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I want to learn about how can the V1 and VR (rotate) speeds be calculated for take off. And which factors are affecting this calculation?

Basically it may depends on the aircraft weight, runway length, engine capacity and flap rates. But what is the other factors and what are the V1 and VR formulas. Is it possible to explain, e.g. for a Boeing 737 which is taking off from a 6,791 ft (2,070 m) runway in good weather conditions?

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There is no general equation/function that you can use to calculate V1/Vr... the manufacturer test out the aircraft's performance during certification, and you then compare your current aircraft and conditions against the known manufacturer's aircraft and conditions to get your results.

V1 is the border between accelerate for takeoff, and STOPPPPPP! by the end of the runway.
* Acceleration is dependent upon the engine thrust being used, the atmosphere providing oxygen for combustion and nitrogen for jetflow mass, the aircraft's mass that has to be accelerated, the drag of the used flap setting, and the possible drag of the runway surface and slope.
* The STOPPP! aspect is dependent upon the drag of the flaps and speedbrakes, the drag of the runway slope and surface, and the mass of the airplane.

Vr is nothing to do with V1... it's just a margin below V2 (5kts? 10kts? depending on thrust setting) that allows you to rotate earlier than V2 but still accelerate (in the air) to then achieve V2 by 15m/50ft. And V2 depends on the weight of the aircraft and the flap setting.

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  • $\begingroup$ Vr is pretty well set, for a given aircraft, by the gross weight and flap setting. It will be the cap on V1 -- if your calculated V1 is 160 but you plan to rotate at 140, "stopping at 150" no longer has meaning -- if you see 150 or 160 knots while still on the runway, you've deviated from the plan. (If you are getting into "improved climb" performance, then relationships get more complicated, but that's a fairly advanced topic beyond this discussion.) $\endgroup$
    – Ralph J ♦
    Commented Dec 13, 2017 at 14:48
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This is a great question that I pondered myself while getting my multi-engine rating.

V1 is determined by several factors but the one most important factor is called the "accelerate / decelerate distance". In other words, the distance it will take to stop the aircraft before the aircraft runs out of runway or continue safely into the air.

Ordinarily you might think V1 is the same, but "lower" performance and "higher" performance aircraft behave differently during takeoff. A comparatively low performance aircraft such as single engine recip will have V1 and VR equal. But high performance aircraft such as a turbo prop or jet will have a V1 quite different from VR.

Acceleration is a speed - or more accurately a change in speed over time designated by delta-a meaning that an aircraft accelerating before stall speed will continue to INCREASE in speed for a short period of time even if there is an engine thrust failure - in layman's terms this is called momentum.

Therefore, in moderately or high performance powered aircraft V1 and VR will be different and the manufacture will base the value on flight tests.

Lets consider two examples. A single engine airplane like like a Cessna 150 will have V1 and VR at the stall speed. so the pilot pulls back at V1 and by the time the aircraft lifts off the ground it has gained an additional 5kts an climbs at V1+5kt. In a jet, the pilot will reach VR 10 or 20kts before V1 but even if the engine fails, momentum will carry the aircraft past V1 and safely the aircraft will reach flying speed without additional thrust.

Therefore, depending on what we might call excess (yes a slightly inaccurate term) thrust, VR occurs prior to V1 because of delta-a (high acceleration).

For more powerful aircraft, VR is a moving target. B747 have so much power that if they are light weight or have a long runway, they are authorized to use less than full power for the takeoff to conserve engine wear and noise abatement. Therefore VR is different for each takeoff.

Update: As noted below, for the single engine scenario I should have used VS not V1. And, yes powerful aircraft have a Delta-A - change in acceleration, that is different from their velocity. As an example, earths gravity (32ft/sec/sec) is an acceleration not a pure velocity.

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  • $\begingroup$ "A single engine airplane like like a Cessna 150 will have V1 and vr at the stall speed." This doesn't make sense, at least if V1 (as I have understood it) is the speed at or above which takeoff rejection will result in a runway overrun no matter what else you do. That would be a function of speed, breaking capability (including runway surface conditions), and weight. $\endgroup$
    – user
    Commented Dec 13, 2017 at 13:18
  • $\begingroup$ With a sufficiently long runway, then, you might at best have to try really hard simply to keep the plane on the ground long enough to even reach V1. If you can get back down on the runway without turning around, and land safely, then surely by definition you never exceeded V1 because the remaining runway length provided adequate stopping distance? $\endgroup$
    – user
    Commented Dec 13, 2017 at 13:18
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    $\begingroup$ I'm sorry, but "V1" is undefined for a single engine aircraft, since the ability to lose the critical engine & continue the takeoff only applies when the aircraft has multiple engines. A single might have a refusal speed above which a stop on the remaining runway cannot be completed -- don't abort above this speed for a system failure, perhaps; but if THE engine fails on the runway, you're stopping, no matter what your speed! $\endgroup$
    – Ralph J ♦
    Commented Dec 13, 2017 at 14:12
  • $\begingroup$ @RalphJ To a first order approximation, I would expect "delta-a" to be the second derivative of position, where velocity is the first derivative. Change in position (over time) is velocity; change in velocity (over time) is acceleration (or deceleration). It's also worth keeping in mind that velocity is a vector, while speed is a scalar quantity. $\endgroup$
    – user
    Commented Dec 13, 2017 at 14:36
  • $\begingroup$ @MichaelKjörling Delta A would be the delta in acceleration, just as delta V is the delta in velocity (a.k.a. acceleration). So Delta A would be more like the THIRD derivative of position. If the aircraft has a (more or less) steady acceleration with takeoff thrust set, you'd get a drop in acceleration (and thus a large, negative, delta A) at the point an engine fails. Beyond that, I doubt operators have much use for delta A. Engineers, probably have more. You can see "V" on instruments, you can see "A" on a HUD; delta A, not so much. $\endgroup$
    – Ralph J ♦
    Commented Dec 13, 2017 at 14:43

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