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## AP®︎/College Physics 1

### Unit 4: Lesson 2

Centripetal acceleration- Race cars with constant speed around curve
- Visual understanding of centripetal acceleration formula
- Deriving formula for centripetal acceleration from angular velocity
- Change in centripetal acceleration from change in linear velocity and radius: Worked examples
- Predicting changes in centripetal acceleration
- Centripetal acceleration review

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# Race cars with constant speed around curve

When acceleration could involve a change in direction and not speed. Created by Sal Khan.

## Want to join the conversation?

- I dont understand how change of direction can effect acceleration? Formula a=v/t does not indicate that?(80 votes)
- night3x: You are right in the last statement. Direction has nothing to do with speed.

Speed and Velocity are not the same thing, so direction is an integral part of velocity, it is needed to describe velocity, whereas speed is defined only by a quantity, a number if you will, saying how fast something goes - but not to where.

Velocity on the other hand, has to say how fast and which way.

This is why you can only have speed which is a positive number (or zero), but velocity can be a negative number as well (moving at some speed in an opposite direction).

I hope I did not introduce more confusion in all this :)(122 votes)

- Is the concept similar, when the second hand on a clock makes one complete revolution in 1 minute? is the tip of the second hand accelerating?(33 votes)
- yes it is, just like the race cars the magnitude is the same but the direction is constantly changing due to an inward acceleration(24 votes)

- Would the driver be depressing the gas pedal to accelerate and therefore maintain his speed through the turn even though the speedometer still reads 100 mph?(20 votes)
- No. Two different accelerations in that case. The centripetal (center seeking) acceleration is what you feel when you round a curve and you're thrown outward. It is always inward or perpendicular to the curve.

If the drivers speed is constant then there is no acceleration that is tangential to the curve. You'd experience this acceleration, if there was one, by being pushed back into the seat.

The most likely reason the driver would need to depress the gas pedal is that there is some loss due to the friction between the tires and the road and/or the air resistance of the car.

Goes back to Newton's first law if there were no external forces acting on that race car it would speed along happily at 100 mph forever without any need for added force.(20 votes)

- could this also be represented as a planetary system where the cars are planets, the center of the track is the sun, and the acceleration towards the center is gravity? If this could be represented this way it would be a lot easier for me to understand.(11 votes)
- suraj1997pisces is on the right track. Gravity does half the work of keeping the planets in orbits, accelerating toward the sun. The other half of the equation, so to speak, is the angular momentum caused by the planet racing around the orbit. This momentum is best demonstrated by swinging a ball on a string around your head. So long as you hold onto the rope (gravity) the ball stays in its orbit, though it tugs outward on the string (angular momentum). If you let go of the string, the ball flies away from you in a straight line. This line is a tangent to the orbit, which is always a circle or ellipse.

So long as the two forces (gravity and angular momentum) are stable, the orbit is stable. The car or planet or ball goes around and around forever. But once one force or the other gets out of balance, the orbit decays. If gravity > angular momentum, the object accelerates toward the center of the star or whatever. If gravity < angular momentum, the object accelerates away from the center.(12 votes)

- If centrifugal force is a "fictitious" force, how are we able to feel it?(6 votes)
- If you start moving on a straight line (say with constant velocity), by the inertial principle, you'll tend to continue with that movement!

When, in a car, you take the curve the velocity changes... but your body still wants to move in a straight line. Respect to the car, your body goes in the oposite direction than the curve... So it is interpreted as if a "force" is pushing you outside the car. (This is what we know as centrifugal force)

Nonetheless, the real thing is that the sit, sitbell, the door, and any part of the car you are sticked to, are pulling you inside the car! (This is the real deal... the centripetal force)

Cheers!(14 votes)

- Why is the acceleration vector towards the center of the circle if we feel a push towards the outside of the circle?(3 votes)
- The acceleration vector is towards the center because we are being pushed by the sides of the car towards the center of the curve. We feel as though we are being pulled outwards because our body wants to continue traveling in a straight line (due to inertia).(10 votes)

- I have a quick question what would happen if it was an oval would it accelerate?(5 votes)
- Good question!

