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## Physics library

### Course: Physics library > Unit 4

Lesson 1: Circular motion and centripetal acceleration- Race cars with constant speed around curve
- Centripetal force and acceleration intuition
- Visual understanding of centripetal acceleration formula
- What is centripetal acceleration?
- Optimal turns at Indianapolis Motor Speedway with JR Hildebrand
- Calculus proof of centripetal acceleration formula
- Loop de loop question
- Loop de loop answer part 1
- Loop de loop answer part 2

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# Optimal turns at Indianapolis Motor Speedway with JR Hildebrand

Created by Sal Khan.

## Want to join the conversation?

- Fascinating discussion! I'm curious what the relative radii of these three circles around the track are.(40 votes)
- The increased radius of the much larger circle really opened my eyes to the problem. Its like the driver is performing the best "trade-off" or "compromise" between centripetal force and the work required to complete the turn AS WELL AS setting up the car to then accelerate down the straight line as quickly as possible. Another instance of mathematics underlying everything that happens, even if we are unaware of it while we do it.(47 votes)

- Cool! But what about if you were running, what is faster?(8 votes)
- I guess the Inner circle would be a better option then.

Because when running, you aren't at much velocity. So you'll not face much Centripetal Force because in the formula,`Centripetal Force is directly proportional to the Square of Velocity.`

And also You can observe in Olympic and other races the players choose the Inner circle always.(44 votes)

- As the car rounds the turn, does it lose speed due to the extra work needed to maintain the turn including friction; and if so, though the pedal is to the metal, is the car losing speed though accelerating?(5 votes)
- Even at full throttle around the corner the car will lose speed due to tire drag as the inertia of the car would love to continue straight ahead, yet the input from the driver (steering wheel through to tire contact patch) is asking the car to turn. The result of these forces, if the driver is correct, will be that the car changes direction (success), but then also loses speed (a clear trade off) as the tires themselves twist and flex to balance the forces interested in continuing on said path (inertia) and the driver's interest in changing direction. The driver turns the steering wheel; this continues down through the metal steering rod, rack/pinion, and out to the steel wheels. Everything is a solid metallic connection. Then, you reach the tires. It is the tires that mediate between the massive momentum (and therefore inertia) and the driver's interest in turning. In order to make it all work, the tires must twist (slip angle) between what the driver is asking the car to do, and what the pavement and grip level of the tires will allow. I'm sure there are plenty of other discussions about slip angle; I'm not an engineer, but I am a supporter of Mr Hildebrand, whom I've known well over a decade and believe in immensely; good luck JR! -DMc.(7 votes)

- Can any other car take such sharp turns if they used the same tire?(1 vote)
- No. Cornering ability is also a function of the vehicle's suspension, weight, downforce, and many other things. Also, race tires are designed to heat up at speed and become more "sticky". A regular vehicle would not drive fast enough to heat up race tires properly.(8 votes)

- I am a little confused weather the centripetal acceleration decreases the speed of car. Becouse in the previous videos Sal told about the constant speed the object has during circular motion.(4 votes)
- The centripetal acceleration does not decrease the speed of the car. The problem with having more centripetal acceleration is that you have to accelerate the car very quickly over an even shorter period of time and most cars just won't be able to make that acceleration fast enough. Hope this helps!(2 votes)

- Wait, so is a higher centripetal acceleation better for a race car driver?(3 votes)
- He needs the right amount for the speed he has and the radius of the turn he is making.(3 votes)

- Where does Centripetal Force of car travelling on a banked curve come from? It has Fgy and Normal Force in vertical direction, and Friction and Fgx in horizontal direction all cancelled out right?(2 votes)
- No. Friction + the component of the normal force that is parallel to the ground give you a net centripetal force.(3 votes)

- can anyone explain me the difference between centripetal and centrifugal force?(1 vote)
- Centripetal force is toward the center

Centrifugal force is away from the center

In circular motion, a centripetal force is required.

