Best Climb Speed⚠ unverified

Understanding the Parameters

• The minimum-drag speed $V_{md}$ — the speed at which total drag is least, which is also the speed for maximum lift-to-drag ratio (see Lift-to-Drag Ratio). At $V_{md}$ the two halves of drag balance: induced drag (dominant at low speed) equals parasite drag (dominant at high speed). It is the natural reference speed for climb, glide, and endurance analysis.

• Why best climb is faster than $V_{md}$ for a jet — climb rate is excess power per weight, and excess power is $(T-D)V$. A jet's thrust is roughly constant with speed, so $(T-D)V$ keeps growing past $V_{md}$ as the extra $V$ outweighs the slowly rising drag — the maximum lands at $\sqrt3\,V_{md}$. The plane climbs best by flying faster than its most efficient cruise speed.

• Why a propeller aircraft is different — a propeller delivers roughly constant power, so its excess power is $P_a - DV$, maximised where $DV$ (the power required) is minimum — the minimum-power speed, which sits below $V_{md}$ at $\approx 0.76\,V_{md}$. A prop aircraft climbs best by flying slower than $V_{md}$. The single $1.3$ factor cannot capture both regimes.

• $V_x$ vs $V_y$ — this calculator estimates $V_y$ (best rate, max altitude per time). The distinct $V_x$ (best angle, max altitude per ground distance) is slower and used for obstacle clearance; near $V_x$ and $V_y$ the climb performance is flat, so precise speed-keeping matters less than being in the band.

Derivation (Approaching a Proof)

Model drag with the standard parabolic polar in terms of speed. Total drag has a parasite part rising as $V^2$ and an induced part falling as $1/V^2$:

$$D(V) = \underbrace{\tfrac12\rho V^2 S\,C_{D0}}_{\text{parasite}} + \underbrace{\frac{2 k W^2}{\rho V^2 S}}_{\text{induced}},$$

where $k = 1/(\pi\,AR\,e)$. Minimum drag occurs where the two terms are equal (a standard result), defining

$$V_{md} = \left(\frac{2W}{\rho S}\sqrt{\frac{k}{C_{D0}}}\right)^{1/2}.$$

Jet best climb. Rate of climb $\propto$ excess power $= (T - D)V = TV - D(V)\,V$. With jet thrust $T$ approximately independent of speed, maximise $f(V) = TV - D(V)V$ by setting $df/dV = 0$. Substituting the polar and differentiating, the parasite term contributes $+$, the induced term $-$, and the stationary condition works out to

$$V_y^{\text{jet}} = \sqrt{3}\;V_{md} \approx 1.73\,V_{md}.$$

The $\sqrt3$ arises because the excess-power maximum shifts the parasite/induced balance by a factor of three relative to the drag minimum.

Propeller best climb. Here $P_a$ is roughly constant, so excess power $= P_a - P_r$ is maximised where the power required $P_r = D(V)\,V$ is minimum. Minimising $DV$ (rather than $D$) shifts the optimum the other way:

$$V_{mp} = \frac{V_{md}}{\sqrt[4]{3}} \approx 0.76\,V_{md}.$$

So the true best-climb speed brackets $V_{md}$ dramatically — from $0.76\,V_{md}$ (prop) to $1.73\,V_{md}$ (jet). The registry's fixed $1.3\,V_{md}$ sits between, a serviceable single-number compromise but tied to neither exact result. This is the flag.

Dimensional check. $1.3\,V_{md} = (\text{–})\times\text{m}/\text{s} = \text{m}/\text{s}$ ✓ — a dimensionless multiplier scaling a speed.

History and Development

• The parabolic polar. The parabolic drag model underlying these results traces to the finite-wing theory of Ludwig Prandtl (induced drag $\propto C_L^2$, see Induced Drag) combined with parasite drag; it made min-drag and best-climb speeds computable from a handful of coefficients, and is the workhorse of performance analysis to this day.

• Rules of thumb in flight training. Practical aviation is full of such multipliers — $1.3\,V_s$ for approach speed, $1.2\,V_s$ for lift-off — because pilots need memorable numbers, not calculus. The $1.3\,V_{md}$ here is that kind of heuristic: close enough for a first estimate, not a substitute for the aircraft's actual flight manual $V_y$.

• The $\sqrt3$ result. The clean $V_y = \sqrt3\,V_{md}$ for an idealised constant-thrust jet is a favourite of performance textbooks (Anderson, Hull) precisely because the messy calculus collapses to a memorable constant — a reminder that the "best" speed depends fundamentally on whether the engine delivers constant thrust or constant power.

Related Concepts: Rate of Climb, Excess Power, Lift-to-Drag Ratio, Stall Speed, Drag Force, Induced Drag, Parasite Drag

Notes: $1.3$ is an empirical rule of thumb, not derived. True $V_y$: jet $\sqrt3\,V_{md}\approx1.73$; propeller best climb near min-power $V_{md}/\sqrt[4]{3}\approx0.76\,V_{md}$ (below $V_{md}$). Estimates $V_y$ (best rate), distinct from $V_x$ (best angle, slower). $V_{md}$ = min-drag = max-$L/D$ speed. Default $V_{md}=1$ placeholder.