this video is sponsored by brilliant
when it comes to solving crime there's a
lot more math physics and statistics
going on in the background than you may
realize and this is need for things like
making predictions or reconstructing a
crime scene and sometimes the underlying
analysis can be as simple as looking at
the outcomes of a coin toss if I take a
coin out of my pocket and I flipped it
ten times and every single time it
landed heads what would you think would
you think it's a fair coin or maybe it's
a little bit biased or weighted
well you doesn't be right to have your
suspicions because I mean the chance of
that happening for a fair coin is about
one in a thousand but hey it could still
happen
well statisticians use something called
hypothesis testing to determine the odds
of some hypothesis being true like where
the odds of this coin being fair based
on the outcome of a certain number of
flips because even if you got a hundred
heads in a row there's still no
guarantee that the coin is biased but as
you can see the chance that it's biased
clearly changes based on the outcome now
this type of analysis was actually used
to sentence Kristen Gilbert to prison a
few decades ago
Kristen Gilbert was a nurse who was
often known for being the first person
in the hospital to recognize when a
patient was going into cardiac arrest to
which she would then give the necessary
medicine while an emergency team arrived
and this saved several lives but
suspicion started to grow due to certain
factors and people began to think that
maybe she was inducing cardiac arrest in
these patients so she could then catch
it and quote save the day and thus an
investigation was started which involved
a statistician being brought in now what
this person did was looked at the number
of deaths that occurred while Kristen
was on duty versus how many occurred
when she was not at the hospital and
they compared these numbers to see if
there was an obvious difference now if
one or two more deaths happened on her
watch that could just be a coincidence
it doesn't prove anything so how many
more deaths would need to occur first to
have sufficient evidence that something
criminal was going on well now we need
statistics to answer this question and
using hypothesis testing the
statistician determined that there was
less than a one in a hundred million
chance that that many deaths occurred
just by chance on her watch meaning it
was almost a guarantee that something
criminal was go
although this was not the determining
factor in your sentence since again
she's probability this did help provide
enough evidence that she was guilty and
she's currently serving life in prison
now another interesting use of math to
solve crime which actually isn't based
on a true story as far as I know has to
do with looking at the tangent line to a
curve so let's say you come across a
crime scene and notice bike tracks going
in each direction the question we want
to answer is can we determine which way
the bicycle is traveling and can we
determine how large the bike is just
from the tracks and the answer is yes to
both of these now if the cyclist moved
in a perfectly straight line and left
these tracks we're out of luck but since
that probably would not happen we can
solve this mystery since bikes have two
tires we're going to see two sets of
tracks so it could look like this if
they had to make some weird turn but
just help really analyze the differences
I'm going to use this more chaotic set
of tracks found on brilliant site to
help the visualizations the first
question we need to answer is which of
the two sets of tracks is the back tire
we'll think about how a bike works the
front tire always points in the
direction of motion at that instant in
time if the bike looks like this and you
start pedaling the direction of motion
would be in this direction the way the
front tire is pointed so when we look at
the tracks created by the front tire the
tangent line points in the direction the
bike was going at that instant the back
tire on the other hand is fixed it
cannot rotate and the way it's fixed it
is always pointing at the front tire
this means that if we draw a tangent
line at any point on the back tires
tracks it should intersect the front
tires tracks since at that point in time
the back tire was again pointing at the
front tire so looking back at this
picture clearly the dashed line cannot
be the back tire since a tangent line
here for example does not intersect the
other curve or what would be the front
tire so the solid curve must be the back
tire since at any point I can draw a
tangent line that intersects the other
curve okay but we still don't know which
way the bike is moving but here's how
we're gonna find that out again the
tangent line to the back tire will
always intersect the front
tires tracks but the length of that line
has significance it's the distance from
the back tire to the front since the
bike obviously does not change sizes
that moves that length does not change
so the line segment that is tangent to
the back tires tracks and intersects the
front should always be the same distance
this means we know the bike was moving
to the right to see that if we picked a
