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Equation 3 · Flight at the Energy Floor: Transport Selection to 2100

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MM

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cruise Mach number. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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MM

Symbol M

cruise Mach number.

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where a is the local speed of sound, M is cruise Mach number, g is standard gravity, cTc_T is thrust-specific fuel consumption, L/D is the lift-to-drag ratio, and W1W_1 and W2W_2 are the aircraft’s weight at the start and end of the cruise segment [ 1 ] . Three of those four factors — speed, engine efficiency, and aerodynamic efficiency — are where a century of aeronautical engineering has already done most of its work and where further gains now come in single-digit percentages per generation, the flattened tail of aviation’s S-curve. The fourth factor, the mass-ratio term ln⁡(W1/W2)\ln(W_1/W_2) , is where the aviation lineage’s entire battery-versus-kerosene argument actually lives, because W1W_1/W2W_2 is set…
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where a is the local speed of sound, M is cruise Mach number, g is standard gravity, cTc_T is thrust-specific fuel consumption, L/D is the lift-to-drag ratio, and W1W_1 and W2W_2 are the aircraft’s weight at the start and end of the cruise segment [ 1 ] . Three of those four factors — speed, engine efficiency, and aerodynamic efficiency — are where a century of aeronautical engineering has already done most of its work and where further gains now come in single-digit percentages per generation, the flattened tail of aviation’s S-curve. The fourth factor, the mass-ratio term ln⁡(W1/W2)\ln(W_1/W_2) , is where the aviation lineage’s entire battery-versus-kerosene argument actually lives, because W1W_1/W2W_2 is set by how much of the aircraft’s total weight has to be energy-carrier to deliver a given amount of usable energy — and that, in turn, is set directly by the energy-carrier’s specific energy in watt-hours per kilogram. A carrier with low specific energy needs more of the aircraft’s weight budget just to hold the same energy, which shrinks ln⁡(W1/W2)\ln(W_1/W_2) for any given range target and forces a designer to trade range against payload, payload against range, or accept a shorter mission than the same airframe could fly on a denser fuel. Nothing about propeller efficiency, wing aspect ratio, or engine bypass ratio changes that trade; it is set upstream of all of them, by chemistry.

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