Happy Pi Day everybody this year we're going to calculate pi experimentally
with a vehicle losing control
we are at a testing track at an undisclosed location in North America because a viewer
reached out to say that they work in incident reconstruction so they show up after there's
been some kind of traffic collision and they look at the markings on the road and any other evidence
and try and reverse engineer how fast the vehicles must have been going and in doing that they use
Pi so instead of starting from the beginning knowing pi and then calculating the speed a
vehicle was going if we already know the speed of the vehicle we can work backwards and get pi
this can't go wrong so yeah I'm a trained Reconstructionist for three years now so
we specialize in training and math formulas for Collision reconstruction so when you have
a construction you have the before during and after When we arrive at the sea and The Afters
there so you have the vehicles in place based on the physical evidence on the scene and the actual
damage on the vehicles you can do the during what happened during the collision and what
we do is use formulas to try and bring it back to the before to find out where people were and
how fast they were going so I was watching one of your pi videos and I was thinking to myself as a
collision Reconstructionist surely somewhere there is pi in these formulas and lo and behold there is
before we do find pi let's have a look at the map that Eric actually uses so you've
got a car it's getting out of control while it's turning and the tires leave like an art skid mark
as the car continues to turn while skidding and by measuring that skid it's possible to
work backwards calculate the initial speed and that is where the map comes in and you
know what this feels like a perfect time for some calculations calculation time all right
so we need to derive the equation which takes a cast getting out of control and gives us
the speed at which it must have been going originally and then later on we'll squeeze pi
out of that don't worry so here's what we know if we've got a car that started to slide as it
goes around a corner so I'm going to draw my car I'm just that's the car that's the ground I know
that's going to make car people very angry I'm simplifying things a lot in fact this car is on
a single tire so I'm going to draw one tire right in the middle of the car there you are
classic classic car with one tile because it's mathematically equivalent to what we're trying to
do here now if that car is going around a corner it needs a force to like push it like there's got
to be a turn turning force and if the car is skidding the only turning Force available to
keep it going in a corner is the friction between the tire and the road so our centripetal force
to keep it going in this way that's the well it's the velocity of the car squared times its
mass divided by the radius of the corner it's curling around and the friction you're getting
out at the other side here that's whatever the the weight whatever the mass of the car
times gravity pushing it down into the road times whatever the coefficient of friction is that's
that situation car is sliding that's the force from the skidding tires which I've approximated
as one tire and that's the force we know it must have been experiencing because we know the radius
it was curling around we're just going to put all this together we can already get rid of the mass
and the mass we can move the radius over there and so we end up with an equation which is the
velocity the car was originally going equals the square root of gravity times radius times friction
and that that is the number one equation for what we're doing here if you know what gravity is which
you should you know the radius that the car was following as it was spinning out of control you
know the coefficient of friction on the asphalt multiply them together take the square root and
you get the original velocity however Collision reconstruction let's go one step further they
make this even more user-friendly by messing around with the value of G allow me to explain
let's say instead of velocity which at the moment we're going to have in feet per second
you want speed in miles per hour well there are 5 280 feet per mile and there are 3 600 seconds
per hour so that conversion will go from feet per second into miles per hour so we need to get our
original formula from before instead of G being in meters per second per second it's going to be
32.185 feet per second per second I know I'm upset as everyone else okay and then there we
now got the radius and the coefficient friction so we can do is take all of this Chuck it inside
the square root here by squaring it cancel it all down and that actually comes out to be
14.962 it's so close to 15 it's ridiculous so what we can actually do is say say that
the speed in miles per hour is just the square root of 15 times the radius in feet times the
coefficient of friction you just plug the numbers in and everything drops out neat and tidy because
we've messed around with this value in here so that is exactly what we're doing today we have
a professional driver in the car and they are going to Skid the car out of control in this
exact way inside the car we have a camera filming the speedo so we can let it go back look at the
footage and calculate the exact speed the car was going when it lost control and the point where it
loses control is defined as when the skid marks from the back tires cross the front tires and
so that's the point we're looking for in the markings on the tarmac once we've found one of
these skids we can then get down to work and up until now this is what Eric would normally do in
one of these situations here however we're going to deviate slightly because we're kind of avoiding
accidentally either using or going past pi along the way let's say you've got this nice skidding
out of control tire mark like that and you want to work out what's the radius of the curvature what
Eric would normally do is take a cord back over here like this measure that distance also measure
this little distance here and using those two distances it's a reasonably simple bit of geometry
to go back to whatever the radius must be if you completed that entire curvature and Eric was kind
enough to show me how they would normally do this the marks they look for the measurements they take
however we don't want to do any of that no no no no no what we're gonna do instead is complete the
circle so we're going to fill in the rest of the circle as if this I know the car did a full donut
we're going to measure that complete circumference which did involve putting a string into the ground
and then measuring around but we're avoiding that the point is we now have the circumference
we can then use the speed and work backwards to get whatever that radius was purely from
the calculation not from measuring it we divide one by the other give us some pi we use one of
those wheelie measuring out things that they would normally use to measure the cord to go all the way
around the circumference once we had that value we could now take the speed of the car which we
can get from the footage and use that to calculate what the radius must have been oh as long as we
know what the coefficient of friction was we've extended the skid marks into the full circle
measure the circumference although you wouldn't normally do that would you no no no no no that's
like the worst way of doing it what we're trying to do is avoid accidentally getting the radius
along the way because we wouldn't we don't want to accidentally use pi and traditionally you just
calculate the radius and then carry on with your life correct right so we and and we could have
just measured this accomplishment already so that big circle but that's not the point
we're using the cast skid marks and the speed to get Pi now the other factors we have to include
uh first of all is the coefficient of friction where the car lost control so here we've got a
40 pound test mass and what are we going to do with this so you put it down on the pavement and
the direction that the vehicle was going you put the scale on there attach it right and we want the
kinetic coefficient of friction correct so there's actually a little I remember oh on
that yeah and do I have to go like perfectly so you want to get it as level with the ground as you
possibly can like this you don't want sine and cosine getting in the way exactly story energy
takes more energy to get going than just once we get it going I get into a steady state yeah and
you read it off and you try to get it as close to the ground once I get into it there you go
so it's about looks like 26 26 ah and it weighs 40. so 40 pound weight 26 so it's 26 over 40.
