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The best glide speed is the airspeed giving the greatest horizontal distance per unit height lost in a specified steady glide condition. In still air this is the speed for maximum lift-to-drag ratio. It differs from the minimum sink speed, whose objective is longest time aloft for a given height loss.^[ref]

Write a glider sink polar as downward rate $w(V)>0$. The horizontal air-relative speed is $u(V)=\sqrt{V^2-w(V{)}^2}$, so the exact still-air range per unit height is $u/w$. For a shallow glide angle, $u\simeq V$ and maximizing range is approximately minimizing $w/V$. At an interior optimum,Geometrically, a line from the origin is tangent to the positive-downward-rate polar. This is not the horizontal tangent defining minimum sink speed.

For an original calculation, take a parabolic drag polar $C_D=C_{D0}+k{C}_L^2$ and approximate $L\simeq W$. At fixed air mass density $\rho$, wing area $S$ and gravitational weight $W$,Minimizing $w/V\simeq D/W$ givesStipulate $W=3000\mathrm{N}$, $S=10{\mathrm{m}}^2$, $\rho =1.2kg/{\mathrm{m}}^3$, $C_{D0}=0.020$ and $k=0.045$. Then $A=0.12$, $B=67500$ in the corresponding SI units, $V_{\mathrm{BG}}\simeq 27.39m/s$, and the model gives $w\simeq 1.643m/s$. Its shallow-glide range ratio is $V/w\simeq 16.67$. These parameters are invented; an attainable lift coefficient and a valid drag polar are prerequisites, not consequences of the optimization.

Wind changes the ground-range objective to $(u+U_w)/w$ for an along-track wind component $U_w$ and no vertical air motion. Therefore the still-air optimum need not maximize ground distance. The FAA constructs the corresponding shifted tangents on its performance polars.^[ref] A controlled performance measurement records steady airspeed and height loss at several speeds; wind, vertical air motion and changes of loading must be distinguished from the polar itself.

A drag polar relates drag coefficient $C_D$ to lift coefficient $C_L$ for specified geometry, configuration and flow conditions. It is commonly displayed with $C_D$ horizontally and $C_L$ vertically. The slope of a ray from the origin to a measured point is then $C_L/C_D$, the lift-to-drag ratio when common references are used.

A wind tunnel can obtain a polar by varying angle of attack while recording both forces and the reference dynamic pressure. Reynolds number, Mach number, surface condition and deployed devices must also be recorded; changing one of those can change the curve. A polar is not a geometry-only fingerprint.

For an original illustrative dataset, the pairs $(C_L,C_D)=(-0.2,0.024),(0,0.020),(0.4,0.028),(0.8,0.052)$ lie on $C_D=0.020+0.050C_L^2$. Their positive-lift ratios at $C_L=0.4$ and $0.8$ are about $14.29$ and $15.38$. The larger ratio among those samples does not establish the continuous maximum lift-to-drag ratio; the unsampled neighborhood near $C_L=0.63$ performs better in this assumed model.

A parabolic drag polar summarizes such data with a small number of parameters. A shifted parabolic drag polar allows minimum drag to occur at nonzero lift coefficient. Flow separation and changes of flow state can also produce branches or hysteresis that one parabola does not describe. A fit supplies a model whose residuals need inspection, rather than evidence that every unmeasured state lies exactly on the fitted curve.

A glider sink polar relates a glider's steady downward speed relative to the air to its true airspeed, for specified loading, configuration and atmospheric conditions. Write it as $w(V)>0$ with downward motion positive. Some plots instead put descent below the horizontal axis; the sign convention must accompany the curve. The minimum sink speed and best glide speed select different points on this same performance relation.^[ref]

The curve is a measurement or model of performance, rather than a drawing of the flight path. At steady speed, its horizontal air-relative component is $u=\sqrt{V^2-w^2}$. Thus time per height lost is $1/w$, while horizontal distance per height lost is $u/w$. The best glide speed article derives the tangent construction and an original parabolic drag polar example; those results need not be recomputed to interpret a measured polar.

