Takeoff - flaps settings
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Takeoff - flaps settings
Goodday gentlemen,
it's my first post in this forum, though I've been reading it for quite some time. It's great to see professionals taking time to post here. I wish one day I'd have pair of my own wings. Enough of sweet words and here is my question.
A takeoff with more than normal takeoff flap settings will result in:
a) longer takeoff distance
b) shorter takeoff distance
c) the same takeoff distance
It seems easy and normally i'd think that the answer is "B" but my book says it's "A" but it's not explaining it. Would you please tell me which one is the correct answer and most importantly "why?" Thanks a lot in advance!
it's my first post in this forum, though I've been reading it for quite some time. It's great to see professionals taking time to post here. I wish one day I'd have pair of my own wings. Enough of sweet words and here is my question.
A takeoff with more than normal takeoff flap settings will result in:
a) longer takeoff distance
b) shorter takeoff distance
c) the same takeoff distance
It seems easy and normally i'd think that the answer is "B" but my book says it's "A" but it's not explaining it. Would you please tell me which one is the correct answer and most importantly "why?" Thanks a lot in advance!
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I think its A because increased flap increases drag.
A small amount of flap increases lift but with a small drag penalty. With larger amounts of flap, the increased drag outweighs the increased lift.
With lots of drag, accelleration will be slower and there for take off distance will be greater.
A small amount of flap increases lift but with a small drag penalty. With larger amounts of flap, the increased drag outweighs the increased lift.
With lots of drag, accelleration will be slower and there for take off distance will be greater.
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Poorly defined question! What is "normal" flap setting? It will vary with weight, runway length, thrust setting, a/c type etc...
A "high" flap setting will give an earlier lift-off (TOM vs. RWY), but that extra flaps, regardless of how much, will always give a drag penalty (no, that extra lift will never compensate for it), thus resulting in less performance and you'd perhaps have a hard time reaching the 35ft screen (since take-off distance is measured from brake release to 35ft for JAR25 a/c).
However; it also depends on whether or not you alter your Vee-speeds. Next to impossible to answer that question without further information :-)
A "high" flap setting will give an earlier lift-off (TOM vs. RWY), but that extra flaps, regardless of how much, will always give a drag penalty (no, that extra lift will never compensate for it), thus resulting in less performance and you'd perhaps have a hard time reaching the 35ft screen (since take-off distance is measured from brake release to 35ft for JAR25 a/c).
However; it also depends on whether or not you alter your Vee-speeds. Next to impossible to answer that question without further information :-)
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I agree with crossunder. A whole lot more information is necessary to answer that question. With the 747 when I define Noise abatement procedures for the greatest height over the monitors I suggest Flaps 20 takeoff vs Flaps 10, but it does not work in all situations. It also makes the close in noise greater as the duration is longer due to the slower V2 speeds for Flap 20.
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thank you gentlemen,
these were thorough answers.
i'm not quilty of the question,
it's taken from cathay pacific's enterance exam.
shoot cathay guys
there're couple of this kind of tricky / unclear questions, that's why i wanted to ask pro's.
once again your comments are highly appreciated.
these were thorough answers.
i'm not quilty of the question,
it's taken from cathay pacific's enterance exam.
shoot cathay guys
there're couple of this kind of tricky / unclear questions, that's why i wanted to ask pro's.
once again your comments are highly appreciated.
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If we were to plot the take-off distance for any given mass, against the flap setting from zero to maximum (landing flap) we would see something akin to a drag curve. High at each end and lowest somewhere in between. With zero flap the lift-off speed is high, so it takes a long time to accelerate to Vlof. With landing flap Vlof is lowest but there is a lot of drag so the acceleration rate is reduced. This again means that a long distance is required to accelerate to Vlof The minimum distance would be required when using the flap setting which gives the best trade-off between reduced Vlof and increased drag.
If we assume that we normally use this optimum flap setting, then any other setting wil increase take-off distance. So the answer is option A.