point then drew a tangent line towards
the right we'd get a length pick another
point and we could find an intersection
point Yuling the same like same with
another point and so on you won't find
that going the other way though so there
you go the bike was traveling to the
right the salted line is the back tires
tracks and the bike is about this long
from the back to the front tire so
that's a lot of information from just
two sets of tracks and now if you were
asked what the tire tracks the bike
moving in a circle looked like you'd
know it's not this but rather this where
the front tire always points in the
direction of motion and the back tire
points towards the front tire such that
the length of the line segment is the
same as the distance of separation of
the wheels next up traffic is a
complicated thing maybe you've seen this
animation that shows that traffic can be
caused by just one person cutting off
another now you may not think that much
goes into solving the problem of traffic
besides adding lanes here and there or
controlling through a on rings for
example but oh man there is a lot of
analysis that goes into traffic for one
traffic engineering is itself an entire
branch of civil engineering but also go
look at the traffic flow Wikipedia page
it's about as long as world war two's
Wikipedia page and you'll come across
things like single vehicle dynamics
methods of analysis the hamilton-jacobi
partial differential equation
three-phase traffic theory kinetic wave
model merge models traffic bottleneck
and way more so yeah not as simple as
one may think but we're gonna take it
down and just look at how bad one single
event can be in terms of track let's
take a one lane road where a tree falls
and for this problem there's no going
around or doing a u-turn or anything
like that
now let's say on this road the flow rate
is 10 cars per minute so every minute 10
more cars build up in front of this
fallen tree then let's say on average
the cars are spaced 5 meters apart as
they sit in the traffic jam this means
the physical line of cars is increasing
at 50 meters per minute now once the
tree is removed what happens well the
first person starts moving then there is
a delay just due to reaction time until
the second person starts to move this
repeats for the third then the fourth
and so on if we assume the reaction time
of two seconds on average for each
driver that means the front of this
traffic jam is moving backwards at a
rate of one car every two seconds and
since they're spaced 5 meters apart that
line is moving at 2.5 meters per second
or 150 meters per minute this is known
as the leading edge propagation speed
but note that cars are still arriving to
this traffic jam at the same rate of 10
cars per minute which we said increases
the lines length at 50 meters per minute
that means the front of the jam is
approaching the back at a rate of a
hundred meters per minute if let's say
the line was one kilometer long when the
tree was finally clear that means it
would take ten minutes for the traffic
jam to be entirely cleared and the front
line to catch the back and in that time
the last car to encounter the traffic
would find it 1,500 meters from where
the tree fell so even once the tree is
cleared there's someone several
kilometres away that is going to
encounter the tail end of the traffic
over a kilometer or nearly a mile from
where the tree fell but the crazy thing
is if 29 cars arrive per minute instead
of 10 and still the line was one
kilometer long once the tree was cleared
the jam would propagate 30 kilometers
from that tree over a course of over
three hours from when the tree was
cleared now the line would not be that
long but it would propagate that far
backwards like a wave as people arrived
at the back while others drove away from
the front as then it would take three
hours for that front line to catch the
back
think about that a tree could fall on a
road be picked up not even seven minutes
later and cause over three hours of
traffic that would move 30 kilometers
from where that initial event happened
so if this event happened in Los Angeles
and someone left from San Diego at the
same time the tree was finally cleared
that person leaving would still hit the
traffic caused by that event several
hours later - and at 30 cars per minute
theoretically the traffic jam would
never dissipate until the traffic flow
slowed down or people reacted faster so
there's an overly simplistic analysis of
how traffic works when engineers design
a building there's obviously a lot of
work that goes into it
like there's even people in charge of
analyzing the soil that the building
will rest on top of because when there's
not well we've seen what happens but now
question is how do skyscrapers even
stand up when we've clearly seen how
unstable structures become as height
increases which you can see if you just
stack a bunch of books on top of each
other
well the thing is as height increases
the center of mass of that stack moves
up if all the books have the same weight
then it being the dead middle and
there's always a gravitational force
applied at that point so if there's even
a slight tilt in the structure this will
actually apply a torque about the