A useful extension is glider ballast scaling. Suppose the gravitational weight changes from $W$ to $\alpha W$ at unchanged air mass density and geometry, and suppose the same lift coefficient and drag coefficient remain applicable. At corresponding steady glide conditions, the lift-to-drag ratio and glide angle are unchanged. Since $L=\rho V^{2}S{C}_L/2=W\cos \gamma$,For an original stipulated point $V_1=25m/s$ and $w_1=0.80m/s$, increasing gravitational weight by $25\%$ gives $V_2\simeq 27.95m/s$ and $w_2\simeq 0.8944m/s$. The corresponding distance-to-height ratio is unchanged, but the same height is lost sooner. This scaling assumes the aerodynamic coefficients transfer; changes of Reynolds number, surface condition or trim can invalidate it. The FAA illustrates the loading-dependent displacement of glider polars.^[ref]

To measure a polar, record steady true airspeed and height change at several operating points while documenting configuration and loading. Ground-relative descent $s$ includes vertical motion of the air: with upward air velocity $u_z$, $s=w-u_z$. For example, an air-relative sink of $1.5m/s$ in air rising at $0.5m/s$ appears as $1.0m/s$ ground-relative descent. Repetition can reveal variability, but one trajectory cannot separate these two contributions without additional information. A measured groundspeed is likewise not automatically the horizontal component of the air-relative polar.

A lift-constrained maximum lift-to-drag ratio is the best ratio available within an allowed range of lift coefficients, rather than over every coefficient admitted algebraically by a drag polar. It is a constrained optimization problem whose feasible set must be specified before claiming an attainable optimum.

For the parabolic drag polar with positive $c=C_{D0}$ and $k$, defineIts derivative isThe ratio increases up to $x_{\ast }=\sqrt{c/k}$ and decreases afterward. The exact constrained solution is thereforeThis proof handles an interior optimum and either active endpoint; simply inserting an upper limit into the unconstrained formula does not.

An original example takes $c=0.020$ and $k=0.050$, giving $x_{\ast }\simeq 0.6325$ and unconstrained ratio 15.8114. If the allowed interval is $[0.20,0.45]$, the maximum occurs at 0.45 and is about 14.9378. If the allowed interval is instead $[0.90,1.20]$, it occurs at the lower endpoint 0.90 and is about 14.8760. A lower bound can be active as well as an upper bound.

In steady level flight, a speed interval can induce a lift-coefficient interval. With fixed gravitational weight $W$, mass density $\rho$ and wing area $S$,The endpoints reverse because the function decreases with speed. Additional limits, including the usable range before aerodynamic stall, require intersecting feasible intervals.

The simple finite-wing contribution motivates a quadratic polar, but changing configuration, Mach number or Reynolds number can change its fitted parameters.^[ref] Restricting the mathematical interval does not validate the polar across that interval. Wind and ground-distance objectives also define a different optimization from the aerodynamic ratio considered here.

The lift-to-drag ratio $L/D$ compares the lift supporting an aircraft with the drag opposing its motion. For $L>0$ and $D>0$, a larger ratio means less drag for a given lift. If both coefficients use the same dynamic pressure $q$ and reference wing area $S$, then $L/D=C_L/C_D$. Ratios using mismatched reference areas do not have this interpretation.

A parabolic drag polar provides a useful worked model:Here $C_{D0}$ is the zero-lift drag coefficient. In a simple finite-wing model, $k=1/(\pi eAR)$ relates lift-induced drag to wing aspect ratio and a span efficiency factor.^[ref] A fitted whole-aircraft $k$ can also incorporate additional effects; it need not be determined by geometry alone.

The maximum lift-to-drag ratio follows by differentiation:At this point the two modeled contributions to drag coefficient are equal. With the stipulated $C_{D0}=0.020$, $e=0.8$ and $AR=8$, $k\simeq 0.049736$, $C_L^{\ast }\simeq 0.6341$ and $(L/D{)}_{\max }\simeq 15.85$. The optimum is usable only if the actual wing can attain that lift coefficient in the regime where the model is valid; aerodynamic stall can intervene.

For steady unpowered descent in still air, the force balance gives $\tan \gamma =D/L$, where $\gamma$ is the glide angle. Thus the stipulated optimum corresponds to $\gamma \simeq 3.61{}^{\circ }$ and about $15.85$ units of horizontal travel per unit of height lost.^[ref] Wind and changing flight conditions alter travel over the ground. An airfoil section's ratio also should not be substituted for that of an entire aircraft, which includes other sources of drag.

The maximum lift-to-drag ratio is the largest attainable ratio of positive lift to drag within stated operating conditions. It depends on both the drag polar and the range of lift coefficients the vehicle can actually attain.

For the positive parabolic drag polar $C_D=C_{D0}+k{C}_L^2$, a short proof uses the arithmetic-geometric mean inequality:Consequently
provided that coefficient is attainable. Equality requires equal modeled zero-lift and lift-dependent drag contributions. This bound supplies a global argument, rather than merely locating a stationary point by differentiation.