This is not just a Cathay Pacific problem, it is typical of the kind of thing that is asked in the JAR ATPL(A) Performance exams. But the questions in these exams also tend to include the effects on climb limited take-off mass and/or obstacle limited take-off mass.
If we assume that we normally use this optimum flap setting, then any other setting wil increase take-off distance. So the answer is option A.
This is not just a Cathay Pacific problem, it is typical of the kind of thing that is asked in the JAR ATPL(A) Performance exams. But the questions in these exams also tend to include the effects on climb limited take-off mass and/or obstacle limited take-off mass.
Last edited by Keith.Williams.; 16th Nov 2003 at 02:31.
An increased flap angle will allow a reduced rotate speed which will, despite the greater drag due to the flaps and thus reduced acceleration, result in a reduced take-off ground roll (take-off run). However, the climb gradient after take-off will be reduced by an increase in flap angle, both maximum gradient and all engines operating gradient at scheduled climb speed, typically V2 + 10 kts. (Note that V2 will often be lower with an increased flap angle).
I suspect that the "take-off distance" referred to in this question refers to the distance from brake release to an (unspecified) height. If this height relates to setting climb power or beginning the acceleration for flap retraction, the take-off distance may well be increased due to the reduced climb gradient.
I suspect that the "take-off distance" referred to in this question refers to the distance from brake release to an (unspecified) height. If this height relates to setting climb power or beginning the acceleration for flap retraction, the take-off distance may well be increased due to the reduced climb gradient.
I think you are assuming a continuous slope in the aerofoil-polar, which isn't usually true, there are many aircraft where past a point, the increase in Cl.max with increasing flap deflection becomes negligible, yet the increase in Cd0 becomes quite large.
Thus, whilst L : D is improved to an intermediate flap setting, it is worsened again as further flap is selected - giving a poorer take-off run and climb gradient in most cases. Not a bad thing, we all rather rely upon large flap settings for landing.
This is the reason that many aircraft (including the afforementioned C172 I think, and certainly a PA28) take-off with an intermediate flap setting, and on a go-around the immediate action (following full power selction) is to raise the flaps to that intermediate setting.
G
N.B. I've just discovered that if you type "L","colon","D" you get a very pretty smiley face, and not shorthand for lift-to-drag ratio as intended.
Thus, whilst L : D is improved to an intermediate flap setting, it is worsened again as further flap is selected - giving a poorer take-off run and climb gradient in most cases. Not a bad thing, we all rather rely upon large flap settings for landing.
This is the reason that many aircraft (including the afforementioned C172 I think, and certainly a PA28) take-off with an intermediate flap setting, and on a go-around the immediate action (following full power selction) is to raise the flaps to that intermediate setting.
G
N.B. I've just discovered that if you type "L","colon","D" you get a very pretty smiley face, and not shorthand for lift-to-drag ratio as intended.
I think you are assuming a continuous slope in the aerofoil-polar, which isn't usually true, there are many aircraft where past a point, the increase in Cl.max with increasing flap deflection becomes negligible, yet the increase in Cd0 becomes quite large.
Thus, whilst L : D is improved to an intermediate flap setting, it is worsened again as further flap is selected - giving a poorer take-off run and climb gradient in most cases. Not a bad thing, we all rather rely upon large flap settings for landing.
This is the reason that many aircraft (including the afforementioned C172 I think, and certainly a PA28) take-off with an intermediate flap setting, and on a go-around the immediate action (following full power selction) is to raise the flaps to that intermediate setting.
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I canīt understand how the B747 climbs better with flap 20 than 10, it needs further explanation !!!!! Please 747 FOCAL try your best!!!!!
With a large flap setting the aircraft becomes airbone earlier but with shallower climb gradient. That means that if the obstacle (or noise monitor) is close to the liftoff point, a large flap setting is advantageous. A distant obstacle (or noise monitor) requires a small flap setting, the aircraft becomes airbone later but with steeper climb gradient.
