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# Re: [Phys-L] raptor physics (spoiler)

• 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.