Centrifugal force is somewhat of an illusion. When you go in a circle, it feels like you are being pushed outward, but really you are being accelerated inward by the centripetal force.

Sal has vids about circular motion.(5 votes)

- The thing thats nagging me[stupid thing really] is that why does the path he chooses matter cause if he chooses the shortest path i.e. the inner one then by mv^2/r he has more centripetal acceleration so less chances of overturning or skidding so isnt a small radius better than a larger one.(1 vote)
- For the inner one he doesn't HAVE more centripetal acceleration he NEEDS more centripetal acceleration. Since that acceleration comes from a frictional force applied the tires, his chance of overturning or skidding is HIGHER when he tries to take the tighter turn at the same speed.(4 votes)

- So you need to make an arc with the biggest radius... but on the other hand you should get the shortest distance! So which arc is BEST??(2 votes)
- here's one more factor; G-force

which is mass times acc_centripetal and its direction is opposite of acc_cent, meaning if you have too strong acc_cent, you(r car) must be pushed outward or even flipped over outside the road

that may be the first and foremost concern in this talk

that's why the larger the arc, the faster a car would pass through the corner. cause you don't need to worry too much about flipping over then can keep the greatest velocity while cornering

in this specific case, that is the third option starting from the rightmost, piercing through the leftmost, and exiting to the uppermost part of the road

put in another somehow simpler perspective, that path is closest to the stright path thus you can apply the closest velocity you use in the straight path thus it's the fastest(1 vote)

