So you think you understand… Descent.

Continuing my theme that all is rarely as it seems with how aircraft work, particularly when pilots are equipped only with half truths peddled in early training. And pity the controllers who are typically left even more in the dark about such things.
Let’s think about how fast aircraft descend. The ab initio trainers pilots first encounter operate at pretty much the same weight day in, day out. The biggest change is when the instructor gets out for that nail biting first solo, but he (she?) typically accounts for less than 10% of the aircraft mass. The student, if not too terrified to notice, will appreciate a much better rate of climb, and may spot that the aircraft glides a little further than with two on board. Those trainers typically climb and descend at much the same speed which isn’t very far from their best glide or minimum drag speeds. The student doesn’t encounter really large weight changes and wide speed variations until getting into a real airliner for the first time. At some point, perhaps after the descent planning has gone badly wrong yet again, it dawns that at high speeds, light aircraft descend faster than heavy ones.


This seems pretty counter-intuitive, and I recall asking around members of my course, first to see if what I thought was happening really was, and then to see if anyone knew why. This is when the first great truth of airline training became clear: the Training Captain (otherwise known as God) IS NEVER stuck for an answer.. even if it’s the wrong one. ‘It’s something to do with the centre engine, old boy’ was dismissive reply. This needs a little clarification, we were doing our base training on that queen of the skies, the Trident, and during descent in order the keep the pressurization going, the #2, centre, engine had to be maintained at a minimum rpm while the two outer engines were retarded fully to idle. What this had to do with the rate of descent effect, I couldn’t figure out, but with lots more to learn and worry about, I put the question to one side for a while. But it still bothered me, and one day I sat down and thought, and realised that the answer is very simple (well it had to be for me to work it out), but I don’t think I have ever seen an explanation elsewhere.
Let’s just recap what needs explaining: at a given high speed, let’s say 350kt IAS, an empty aircraft will typically come down about a third faster than a full one (and the difference may be even bigger for a long range type with a wider weight range). At approach speeds, there is a small effect of weight, but the lighter aircraft does descend slightly more slowly.
To understand what is going on we have to go back to the basic lift and drag forces acting on the aircraft. As you may still remember from lectures long ago, the aircraft’s drag can be thought of as made up from two components: one, ‘profile’ drag which varies with (speed)² and is more or less independent of weight, and the other, ‘induced’ ( or lift dependent) drag, which varies with (weight)² and is proportional to 1/(speed)². The total drag is the sum of these two, one rising with speed, the other decreasing (see the illustration). As a result, there is a speed at which, for a given weight, total drag is at a minimum, and not surprisingly that is known as the minimum drag speed usually abbreviated to V_MD. This speed varies with weight, but for most current transport aircraft at typical approach weights it is around 220kt. It is worth remembering that although long range aircraft can have very different take-off weights because their fuel load depends obviously on how far they are going to fly, when they come back to land, they tend to be in a very much narrower weight range, having at that time fuel on board for, holding, approach, divert and final reserve in addition to their payload, just as any short or medium range aircraft does. So most of the weight difference between one descent and the next is the payload. The figure illustrates how the drag varies with speed, Note that at min drag, the two components are in fact equal – this isn’t chance as some simple calculus can show, it is a feature of two curves where one is proportional to the reciprocal of the other…. well, you did ask… In the figure, I have drawn the drag curves for a hypothetical aircraft with a V_MD, at a given weight W of 200kt to keep the arithmetic simple, and with a V_MO of 350kt. I have added the curves for the same aircraft for a lower weight of 0.8W. The point to notice is that at high speed the two total drag curves are virtually identical i.e. the drag doesn’t vary much with weight.


So let’s consider what happens with the aircraft cruising at V_MO at top of descent. The engines are reduced to idle and the aircraft starts to descend. The work done against drag, which we have just agreed is much the same at both low and high weights and which has to come from somewhere is supplied by the reduction in the aircraft’s potential energy … and there you have it, the lower the weight, the further it has to fall to do the same amount of work. QED. At low speed, as the drag depends partly on (weight)², the drag decreases faster than the weight does, so the lighter aircraft does descend more slowly (which corresponds to our experience on basic trainers), though on my simplifying assumption of zero idle thrust, which isn’t totally realistic, the effect isn’t noticeable until well below V_MD.
I hope that puts some doubts out of some pilots’ minds, and perhaps clears up some controller puzzlement.