Taking off with F20 allows a greater height over close in monitors, but this doesnīt mean at all that your angle nor your rate of climb are higher than with F 10. Am I right?
Flap setting for maximum TOW vs OBST depends on the position of the obstacle (same criteria for noise monitor).
Flap setting for maximum TOW vs RWY is between 25 to 50% of the full deflexion. The larger the flap setting (approved for take off) the higher TOW vs RWY.
Flap setting for maximum TOW vs CLIMB( 2š segment) is the smallest (even 0š if approved for take off).
With a large flap setting the aircraft becomes airbone earlier but with shallower climb gradient. That means that if the obstacle (or noise monitor) is close to the liftoff point, a large flap setting is advantageous. A distant obstacle (or noise monitor) requires a small flap setting, the aircraft becomes airbone later but with steeper climb gradient.
Taking off with F20 allows a greater height over close in monitors, but this doesnīt mean at all that your angle nor your rate of climb are higher than with F 10. Am I right?
Flap setting for maximum TOW vs OBST depends on the position of the obstacle (same criteria for noise monitor).
Flap setting for maximum TOW vs RWY is between 25 to 50% of the full deflexion. The larger the flap setting (approved for take off) the higher TOW vs RWY.
Flap setting for maximum TOW vs CLIMB( 2š segment) is the smallest (even 0š if approved for take off).
Genghis Mater, who has Engineers for a father, husband, and son (me) maintains that an Engineer can be identified as somebody who when all other avenues have been explored will finally reach for the instructions.
In that spirit, I thought I'd get an aerodynamics textbook off the shelf. This, as is traditional, divides the take-off into three segments, as follows:-
Ground roll
S1 = integral from 0 to Vr of [W / 2g (T - D ) ] dVē
Assuming that's correct, then for most of that ground roll, lift isn't a significant player, but drag is. So, for most of the ground roll, so long as the flap setting used isn't significantly increasing drag it's not a player.
But the boundary condition, Vr (rotation speed) is clearly a function of CL.max and hence increased flaps reduces take-off roll until the increase in Cdo (or Cdi) doesn't become sufficient to overwhelm that.
Rotation segment
The book cops out this time, suggesting that the S1 formula should be used over 1 second segments, based upon a known and assumed constant rotation rate. So, on that basis, the same argument applies.
Initial climb segment
The initial part of this is the curved path from parallel with the runway to the steady climb gradient, this is defined by...
S2 = Vē * (T - D) / (delta-n x g x W)
So, what are the players here...
V - clearly the slower you can go (which is going to be a factor of Vs) the shorter the run, and since this is a squared term it's probably the most significant.
(T - D), no surprises there, as before you want to keep drag down. But, since we're at the left hand side of the drag curve it's mostly induced drag that you want to reduce, Cdo shouldn't be too big a player. Flaps *probably* help here by shifting the main spanwise centre of lift inboard, reducing tiplosses.
delta-N, the increase in g during rotation implies that high-lift devices are helping again, since the more g you pull in the rotation, the more S2 is reduced.
I have to say, seeing W on the bottom of that equation is rather anti-intuitive, higher weight should surely be increasing S2. However, thinking through it, greater W will probably reduce delta-n by the same factor, so I don't think that one should get too worked up about it, and anyhow it's not flaps dependent.
Which all seems to come down to the principle of use flaps to keep speed down, and they reduce take-off distance, until you hit the law of diminishing returns and increases in CL.max are more than compensated for by much greater increases in Cd
And on the 747 thing
I'll start by admitting that I have absolutely no knowledge of the type at-all, I'm just playing with equations here. Climb rate in a steady climb is given by...
RoC = V . (T - D ) / W
We can obviously assume that the weight is the same between 10° and 20° flaps, so presumably the better climb rate you are getting at flaps 20 is down to one or more of...
(1) Your best climb speed is faster at flaps 20, or
(2) Total drag is reduced at flaps 20 (seems a little unlikely to me), or
(3) In some way the engine is put into a higher thrust regime at flaps 20 (seems even more unlikely), or
(and I'm sticking my neck out a little here)...