## Video transcript

SAL KHAN: This is Sal here
with famous Indy car driver-- smiling when I said
famous-- JR Hildebrand. And since you're
here, I thought I would ask a question that's
always been on my mind. JR HILDEBRAND: Yeah. SAL KHAN: We have a picture
here of the Indianapolis Motor Speedway. And I've always
wondered how you-- it seems like turning is a
very important part of the-- JR HILDEBRAND: It's
absolutely an important part of what we're doing. SAL KHAN: --of the race. JR HILDEBRAND:
People get fixated on the car going straight. But the turning part
is pretty important. SAL KHAN: Turning seems
to be the part where a lot of the skill
comes into it. And I've always wondered,
what is optimal? Do y'all try to minimize your
distance and kind of take the turn as quickly or as in
short of a distance as possible by really hugging the
corner, by going like that? But when you do that,
you have to turn more. There's more g-forces. There's more kind
of centripetal force that your tires
have to deal with, the human has to deal with. Versus taking the
outside where you have to cover more distance, but
the centripetal acceleration, the g-forces aren't
going to be as dramatic. So how do you think about that? JR HILDEBRAND: Well,
every track ends up being a little bit different. But when we take Indianapolis
here as the example, if you're already
on the inside-- it's like the 800 meter
runner's kind of path. It's the shortest distance. You can kind of get
from point A to point B. The lap is the same every
time, so it doesn't actually depend on you running a
specific distance or not. For us, in this example, the
car actually just won't do that. If you think about being
all the way on the inside, being all the way on the
inside through the corner, and then exiting all
the way on the inside, it's having to do the most
work to follow that path. And in Indianapolis,
we're approaching turn one at upwards
of 240 miles per hour. And that turn one is
not-- it's hardly banked. It looks quite flat in person. So as opposed to NASCAR running
at Talladega or Daytona, these big, giant
super speedways, the car is having to
do quite a lot of work to get through the corner here. SAL KHAN: So how do you--? Do you take the outside or--? JR HILDEBRAND: So
then you look at that. And I think if you
noted the radius-- if you drew a full circle
out of each of those arcs-- SAL KHAN: Let's do that. So let's say that this is
the shortest distance path. This is kind of a circle that
looks something like this. Let me scroll over
a little bit so we can see a little bit better. So this would be
a circle like this if you were to keep that arc. It would be a circle that
looks something like this. JR HILDEBRAND: So that's
a pretty small circle in the grand scheme
of things here, yeah. SAL KHAN: That's a small circle. And for the larger
one, the circle would look something like this. So you have a larger radius,
a larger turning radius. So you would have to have
less centripetal acceleration, inward acceleration,
and fewer g-forces on this outside one, the
larger the circle is. JR HILDEBRAND: Right. And a different
way to look at it, if you looked at the car trying
to just go around these two different circles,
and it's going to be going the same
speed on either one, it's doing a lot less work to
get around this outside circle. And therefore the speed that
you could carry around that, that sort of goes up. The car has a limited ability
to stick to the racetrack. So opening that up definitely
makes a difference. SAL KHAN: But that's
an important point. At least in Indianapolis, you're
full throttle the entire way. I mean, obviously, if
you hit the brakes, the car could do a very
small turning radius. But you're at full throttle. You're not going to have any
chance if you at all let off the gas. JR HILDEBRAND: That's right. When you qualify
at Indianapolis, you've got to put in four
laps, four of your best laps of the season, of your career
in Indianapolis to qualify. And that you are
absolutely flat trap all the way around
the racetrack. There's no lifting. There's no braking. SAL KHAN: And so
that's why you're saying the car just
wouldn't do that. If you're going all
out, the car just wouldn't even be able
to make this path. JR HILDEBRAND: Exactly. That's a good point. From the driver's
perspective, you have to stay flat out if
you're going to go fast. If you're going to set a
lap time that's relevant, you have to be able
to stay flat out. And so at that point,
you're searching for the line around
the race track that you can do that
most efficiently. And so then, in this
example, increasing that radius by going
from our green circle out to the purple circle
does that rather effectively. SAL KHAN: I see. We're going for the
purple to the green back to-- so you're saying like this. JR HILDEBRAND: Well, yeah. And so then to find the actual
optimal line, what we end up doing is starting out on
the outside of the track, then bending the car into
the inside of the track, and going back to the
outside of the track, really using all of the
road that's available to us. SAL KHAN: Right. So that's interesting. So when I posed the
question, it was kind of like my brain was just
looking at these two circles. But you realize there's a bigger
circle that you could fit here, that there's an arc like this. And this would be,
if you imagine, this would be a part
of a circle that's way huger than even that purple
circle that we're drawing. So that center of that circle
is like here or something. So you have a lot less
centripetal acceleration that you have to place,
inward acceleration that you have to
place on the car. JR HILDEBRAND: Exactly. And therefore, the
car is able to carry a massively increased level
of speed through the corner. And that's really what
we're looking for. And you consider, I think
it's a very interesting-- when I think about what I'm
doing as the driver, I don't think I
really am consciously thinking that much about
the mathematics that go into finding this
optimal racing line. You sort of instinctually
just gravitate towards what the car
feels like it wants to do. But when we look at it
from this perspective, you've got the car going
down the straight away here. It's at 240 miles per hour. That's almost as fast as
the car is going to go. So it's just this sort
of terminal velocity. The drag of the
air hitting the car won't allow it to go
much faster than that. SAL KHAN: The engine's
giving all the power it can. JR HILDEBRAND: Yeah. You're absolutely flat out. SAL KHAN: And that's just
offsetting the drag of that, so that you can't accelerate
to that top speed. JR HILDEBRAND: Exactly. It's almost like you're hitting
a wall of air at that point. You're not going to be able
to accelerate any faster. And so what you're
really trying to do is you're trying to-- in
order to set that fastest lap time, which ends up
equating to the highest average speed around the
lap, that's what's the lowest number in terms of lap time
perspective-- you're trying to get the car to
most efficiently get through the corners so that you
can allow it to accelerate down the straights as
much as you can. You're getting it to diverge
from this intended course that is going on here as
efficiently as you can. And so by creating the largest
radius around the corner, that's how we end up
finding that optimal line. SAL KHAN: That's fascinating.