A lift-constrained maximum lift-to-drag ratio can be lower. With $C_{D0}=0.020$, $k=0.050$ and allowed $0<C_L\le 0.50$, the unconstrained optimum $C_L^{\ast }\simeq 0.6325$ is excluded. Since the ratio increases up to that point, the constrained optimum is at $C_L=0.50$, giving $L/D=15.3846$, below the unconstrained $15.8114$. A stall-limited interval should therefore be applied before claiming the fitted optimum is reachable.

Parameter bounds on maximum lift-to-drag ratio follow from monotonicity. If $C_{D0}\in [c_{-},c_{+}]$ and $k\in [k_{-},k_{+}]$ are positive intervals, then the unconstrained maximum lies in
This interval arithmetic bound is conservative when the fitted parameters are correlated and not all interval combinations are jointly possible. It quantifies uncertainty in the assumed polar; it does not include failure of the polar model itself.

The minimum-drag speed $V_{\mathrm{MD}}$ minimizes the dimensional drag required for steady level flight at fixed gravitational weight, configuration and atmospheric conditions. It is not the speed at the smallest drag coefficient: changing speed changes the dynamic pressure and the lift coefficient needed to support the load.

For the parabolic drag polar $C_D=C_{D0}+k{C}_L^2$, assume $C_{D0}>0$, $k>0$ and constant mass density $\rho$. Substituting $L=W=qS{C}_L$ givesThe ideal finite-wing induced term motivates the quadratic dependence on lift coefficient; a measured whole-aircraft fit needs its own regime assessment.^[ref]

The arithmetic-geometric mean inequality proves a global bound for every $V>0$:Equality holds when $a{V}^2=b/V^2$, so the two modeled drag contributions are equal. Equivalently, the optimum lift coefficient is $C_L^{\mathrm{MD}}=\sqrt{C_{D0}/k}$.

An original model uses $a=0.20$ and $b=512,000$ when $V$ is in m/s and $D$ in newtons. It gives $V_{\mathrm{MD}}=40$ m/s and $D_{\min }=640$ N, with 320 N from each term. At 30 m/s, the terms are 180 and about 568.89 N, giving a larger total of 748.89 N. The slower state nevertheless requires less mechanical power, illustrating the distinct minimum-power speed objective.

At fixed weight, minimizing drag maximizes the lift-to-drag ratio. Under a shallow, steady-glide approximation this relates to best glide through the air, whereas minimum sink speed concerns altitude loss per time. The FAA polar explicitly distinguishes the corresponding operating points.^[ref] Wind, variable coefficients or an unattainable lift coefficient can change the applicable optimum. These model results are not flight instructions or substitutes for measured aircraft performance.

The minimum-power speed $V_{\mathrm{MP}}$ minimizes the mechanical propulsive power required to sustain the specified flight state. For steady level flight with drag $D(V)$, the required power is $P(V)=D(V)V$, using speed through the air. Minimizing this product is a different problem from finding the minimum-drag speed.

With the fixed-coefficient parabolic drag polar, writeDifferentiation givesThus the stationary point is the unique global minimum for positive speed:At this point, $b/V^2=3a{V}^2$: the decreasing drag contribution is three times the increasing contribution, rather than equal to it. In lift-coefficient form, $C_L^{\mathrm{MP}}=\sqrt{3C_{D0}/k}$.

For an original numerical model with $a=0.20$ and $b=512,000$ in consistent metre-second-newton units, $V_{\mathrm{MD}}=40$ m/s and $V_{\mathrm{MP}}\simeq 30.393$ m/s. The power minimum is about 22.461 kW, with drag about 739.01 N. At the drag minimum, drag is only 640 N but power is 25.600 kW. The lower power comes from the combination of force and speed, despite the greater force.

In shallow steady unpowered descent, the gravitational energy-loss rate approximately balances $DV$, so minimizing required power relates to minimum sink speed when the same assumptions hold. This is consistent with the FAA's distinct minimum-sink and best-glide points.^[ref] A changing propulsive efficiency changes the input-power objective: if $P_{\mathrm{input}}=P/{\eta }_p(V)$, a stationary point satisfies $P^{\prime }/P={\eta }_p^{\prime }/{\eta }_p$, not generally $P^{\prime }=0$.

Attainability must also be checked. The model's optimal lift coefficient can exceed the allowed range before aerodynamic stall. The formula concerns the stated required mechanical power, not a universal minimum of fuel consumption or an operational speed recommendation.