(4) The pressure errors are altered in flaps 20 such that the static tends to indicate a higher rate of climb on the VSI ?
I must admit, I find it hard to hang my hat on any of these. Type-knowledge anybody?
G
In that spirit, I thought I'd get an aerodynamics textbook off the shelf. This, as is traditional, divides the take-off into three segments, as follows:-
Ground roll
S1 = integral from 0 to Vr of [W / 2g (T - D ) ] dVē
Assuming that's correct, then for most of that ground roll, lift isn't a significant player, but drag is. So, for most of the ground roll, so long as the flap setting used isn't significantly increasing drag it's not a player.
But the boundary condition, Vr (rotation speed) is clearly a function of CL.max and hence increased flaps reduces take-off roll until the increase in Cdo (or Cdi) doesn't become sufficient to overwhelm that.
Rotation segment
The book cops out this time, suggesting that the S1 formula should be used over 1 second segments, based upon a known and assumed constant rotation rate. So, on that basis, the same argument applies.
Initial climb segment
The initial part of this is the curved path from parallel with the runway to the steady climb gradient, this is defined by...
S2 = Vē * (T - D) / (delta-n x g x W)
So, what are the players here...
V - clearly the slower you can go (which is going to be a factor of Vs) the shorter the run, and since this is a squared term it's probably the most significant.
(T - D), no surprises there, as before you want to keep drag down. But, since we're at the left hand side of the drag curve it's mostly induced drag that you want to reduce, Cdo shouldn't be too big a player. Flaps *probably* help here by shifting the main spanwise centre of lift inboard, reducing tiplosses.
delta-N, the increase in g during rotation implies that high-lift devices are helping again, since the more g you pull in the rotation, the more S2 is reduced.
I have to say, seeing W on the bottom of that equation is rather anti-intuitive, higher weight should surely be increasing S2. However, thinking through it, greater W will probably reduce delta-n by the same factor, so I don't think that one should get too worked up about it, and anyhow it's not flaps dependent.
Which all seems to come down to the principle of use flaps to keep speed down, and they reduce take-off distance, until you hit the law of diminishing returns and increases in CL.max are more than compensated for by much greater increases in Cd
And on the 747 thing
I'll start by admitting that I have absolutely no knowledge of the type at-all, I'm just playing with equations here. Climb rate in a steady climb is given by...
RoC = V . (T - D ) / W
We can obviously assume that the weight is the same between 10° and 20° flaps, so presumably the better climb rate you are getting at flaps 20 is down to one or more of...
(1) Your best climb speed is faster at flaps 20, or
(2) Total drag is reduced at flaps 20 (seems a little unlikely to me), or
(3) In some way the engine is put into a higher thrust regime at flaps 20 (seems even more unlikely), or
(and I'm sticking my neck out a little here)...
(4) The pressure errors are altered in flaps 20 such that the static tends to indicate a higher rate of climb on the VSI ?
I must admit, I find it hard to hang my hat on any of these. Type-knowledge anybody?
G
747 Focal,
I used to fly 747-100/200 with both PW and RR engines until a couple of years ago, and I have a different understanding to you. We used flap 20 for normal take-offs but flap 10 for noise abatement (with climb power set at 1500 ft agl) as, at V2 + 10 kts for the appropriate flap setting, flap 10 gave a steeper climb gradient after lift-off and thus reduced the noise footprint. HOWEVER, we had a special noise abatement profile for LHR and LGW to beat the close in noise monitors. This was a flap 20 take-off with climb power set at 1000 ft agl. These meant that we had climb power set before the monitors and, although we were lower over them than on a "normal" flap 10 noise abatement profile, the dBs were lower as with the flap 10 profile we still has TO power set. alatriste, you were on the right lines.