52 65. it should be about 0.7 okay point so 0.65.7 dry asphalt like this is about 0.7 oh
okay but it's kind of an angle so sometimes it'll the ground we're going down the grid
okay right sometimes in the great the grade you actually use in the equation sometimes right but
if you're doing the drag or sometimes we'll actually get the vehicle itself and skid it
out and find out how much you know if you got the vehicle why not just use the vehicle to find out
you know at 30 miles an hour breaks how far yep reverse back if you have that option but if we've
done that measurement like going down the slope and everything that sort of kind of factored in
the gradient and this correct assuming the tires are this you know there's little variables because
that tire may not be the same tire that was on the car stuff like that but it's an approximation
that we use yeah I don't allow me but for the uh historical level of precision calculations
we good we good right and do we need to worry about the gradient this way so normally you
would but it's such a it's it's not as bad like you do the super elevations what it's called
yeah really banked curves right for his it's not terrible because it's a test track we kind of try
to keep it controlled well I tell you what let's just don't worry about it that's fine with me okay
so we got our coefficient of friction we're going to ignore the gradient yep let's calculate pi
we now have all of the results we ran out of time to do this properly on the day but it's okay I'm
joined uh by Eric through the magic there we go uh the magic of East telecommuting in to this pi
day okay so Eric I've got here the uh the classic equation yes sir you've now got all the values for
this yep we're going to fill them in we're going to get a pi okay right so uh what was the speed
in miles per hour that we had at the at the moment of skidding so the speed at the moment of skirting
was uh calculated to be 25 miles an hour okay so we've got 25 I'm gonna put the units in even
though they make me a little sad uh 15 is 15. now now radius we're not doing anymore correct because
we want to get pi and we've got the circumference so what did we get for the total circumference if
you continue if you like expanded that skid all the way around correct so yeah when the when the
car continues to Skid around the circumference of the whole circle was 382 feet 382 feet all the way
around okay now ah friction so the coefficient of friction we dragged that heavy thing what
were our units oh what were our values for that so the dragster good mass weight 38 pounds and then
the force that the force it took to drag it across the asphalt was 26 pounds so 26 over 38 will give
you 0.68 26 over 38 0.5 68 yeah which is unitless I will yeah because the codependent of friction
is just what you multiply the downfalls by to get the sideways force so we're just dividing
one by the other right give us our coefficient of friction okay so you know what I'm going to
rewrite uh everyone's favorite equation in a way that's going to spit Pi out for us a bit
easier so we're going to do now is put our values in there and we get this Pi days value of pi okay
uh Moment of Truth we've got 15 times 382 times .6 I don't know it doesn't matter how many times
I do this I get nervous every time yeah I know divided by you know what scientific mode there
you can do it all at once in Brackets 2 times 25 squared close brackets equals 3.117 that is way
I'm glad I resisted working it out between now because I love it I was like if it's between
two and four right very very happy 3.12 I think that might even drag my
um my average track record closer to Pi uh there we go that's amazing
everyone also shows that just how useful these equations are for calculating the speeds of
vehicles because they're really Just Energy equations which conserving energy is all we're
doing so yeah exactly and I mean obviously we had to dance a lot to avoid accidentally getting or
using pi along the way right so it's include physical this is ridiculous yeah exactly right
so if we just done your process of measuring that chord and getting the radius straight away then
this drops out and what I like actually is having that 15 I know I'm very disparaging about this
but having that 15 with all the unit conversions like neatly packed up in one number right yeah all
you got to do is measure the chord to get the radius drag a heavy thing yeah take the square
root and it drops out already in miles per hour I think that's right which is very disappointed as
possible yeah exactly that is absolutely amazing I can't believe in what arguably sounded like
the most ridiculous way I've ever attempted to calculate pi that we got that close that's so
good all right we're gonna uh let Eric get back to their real job which is a bit more useful um than
whatever my real job counts has Eric thank you so much you can go back to reconstructing collisions
3.12 everyone please use that for the remainder of the year and those of you who keep track of
how good or bad my Pi Day calculations are you can put 3.12 into the mix and rounding
up to two two decimal places there you go excellent huge thanks to Eric and all the
other undisclosed people at the undisclosed location they were so accommodating absolutely
amazing they can get us in on their test track and thank you to you I'm talking
to the people who are going to volunteer to help me out next year little heads up
it's going to be doing pi by hand year and I might be over ambitious which will require a
lot of volunteers people who want to come and hang out with me and do a lot of tedious calculations
possibly for several days if that sounds like you there is a link to the volunteering form below in
all honesty thank you very much to my patreon supporters they are the reason why I could
travel to undisclosed location via undisclosed means of transportation at undisclosed cost but
I can disclose it was thanks to my patreon supporters that this was able to happen and
indeed next year will happen so thank you so much to them um here's some of their names