The minimum sink speed is the airspeed at which a glider has its smallest steady downward speed relative to the air, for a specified configuration, loading and flight condition. It maximizes time aloft from a given height in still air. It does not generally maximize horizontal range, which defines best glide speed.^[ref]

For a differentiable glider sink polar $w(V)>0$, an interior minimum satisfies $w^{\prime }(V)=0$, with a positive second derivative sufficient for a strict local minimum. For the original shallow-glide model developed in best glide speed,With $A,B>0$, this givesIndeed $w^{\prime \prime }=(6AV+2B/V^3)/W>0$ for $V>0$. The speed ratio is about $0.7598$, within this stipulated parabolic drag polar approximation. A real polar need not follow that numerical ratio, especially near aerodynamic stall.

Using the same original $W=3000\mathrm{N}$, $A=0.12$ and $B=67500$ gives $V_{\mathrm{MS}}\simeq 20.81m/s$ and $w_{\min }\simeq 1.442m/s$. From a stipulated $1000\mathrm{m}$ height, the model's times are about $693.6\mathrm{s}$ at minimum sink and $608.6\mathrm{s}$ at best glide. Yet the approximate range ratios are $14.43$ and $16.67$, respectively. The slower optimum buys time rather than maximum distance.

A horizontal wind changes range without changing air-relative sink at a fixed condition. Vertical air motion changes ground-relative height loss as well: descending at $w$ through air rising at $u_z$ gives ground-relative descent $w-u_z$. Observing constant height therefore does not establish zero air-relative sink. A bank angle also changes the required lift and hence the appropriate polar; the straight-glide optimum cannot simply be reused for a tight turn.

Parameter bounds on maximum lift-to-drag ratio describe how an allowed set of drag polar parameters limits the optimum predicted by that model. For the positive parabolic drag polar, write $c=C_{D0}>0$ andThis is the unconstrained model optimum; an allowable range of lift coefficients can reduce the attainable value.

The derivatives $\partial F/\partial c=-F/(2c)$ and $\partial F/\partial k=-F/(2k)$ show that $F$ decreases with either parameter. Consequently, positive bounds $c\in [c_{-},c_{+}]$ and $k\in [k_{-},k_{+}]$ implyThis interval arithmetic argument is global over the rectangle, without a small-error approximation.

An original example uses $c\in [0.018,0.022]$ and $k\in [0.045,0.055]$. The predicted optimum ratio lies between about 14.374 and 17.568. Its optimizing lift coefficient lies in the different intervalor approximately $[0.5721,0.6992]$. A range of performance values and a range of operating points answer different questions.

A rectangular enclosure can be conservative. For a separate joint-parameter example, suppose fitting constrains $k=0.001/c$ exactly over a positive interval of $c$. Then the product $ck$ is constant and every allowed pair gives $F=15.8114$, even though each parameter varies. Combining independent marginal extremes would include pairs excluded by that relation.

For small random parameter errors, first-order propagation instead givesThe covariance term is needed when parameters are correlated, consistent with the primary metrological propagation law.^[ref] A standard uncertainty is not automatically a hard interval bound. Neither calculation includes failure of the assumed polar outside its fitted regime, and neither by itself identifies one operating point that is optimal for every parameter pair.

The zero-lift drag coefficient $C_{D0}$ is the drag coefficient at $C_L=0$ for a stated configuration and flow condition. It is not necessarily the drag at zero angle of attack: a cambered airfoil can have a nonzero zero-lift angle of attack. In a simple parabolic drag polar, $C_D=C_{D0}+k{C}_L^2$, the lift-dependent term vanishes there.^[ref]

The zero-lift drag coefficient also need not be the smallest drag coefficient on a general polar. An original shifted parabolic drag polar,has minimum $0.018$ at $C_L=0.20$, while $C_{D0}=0.020$. Confusing the intercept with the minimum loses the shift caused by the particular aerodynamic configuration.

A measurement should locate the state where the measured lift crosses zero, then evaluate or interpolate drag at that state using compatible references. Zero readings can be affected by balance tare and systematic error. Measurements at several nearby angles of attack distinguish a crossing from one apparently small reading.

For a complete aircraft, the coefficient can include skin-friction, pressure and component-interaction effects under the selected decomposition. A finite-wing term proportional to $C_L^2$ can vanish at zero lift without making those other sources of drag vanish. Holding $C_{D0}$ constant while changing speed is itself an approximation if the Reynolds number, Mach number or flow state changes.