Genghis,
You need to appreciate that the philosophy behind rotate speed and initial climb speeds in light aircraft is different to "Perf A" aircraft. In, say, a 747 you are no where near Cl max during rotate. In fact, Cl max/Vstall are not factored into Vr. It is all related to achieving minimum climb gradients (i.e. V2, although this can be capped by Vstall) rather than the maximum gradients for light aircraft, where Vstall does become important for rotate. Do not confuse different take-off performance categories.
I used to fly 747-100/200 with both PW and RR engines until a couple of years ago, and I have a different understanding to you. We used flap 20 for normal take-offs but flap 10 for noise abatement (with climb power set at 1500 ft agl) as, at V2 + 10 kts for the appropriate flap setting, flap 10 gave a steeper climb gradient after lift-off and thus reduced the noise footprint. HOWEVER, we had a special noise abatement profile for LHR and LGW to beat the close in noise monitors. This was a flap 20 take-off with climb power set at 1000 ft agl. These meant that we had climb power set before the monitors and, although we were lower over them than on a "normal" flap 10 noise abatement profile, the dBs were lower as with the flap 10 profile we still has TO power set. alatriste, you were on the right lines.
Genghis,
You need to appreciate that the philosophy behind rotate speed and initial climb speeds in light aircraft is different to "Perf A" aircraft. In, say, a 747 you are no where near Cl max during rotate. In fact, Cl max/Vstall are not factored into Vr. It is all related to achieving minimum climb gradients (i.e. V2, although this can be capped by Vstall) rather than the maximum gradients for light aircraft, where Vstall does become important for rotate. Do not confuse different take-off performance categories.
I didn't think that we were mentioning any particular performance category, and to be frank I was just playing with equations to relieve the monotony of, well playing with equations at the moment.
In any case, if your proverbial flying-bus has a fixed value of Vr (which is fair, it is usually a function of VMCG for multi-engined aeroplanes) which is irrespective of flap setting and thus CL.max the only affect that has on my argument above is that the threshold condition of the first integral, Vr becomes fixed (so T - D becomes the only player), which is true for multi-engined aircraft only. For the second integral, if you are arguing that the delta-n value remains fixed because the flightpath isn't influenced by CL.max, then again the only real variable becomes T - D again.
Perf-A requirements include that V2 >=1.2Vs for singles and twins or >=1.15Vs for 3+ engines, (and it's always essential to reach V2 by screen height) so stall speed does remain a player for all take-off calcs unless a company has elected to use fixed values of Vr and V2 for all weights and flap settings, based upon minimum flap and MTOW - which seems unlikely. (Incidentally, Vr is also a function of Vs for singles, it's only for multi-engined aeroplanes that it's a function of VMCG alone.)
G
In any case, if your proverbial flying-bus has a fixed value of Vr (which is fair, it is usually a function of VMCG for multi-engined aeroplanes) which is irrespective of flap setting and thus CL.max the only affect that has on my argument above is that the threshold condition of the first integral, Vr becomes fixed (so T - D becomes the only player), which is true for multi-engined aircraft only. For the second integral, if you are arguing that the delta-n value remains fixed because the flightpath isn't influenced by CL.max, then again the only real variable becomes T - D again.
Perf-A requirements include that V2 >=1.2Vs for singles and twins or >=1.15Vs for 3+ engines, (and it's always essential to reach V2 by screen height) so stall speed does remain a player for all take-off calcs unless a company has elected to use fixed values of Vr and V2 for all weights and flap settings, based upon minimum flap and MTOW - which seems unlikely. (Incidentally, Vr is also a function of Vs for singles, it's only for multi-engined aeroplanes that it's a function of VMCG alone.)
G
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I have no problem with the validity of the equations but they will mean little or nothing to many of the readers of this forum. I know because I teach a good many of them and any talk of integrals and the like will immediately make them tilt.
