In this article I will list the formula’s required to be used within the principles of Flight subject.
The simplest ones are things like
Force = m x g
Power is (Force x Distance)/Time which is F x D/T(V) = Power required is D x V and Power available is T x V
Dynamic Pressure is ½rV2
Lift is ½rV2 CL S
Drag is ½rV2 CD S
It is important that you know these formulas as they will help you interpret a lot of theory questions and you may even have to calculate the value of lift or drag. If you do ensure that you use the correct SI unit, for example velocity must be in m/s and is the TAS of the aeroplane.
LIFT/DRAG
You may also have to rearrange the Lift formula to calculate the CL as a percentage of CLMAX. To do this we replace the assumed none changing values with 1 and remove most of them from the formula.
CL = L/½rV2 S, this changes to be CL = 1/V2
For example, calculate CL as a percentage of CLMAX if the speed is increased from by 40% above the minimum flight speed
CL = 1/(1.4)2 = 1/0.51 = 51%
You also need to be aware of the CDi formula which is CL2/Aspect Ratio. This is because induced drag is affected by mass and speed, so CL squared and it comes down to the amount of the wingtip involved. So, the higher the aspect ratio the less wingtip and wing area effected by induced drag.
As an example, what happens to induced drag if flaps are deployed in level flight and a constant IAS. Most people would assume lift will increase and so will induced drag but, that is not the case. If we look at the Di formula it is:
Di = ½rV2 CDi S
Wing area, density and velocity are all constant so induced drag is entirely dependent on CDi. Aspect ratio won’t change so we need to examine the lift formula to see if CL changes. Lift must be constant and as we have already said wing area, density and velocity are all constant so as lift must also be constant CL cannot change so Di does not change.
STALLING
There are several formulas in stalling which we must examine and the first is what happens to stall speed if the weight of the aeroplane changes. As weight reduces the stall speed reduces as we need less lift to support the lower weight so we can fly slower before we reach the critical angle of attack. This formula can also be used to calculate the new VA speed for a new weight.
The formula is:
VSNEW = VSOLD Ö(New Weight/Old Weight)
Next, we need to calculate load factor in a turn and for this we need:
L = 1/cosf and load factor = L/W.
The stall speed in a turn formula is:
VSTURN = VS1G Ö(1/cosf)
The value in the square root brackets can be replaced by the increase in lift or the load factor. This formula could also be used to calculate the value of VA by placing the G value in the square root function
There are several formulas from flight mechanics that we need to utilise and those are:
CLIMB ANGLE/GRADIENT
Climb angle formula is Sin g = (T – D)/W
Climb gradient is = ((T – D)/W) x 100
Climb gradients are always greater than the angle and to convert between the two we would sin the angle and multiply by 100, and to go the other way divide by 100 and sin-1.
For example, calculate the climb angle and gradient for a twin engine aeroplane with 60000 N per engine a drag value of 45000 N and a weight of 550000 N. Inputting those values into the formula we have:
120000 – 45000 = 0.136. if we sin-1 this value we have 7.8°and a gradient of 13.6%.
550000
In a climb and a descent, the value of lift is:
L = W cos g.
To calculate the horizontal distance in a glide when we know the L/D ratio we use:
Glide distance = Height to lose x L/D ratio.
If we need to take wind into effect, then multiply the distance by TGS/TAS.
TURNING
For turning we have a few formulas to remember and they are:
Turn Radius = V2/(g tanf), V is TAS in m/s, g is 9.81 unless 10 is given, and f is the bank angle.
There are two formulas for rate of turn depending on what is given in the question:
Rate of turn = (g tanf)/V or
Rate of turn = V/radius.
Both give and answer in radians which needs to be multiplied by 57.3° per radian. This stems from the formula 2pr, where there are 2p radians in the circumference of a circle.
You may have to then calculate the time taken to complete a given turn in which case you would divide the turn by the rate of turn.
For example, if we wanted to know how long it was going to take to complete a 180° turn at a rate of 4° per second then the answer would be 180/4 = 45 seconds.
MACH ANGLE
To calculate Mach angle is simply
sinm = 1/M. Where M is the Mach number.
For example, at a Mach speed of 5 the Mach angle would be 1/5 = 18°, using the sin-1 function.
CHANGE OF G
In limitations there are a couple of formulas we may have to utilise. Firstly, you may need to calculate the change in G for an angle of attack change.
As an example, we have a gust which causes a 5° angle of attack change and for every 1° change CL increases by 0.1. If the original CL value was 0.3 what G value would the aeroplane achieve during the gust.
The CL change will be 5 x 0.1 = 0.5, we then add that to 0.3 and divided by 0.3 this gives 0.8/0.3 = 2.66 G.
Next you may have to calculate the G value during a specific gust at different speeds and to do this we need to work out the gust load the aeroplane would stall at for the old and new weight and the change in G for the new speed.
For example: An aeroplane maintains straight and level flight at a speed of 1.5VS. If
At this speed a vertical gust causes a load factor of 1.75, the load factor
N caused by the same gust at a speed of 1.1VS would be:
To calculate the G value we stall at, we just square the speed and that gives 1.52 = 2.25 and the new speed would give 1.12 = 1.21. To calculate the new change in G we take the change in G due to the gust and multiply by new speed divided by old speed.
This gives (1.75 -1) x (1.1/1.5) = 0.55, this would give a G value of 1 + 0.55 = 1.55 G.
However, the maximum G value would be 1.21 as the aeroplane will stall at this value.