angle-of-attack for the maximum glide ratio, and we then moved the stick or yoke forward to reduce the angle-of-attack (and increase the airspeed), this would reduce the drag coefficient (as well as the lift coefficient), but this would also increase Drag, and would decrease L/D and Cl / Cd.

In comments you indicate that you want to know how best glide speed varies with weight. A good approximation is that it scales up or down in proportion to the square root of any change in your wing loading. In gliders where we don't have drag from a windmilling prop to complicate things-- or in any airplane where we've stopped the prop-- this should be exactly true, except for effects due to changes in Reynolds number.

For completeness, we should say something about where the minimum power-off sink rate occurs. That turns out to happen at the angle-of-attack that gives the maximum value of (Cl cubed) / (Cd squared). This always occurs at a higher angle-of-attack (lower airspeed) than does the best glide ratio.