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*From*: John Denker <jsd@av8n.com>*Date*: Thu, 22 Dec 2022 12:34:54 -0700

The real story is simpler than the recent coverage would

suggest. It can be understood using high-school physics, if

we frame the question properly, steer clear of ambiguous

terminology, and pick apart the various contributions.

1) The 𝘮𝘢𝘪𝘯 reason falcons dive is super simple: In level

flight, a falcon is not particularly fast. Compared to most

of its prey, it is comparable in speed or slower. If it

didn't dive, it would have a hard time catching anything.

It speeds up in a dive for the same reason a car speeds up

when going down hill. Conservation of energy. High-school

physics.

So it's not wrong -- just misleading -- to say that

stooping helps the falcon catch agile prey. It also helps

catch non-agile prey.

2) We now consider maneuverability aka agility. Using high

speed to increase maneuverability is counterintuitive,

because a car is less maneuverable at high speeds. If you

try a sharp turn at high speed, the car will skid or flip

over. Similarly, in an airplane, if you try a sharp turn at

high speed, the wings will break off. The same goes for

birds. Other things being equal, there is only so much

acceleration a bird can take without snapping a wing.

--> It helps to frame the discussion

in terms of acceleration.

Let's compare two scenarios, namely two strategies for the

same bird. Let's assume for simplicity, temporarily, that

nothing changes except the speed, so in both scenarios the

bird can withstand the same amount of acceleration. At this

point we find that there are at least three distinct

notions of maneuverability, and the distinctions matter.

— In terms of the time it takes to complete a full turn,

higher speed means less agility. The time required to

reverse the velocity grows in direct proportion to the

magnitude of the initial velocity (given constant

acceleration). This is basically the definition of

acceleration. High-school physics.

— In terms of turning radius, i.e. distance covered during

a full turn, it's even worse. The distance grows in

proportion to speed squared, because you're going twice as

fast for twice as much time. High-school physics again.

— However (!) neither of those is relevant here. The prey

makes a small jink to the left or right, and the pursuing

falcon wants to do the same. The falcon can choose either a

moderately-high or extra-high speed for the dive. We can

visualize this with the help of a diagram:

https://av8n.com/physics/img48/falcon-constant-acceleration.png

The bird is going twice as fast in the red scenario,

compared to the blue. You can see that the red path covers

twice as much distance in the same amount of time. The rate

of turn (in degrees per second) for the red path is worse

by a factor of 2, and turning radius is worse by a factor

of 4. However (!) the lateral distance covered is the same,

as shown by the green bar in the diagram.

We can quantify this as follows: For a small jink, the

lateral motion is independent of the forward motion, so it

is easy to analyze, as Galileo taught us. The acceleration

(𝘢) is the same for both scenarios, (since it's the same

bird, just a choice of speed). Similarly the available time

(𝘵) is the same for both. Therefore the lateral distance,

namely ½ 𝘢 𝘵², is the same for both, as shown by the green

bar. To avoid confusion, we call this the 𝘫𝘪𝘯𝘬𝘢𝘣𝘪𝘭𝘪𝘵𝘺. It

is a particular type of maneuverability.

The researchers used a big fancy computer program to figure

this out, but you can understand it in terms of high-school

physics ... if you frame the question properly, and apply a

tiny bit of domain knowledge. The key idea is to frame the

discussion in terms of acceleration. There's only so much

acceleration the bird can withstand.

As will be explained in a moment, both of the scenarios

just mentioned are in the high-speed regime. In this

regime, a further increase in speed brings neither an

increase nor a decrease in jinkability. It's a wash.

3a) In an airplane, there is a concept called maneuvering

speed, denoted 𝘝a. Loosely speaking, below 𝘝a the wing will

stall before it breaks, whereas above 𝘝a it will break

before it stalls. Both the red and blue scenarios assume

the speed is above 𝘝a. The wing-breaking acceleration is

the limiting factor.

Domain knowledge: Wing stall corresponds to maximum

coefficient of lift. The force is proportional to

coefficient of lift times velocity squared.

𝘝a is a scalar speed (not a vector velocity), but we will

denote it 𝘝a (rather than |𝘝a|) to conform to convention.

We now consider speeds below 𝘝a. In this regime, wing stall

is the limiting factor. In this regime, even at stalling

angle of attack, the wing cannot produce a wing-breaking

amount of force. The maximum possible acceleration is

proportional to speed squared.

In this discussion, "high speed" means speed |𝘝| greater

than 𝘝a.

The prey can't afford to spend its life flying around at

high speed, and when it comes under attack it doesn't have

time to speed up. So, comparing bird 1 with bird 2,

assuming they are the same except for speed, bird 1 has

worse jinkability by a factor of 𝘜₁² / 𝘜₂² ... where for

each bird 𝘜 is either |𝘝| or 𝘝a, whichever is less.

3b) Compared to its prey, the falcon is sturdier. Heavier

bones and all that. This makes sense; a mid-air collision

that kills a pigeon doesn't kill the falcon. This is one

reason why the falcon is neither fast nor maneuverable at

low speeds; it's just too heavy and clunky. However, this

means it can withstand higher acceleration, so at high

speeds it becomes more maneuverable. At high speeds the

falcon has greater jinkability compared to the prey, even

if the prey could achieve a comparable speed, because wing

breakage is the limiting factor.

3c) When stooping, the falcon tucks in its wings, mostly.

At any given speed, this decreases the bending moment on

the wing bones. Same force but less leverage. So this

increases the magnitude of the acceleration that the bird

can withstand. The coefficient of lift is reduced, so the

bird has to go super-extra fast to produce the desired

acceleration.

3d) Combining (3a), (3b), and (3c) we expect the falcon to

be have better jinkability compared to its prey.

4) We can now consider reaction time. When the prey jinks,

the falcon's response is delayed by its reaction time.

However, it can make up for this by using greater

acceleration during the remaining time.

5) We should not make too much fuss over high-speed

maneuverability, because at the end of the chase, when you

might think maneuverability was maximally important, the

falcon slows down. That makes sense because a mid-air

collision at top speed would kill both birds.

6) There is such a thing as multi-factor causation. You

could perfectly well say that stronger bones improve

jinkability (assuming high speed). Or you could say that

tucking in the wings improves jinkability (assuming high

speed). Or you could say that high speed improves

jinkability (assuming a high tolerance for acceleration).

It's also true that quick reaction times help catch agile

prey.

Or you could forget about maneuverability and say that the

falcon dives in order to go fast enough to overtake the

prey.

Solving this problem requires some skill. In particular, it

requires picking the problem apart to identify the various

factors, each of which is easy to understand in isolation.

After picking them apart and analyzing them, it is crucial

to put them back together again. That is, fixating on any

one factor to the exclusion of the others would be highly

misleading. For example, saying "high speed improves

agility" is misleading, because although it helps in the

low-speed regime, once you're in the high-speed regime

further increases don't help. Also, any mention of

"agility" is open to misinterpretation because it is

ambiguous. Et cetera.

**References**:**[Phys-L] raptor physics***From:*John Denker <jsd@av8n.com>

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