Oggers
If I may take your comments in turn:
The power required is only proportional to TAS cubed if ρ is constant, which of course it won't be at two different altitudes. So at constant IAS the power at FL300 will be 1.5 times that at FL50, not 3.375 times.
You are correct. I made the error of overlooking the fact that in the special case of a change in altitude at constant IAS, the reducing density and increasing TAS squared cancel each other out, so the increase in power required is proportional to the increase in TAS. So it would be more accurate for me to have said:
1. At any given altitude and IAS the power required is proportional to TAS cubed.
2. For any change in altitude at constant IAS the change in power required is
proportional to the change in TAS.
So in my example calculation the power required at FL300 should have been 1.5 times that at FL50.
Bloggs was right to call that error out.
Bloggs did not actually call me out for that error. What he/she actually said was:
Keith, you've confused air miles and ground miles. Power required should be measured in this case as numbers of molecules of air being flown-through, not nautical miles flown over the ground.
My understanding is that the V in the drag formula is velocity with respect to the air the aircraft is in, ie IAS, not TAS.
The power required at Seal levl and FL300 for the same IAS is roughly the same.
All of which is incorrect.
It is the mass of the jetwash/propwash/propellant that is the measure of its inertia, not “stiffness”.
You are correct in saying that the mass is the measure of its inertia. We only need to look at Newton’s Second Law equation F = MA to see that. But Newton assumed that the bodies in question were free to move. And if for example, one of the bodies were moving around a pivot such as your hamster wheel, then the stiffness of the bearing would also affect the outcome.
Let’s imagine that we took your hamster wheel and added an adjustable friction brake to the pivot. With the brake fully off the hamster would accelerate the wheel up to some given speed and then maintain this speed for as long as it kept running. But if we gradually tight the friction brake while the hamster continued to run, the wheel would gradually decrease. At some point the friction would be sufficient to stop the wheel, and if the hamster kept running it would follow a vertical looped path inside the wheel. So although we did not ever change the mass of the hamster or the mass of the wheel we have completely changed the outcomes. In order to determine which situation (wheel spinning / hamster stationary or Wheel stationary / Hamster spinning) gave the better propulsion efficiency, we would need to know what the hamster was trying to achieve. And as neither of us speak hamster we can never know that.
The problem with propellers and/or jet engine sin aircraft is that in order to generate thrust they must exert a rearward force on some object or material which is not itself part of the aircraft or propulsion system. The only obvious candidate is the air. But because the air is a gas, the exertion of any rearward force upon it will cause it to flow rearwards. This results in the propulsion system transferring kinetic energy to the air. Any energy which is transferred to the air cannot be used to do any useful work to propel the aircraft forward, so it represents a loss of energy. But if we were able to create a propulsion system which could exert the required rearward force on something which would not flow rearwards, we would have a more efficient propulsion system. I do not claim to be able to identify such a system, but that does not mean that it is impossible to do so. During my lifetime a great many products have been created which I could not possible have imagined beforehand.
In that case your propulsive efficiency is 100% because the chosen frame of reference is the same thing you are pushing against.
We can of course select any frame of reference we may wish, and the results will often appear to be very different. But in this case we are considering the motion of the car over the surface of the Earth, the Earth is the most appropriate reference frame.
Replace the car and Earth analogy with a hamster wheel and you find the wheel is spinning whilst the hamster is stationary. The hamster wheel is 'stiff' and yet it spins noticeably because it has a relatively small mass compared to the hamster. It is not about stiffness (except insofar as pushing against something stiff may transfer the force to something more massive).
Perhaps stiffness is too loose a term. The hamster wheel is certainly stiff in that it will resist deformation. But it is not stiff in terms of rotation. As I have said above if we increase the friction on the wheel pivot, the resulting motion will become very different. So the motions of the bodies concerned ( in this case hamster and wheel) are not determined entirely by their masses alone.
Well, it would be very strange to argue that the Earth was accelerating relative to a car that was 'maintaining 70mph' relative to the Earth.
I did not suggest that it was a good argument or even that it was a valid argument. I simply stated that some people might choose make it.
But it would not be the least bit strange to point out that the force between tyre and Earth is a de facto torque on the Earth causing a – albeit infinitesimal – change of angular velocity. According to NASA:
"Any worldly event that involves the movement of mass affects the Earth's rotation, from seasonal weather down to driving a car"
I really hope that we do not need NASA to tell us that. But that does not mean that every car moving over the surface of the Earth causes the Earth to accelerate. In many cases (including the example I gave above) there will be similar cars exerting similar forces in opposite directions. Where pairs of these individual forces cancel each other out there will be no acceleration exerted on the Earth but the cars will still be producing thrust.
That would not test the argument anyway. If the cars are pointing in opposite directions the net force is zero so there is no torque on the Earth only compression of the surface and tension in the rope.
But the driver of each car is not interested in the net force and the propulsive efficiency of the car does not depend upon it. He/she simply wants to move forward. The tension in the rope is entirely due to the equal and opposite thrust forces that are being produced by each car. The fact that the cars are unable to move does not mean that there is no thrust. Simply cut the rope and we will see both cars accelerate away. But there will be no acceleration or even net torque applied to the Earth.
The point which I am arguing is that the thrust in our car or aircraft or whatever, is not generated by the rearward acceleration of whatever we may be pushing against. It is simply the equal and opposite reaction predicted by Newton’s third Law. The rearward acceleration of the air caused by our propeller or jet engine is a wasteful by-product of this process. The process of generating thrust would be far more efficient if we could find a means of eliminating this waste product.
If I stand on the ground my feet exert a downward force equal to my weight. If the ground is stiff enough it exerts an equal and opposite upward force which prevents me from sinking into the ground. But this process does not cause the Earth to accelerate downwards.