We all know that in an idealised world all objects accelerate at the same rate when dropped regardless of their mass. We also know that in reality (or more accurately, in air) a lead feather falls much faster than a duck's feather with exactly the same dimensions/structure etc. A loose explanation is that the increased mass of the lead feather somehow defeats the air resistance more effectively than the duck's feather.
Is there a more formal mathematical explanation for why one falls faster than the other?
Answer
We also know that in reality a lead feather falls much faster than a duck's feather with exactly the same dimensions/structure etc
No, not in reality, in air. In a vacuum, say, on the surface of the moon (as demonstrated here), they fall at the same rate.
Is there a more formal mathematical explanation for why one falls faster than the other?
If the two objects have the same shape, the drag force on the each object, as a function of speed $v$, is the same.
The total force accelerating the object downwards is the difference between the force of gravity and the drag force:
$$F_{net} = mg - f_d(v)$$
The acceleration of each object is thus
$$a = \frac{F_{net}}{m} = g - \frac{f_d(v)}{m}$$
Note that in the absence of drag, the acceleration is $g$. With drag, however, the acceleration, at a given speed, is reduced by
$$\frac{f_d(v)}{m}$$
For the much more massive lead feather, this term is much smaller than for the duck's feather.
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