We can get a more inelligible picture of the situation if we imagine an aircraft with flap settings from zero to 90 degrees. The take-off run with zero flap will be quite long, but that with 90 degrees will be even longer. Now if we try 10 degrees and 80 degrees, we will find that our take-off run has decreased in both cases. Try again at 20 and 70 degrees and again the distance will have reduced. If we continue this process we will find an angle at which the take-off roll is minimum. This is the optimum angle to which I referred in my previous post.
The recommended range of flap angles for take-off in any aircraft is unlikely to include any angle greater than this optimum. So in the real world, provided you never use a flap angle beyond the recommended range, the take-off roll will decrease with increasing flap angle. But this will not be the case if we use an angle greater than recommended.
The original question concerned the use of flap angles greater than normal. If we take this to mean greater than the recommended range of angles, then take-off run and distance to screen height will both increase.
If we repeat this process and measure the climb gradient we will find that the gradient is greatest with zero flap and decreases as flap angle is increased. It is possible to create exceptions to this rule but they are probably very rare.
We can get a more inelligible picture of the situation if we imagine an aircraft with flap settings from zero to 90 degrees. The take-off run with zero flap will be quite long, but that with 90 degrees will be even longer. Now if we try 10 degrees and 80 degrees, we will find that our take-off run has decreased in both cases. Try again at 20 and 70 degrees and again the distance will have reduced. If we continue this process we will find an angle at which the take-off roll is minimum. This is the optimum angle to which I referred in my previous post.
The recommended range of flap angles for take-off in any aircraft is unlikely to include any angle greater than this optimum. So in the real world, provided you never use a flap angle beyond the recommended range, the take-off roll will decrease with increasing flap angle. But this will not be the case if we use an angle greater than recommended.
The original question concerned the use of flap angles greater than normal. If we take this to mean greater than the recommended range of angles, then take-off run and distance to screen height will both increase.
If we repeat this process and measure the climb gradient we will find that the gradient is greatest with zero flap and decreases as flap angle is increased. It is possible to create exceptions to this rule but they are probably very rare.
Last edited by Keith.Williams.; 19th Nov 2003 at 05:24.
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alatriste, LOMCEVAK,
Your both right on. It is only for close in monitors that the Flaps 20 is better for takeoff due to noise abatement. The takeoff roll is longer at Flaps 10 and forces the airplane to leave the ground so close to the monitors that you are still at full power as you cross over. The flaps 20 allows you to leave the ground earlier and slower, thus allowing you to obtain sufficient altitude so that you can power back just before reaching the monitor.
I have designed climb profiles that call for a flaps 10 with a slow decel from V2 +20 to V2 + 10 after you reach ~500ft that gets you much better monitor levels even at close in monitors, but alas certain safety issues arise. You want the best noise levels at any monitor at any airport, cutback to MAX CRUISE and climb to 3000 ft and then go to MAX climb and you will see about a 3-4 dB drop on 747s.
Your both right on. It is only for close in monitors that the Flaps 20 is better for takeoff due to noise abatement. The takeoff roll is longer at Flaps 10 and forces the airplane to leave the ground so close to the monitors that you are still at full power as you cross over. The flaps 20 allows you to leave the ground earlier and slower, thus allowing you to obtain sufficient altitude so that you can power back just before reaching the monitor.
I have designed climb profiles that call for a flaps 10 with a slow decel from V2 +20 to V2 + 10 after you reach ~500ft that gets you much better monitor levels even at close in monitors, but alas certain safety issues arise. You want the best noise levels at any monitor at any airport, cutback to MAX CRUISE and climb to 3000 ft and then go to MAX climb and you will see about a 3-4 dB drop on 747s.
Keith Williams,
Thank you for dragging this back to the start of the thread - "normal" flap angle is probably the key to this question. I like your way of explaining the flap issue. Another interesting point to ponder is that during the take-off roll with all wheels on the ground, the angle of attack increase with increasing flap angle due to the change in the mean chord line.
