inverted spin recovery explanation

" An interesting fact, which is not obvious to many folks,
including
some aerobatic pilots (judging by the number of loop into the ground
accidents) is that the radius of any turn, up, down, sideways or
whatever, is a function of the square of TRUE airspeed, which is of
course a function of density altitude and calibrated airspeed.

So, if the density altitude increases your true airspeed by 5mph,
you get a 5mph^2 impact on your radius. This kind of change in radius
can ruin your day if you are playing down near the dirt.

This velocity^2 thing is also why the reverse cuban or loop down is
a real killer. If you start the pull with X knots too many, you will
use X^2 more radius for the 1/2 loop, throw in an increase in TAS of
say Y due to density altitude and you are into (X+Y)^2 more radius ...
not good. If you have not left margin either in terms of available G
or altitude you are either gonna high speed stall on the way down (and
hit the ground) or hit it on the arc.

I think this may need a little more explaining even if only for my
understanding. I am very new to aerobatics.

So if I normally commence a loop at 100 knotts but get the entry speed wrong
and start at 105 knots then my loop (assume horizontal plane and constant
speed for simplicity) will be

New_Loop_Diameter=Old_Loop_Diameter x New_Speed ^2 / Old_Speed ^2
i.e. a factor of 1.103.

A bad entry of 15 knots over speed would have a factor of 1.323.
But a target speed of 200 but entry of 215 would have a factor of 1.156.

If my understanding is not correct then please explain why. I prefer to
understand the physics/maths before I attempt some of these manoeuvres.

Anyone care to formulate what happens when speed ( or "G") are not constant?

Yes unfortunately while the radius is a function of velocity ^ 2, it
is also a function of a great many other things. The main point is to
recognize the non-linear relationship between speed and radius, and to
then recognize that density altitude changes produce non-linear
effects on radius.

I don't recommend you do anything with the math except make a big
'ahhhhhhh' sound and let it cause a sensible sense of dread in you
whenever you pull towards the ground.

By the way, a minimum radius pull is at a bit above the stall speed
so if you pull to just before the buffet you are pretty close
regardless of airspeed.

Cheers,

Peter