balance point so as the structure height
increases and that center of mass moves
up it will become easier and easier for
that structure to tip over but when it
comes to buildings they actually start
far beneath the ground typically a large
hole is dug and filled with dense
concrete where the main support beams
are driven into this portion as well as
the lower sections of buildings are
really heavy
in fact sometimes the center of mass of
a building is beneath the surface and
this distribution of weight almost
removes any risk of toppling from wind
or earthquakes but buildings do not stay
perfectly still even in strong winds
they're subject to swing in fact people
living on top floors can experience
feelings of seasickness their ways
engineer's can reduce us though and this
is actually a big deal in structural
design for example some buildings use
tuned mass dampers to reduce mechanical
vibrations
this is a picture of one in the Taipei
101 located in Taiwan this thing weighs
660 tons which is about 20 fully loaded
18-wheeler trucks or about 8,000 of meat
so yeah it's really heavy but
significantly reduces the chance of
collapse due to strong wind or an
earthquake as you can see here but
another solution is to use slosh tanks
yes just huge tanks of water a place at
the top of buildings can reduce
vibrations this is because when a force
is applied to a tank of water the water
itself applies a resultant force in the
other direction so in placing the top of
a building once the building starts to
sway to the right let's say the water
will slosh to the left and apply a
restoring force to stabilize the
structure if you hold a chain or a rope
at both ends and just let it hang what
shape will it make you may think it
makes a parabola which was the
assumption at one point in time but in
fact it makes a different shape one with
this equation which is also a hyperbolic
cosine curve this is known as a catenary
if you turn the shape upside down you
get the shape of many arches found
around the world and this is used
because of how forces are distributed
throughout the arch as a result next if
you're ever asked at an interview why
manhole covers are round the correct
answer is because no matter how you
orient them they won't fall through
unlike many other shapes but if there's
a follow-up question of what's another
shape that can accomplish this then you
can really impress them by saying the
reloj triangle this is a shape form from
the intersection of three circular disks
along an equilateral triangle the shape
can be seen in architecture or certain
types of machinery for example it's
useful machinery since it can reach into
corners due to its ability to rotate
within a square next up if you want to
throw a baseball as far as possible you
may think you should throw it at a 45
degree angle but this is incorrect
that's only for when the ball will start
and end at the same height when you
throw an object some arbitrary height
this is in fact the equation for the
optimal angle as the height of the
release point increases optimal angle
decreases from 45 degrees so if you want
to optimize your Shoplet throw the
optimal release is around 42 degrees put
also know that this depends on the
release speed this next thing is
probably saying you've never thought of
but if you were looking at a large
statue where would you have to stand for
optimal viewing as in where should you
be such as the angle between the line
connecting you and the bottom of the
statue and you and the top of the statue
is a maximum well using some
optimization we find the optimal
distance is the square root of the
distance from the ground to the bottom
of the statue times the distance from
the ground to the top of the statue this
is also known as the geometric mean of
those distances so as an example for the
Statue of Liberty according to its
measurements you should stand 216 feet
or about 66 meters away and lastly if
you're curious about things like
collision reconstruction the Coriolis
effect the physics of beam and truss
bridges and plenty more
and check out brilliant who I'd like to
thank for sponsoring this video a lot of
content you saw here was pulled from
their site under their physics of the
everyday course which reveals many more
ways you can use math and physics to
understand the world around you really
is a platform that not only teaches you
new concepts within a wide variety of
math and science subjects but also
challenges you with constant practice
problems along the way these problems
really test whether you understand the
underlying math and science but what I
really like about them is they very
often show unique or real-world uses of
the technical topics you go over I
honestly do learn a lot of new
information every time I go through a
course on their site for a video like
the bike tracks problem was probably my
favorite thing I learned for this one
but there's way more on the site for you
to learn as well additionally the first
200 people to sign up by clicking the
link below or going to brilliant org
slash major prep will receive 20% off
their first annual premium subscription
and with that I'm gonna end that video
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