Genghis,
You appear to still have several misunderstandings regarding the derivation of V2 and Vr in large (Perf A) aeroplanes:
1. V2 is the minimum speed at which a specified gross climb gradient can be achieved, subject to an absolute minimum of 1.1 x Vmca and 1.15 (or 1.2) x Vs. The prime criterion is climb gradient. Vs only comes into play to determine an absolute minimum for a given AUW, and even then the factoring means that you are well clear of Clmax. Note that for a given AUW, V2 is different for different flap angles (typically, if not Vmca or Vs limited, V2 decreases with increasing flap angle).
2. Vr is the speed which, with one engine inoperative and rotation at the prescribed rate, gives V2 at screen height. This is subject to an absolute minimum of Vmcg. Therefore, Vs has no direct influence on Vr.
Note that the latest change to JAR 25 also includes a manoeuvre requirment for V2 but that is irrelevant here.
If you do not understand this, please send me a PM to discuss it before posting again in a manner that might lead to confusion, as your last post may.
Rgds
L
Thank you for dragging this back to the start of the thread - "normal" flap angle is probably the key to this question. I like your way of explaining the flap issue. Another interesting point to ponder is that during the take-off roll with all wheels on the ground, the angle of attack increase with increasing flap angle due to the change in the mean chord line.
Genghis,
You appear to still have several misunderstandings regarding the derivation of V2 and Vr in large (Perf A) aeroplanes:
1. V2 is the minimum speed at which a specified gross climb gradient can be achieved, subject to an absolute minimum of 1.1 x Vmca and 1.15 (or 1.2) x Vs. The prime criterion is climb gradient. Vs only comes into play to determine an absolute minimum for a given AUW, and even then the factoring means that you are well clear of Clmax. Note that for a given AUW, V2 is different for different flap angles (typically, if not Vmca or Vs limited, V2 decreases with increasing flap angle).
2. Vr is the speed which, with one engine inoperative and rotation at the prescribed rate, gives V2 at screen height. This is subject to an absolute minimum of Vmcg. Therefore, Vs has no direct influence on Vr.
Note that the latest change to JAR 25 also includes a manoeuvre requirment for V2 but that is irrelevant here.
If you do not understand this, please send me a PM to discuss it before posting again in a manner that might lead to confusion, as your last post may.
Rgds
L
Lomcevak. So far as I can see the only difference between your version and mine is that I've assumed that Vr and V2 are either identical to, or functions of their minimum permitted values. I've made this assumption because it was certainly the case with most C, D and E category aircraft that I've certified where they've virtually always ended up chasing the minimum values since best available gradient and TODR is otherwise usually based upon lower airspeeds than the minimum permitted.
Not having flown or worked on them, I'm more than happy to accept that the climb performance / climbing acceleration performance of perf-A jets may push the best gradient speed during initial climb-out above the minimum permitted value of V2, resulting in a higher value than I was assuming.
In the meantime I agree that Keith Williams explanation is rather more elegant than my mucking about with equations.
G
Not having flown or worked on them, I'm more than happy to accept that the climb performance / climbing acceleration performance of perf-A jets may push the best gradient speed during initial climb-out above the minimum permitted value of V2, resulting in a higher value than I was assuming.
In the meantime I agree that Keith Williams explanation is rather more elegant than my mucking about with equations.
G
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Genghis.
Might I suggest that you stop playing with equations. They are actually irrelevant unless you know the type. In this case the 747/200. LOMCEVAK does as does 747FOCAL. I don't, but it is a waste of time going through something which has no relevance. Especially as you admit you have no knowledge of the type.
As far as I am concerned the question has been very well answered - even I understand.
Light aircraft are light aircraft, your particular field. Leave the heavies to those who know.
Might I suggest that you stop playing with equations. They are actually irrelevant unless you know the type. In this case the 747/200. LOMCEVAK does as does 747FOCAL. I don't, but it is a waste of time going through something which has no relevance. Especially as you admit you have no knowledge of the type.
As far as I am concerned the question has been very well answered - even I understand.
Light aircraft are light aircraft, your particular field. Leave the heavies to those who know.