Lets put it this way, oval has changing radius and velocity vector (direction). The side of the oval has longer radius and the top and the bottom of the oval has less radius; the oval has two sharp turns and two smooth turns each 2(pi) radian (360, full circle). Centripetal force does not have to change since velocity can accelerate to a certain velocity that is compatible to the radius of the oval. But if it changes, it must be in a certain Newtons depending on the radius of the oval. The car therefore accelerate and does not accelerate due to the fact that the Centripetal force changes. And to prove it, here is an example:

F(centripetal)=m(mass will stay the same since mass cannot be neither created nor destroyed)v^2 divided by r (radius)

I have told you that the radius constantly changes so I will plug in two random radius.

1000 N (we're keeping the force same)=100kg(weight of a car)x 20m/s(speed)^2 divided by 40 meters

and

1000N =100kg(v^2) divided by 20 meters

With some algebra, we get:

20000kg(m^2)/s^2=100kg(v^2)

200m^2/s^2 =v^2(m/s)

approx(14.14m/s) =v

And if the Centripetal force changes, speed remains the same.

Check this one out:

1000N =100kg(20m/s^2) divided by 40

2000N =100kg(20m/s^2) divided by 20

Now these works are just one of a gazillion part of the motion I showed you. The reason why I say gazillion is because the radius constantly changes. If the object were to accelerate, velocity would constantly change every Planck time (smallest possible time). And if the Centripetal force were to change, it would constantly change to maintain the velocity of the car.

Hope it helped!(4 votes)

- does it mean that planets orbiting the sun are constantly accerating(4 votes)
- In fact, yes! But only the direction not the magnitude of the velocity.

The planets are not having constant speeds around the sun though. Because we are going in elliptical orbits, the speed varies throughout the season, however, if we were going in circles the speed would be constant.(6 votes)

- how do you accelerate inward, wouldnt you just accelerate from the forward, but you turn your wheels inward. right?? I dont see how you can accelerate inward?(5 votes)
- If you think of the idea of "force = acceleration x mass" then it may help. If the front tires are bald, the cars won't make the turn at any real speed, because of the force on the tires as they try to hold the turn. So you know there is a force, you know the car has mass, thus, you have acceleration.

And because there is a change in direction, that direction becomes an important part of the equation and speed and velocity must part ways from their shared common usage. The velocity must include any changes in direction as well as magnitude (speed).(4 votes)

- How does acceleration allow direction to change?(2 votes)
- Acceleration is change in velocity. Velocity is a vector. If you change its magnitude or its direction, you have acceleration(6 votes)

## Video transcript

So we have some race
cars racing, right here. And I have an interesting
question to ask you. If we assume that these
cars are making this turn right over here,
that all of them are making this turn at
a constant speed of 100 kilometers per hour, my
interesting question for you is, are these cars accelerating
while they make this turn? So is acceleration happening? And you might say, well, gee,
look, my speed was constant, it's not changing. If I looked at the
speedometer for the car here, if I looked at the
speedometer over here, it won't budge, it just stays
at 100 kilometers per hour. I don't have any change
in speed over time. And so then you
might say that you don't have any acceleration. But then you might
be saying, well, why would Sal even
make this video? And why would that question
even be interesting? And your second
suspicion would be true, because these cars
actually are accelerating despite having a constant speed. And you can pause it and
think about that for a second, if want to. But I wanted to point
this out to you, because in an example like this,
the difference between speed and velocity starts
to matter, speed being a scalar quantity
only having a magnitude. And velocity being
a vector quantity, being speed with a
direction, having a magnitude and a direction. And to think about-- let's
take a top view of this thing, and then I think it'll
become a little bit clearer the difference between
speed and velocity and why these things
are accelerating. So if I were to take a top
view of this racetrack-- I'll do my best
attempt to draw it-- so it might look
something like this. This is the top view. I could even draw
this red and white. So red, just to
give you the idea. So this is the red, and
there's some white in between. Obviously I'm not
drawing as many dividers as there are in
the actual picture, but it gives you an idea of
what I'm actually drawing. And then there's
some grass out here, there's some grass
over here, and then there's some grass over here. And let's focus on this orange
car and this red car right over here. And this is a top view, so this
is its path right over here. And we're saying it
has a constant speed of 100 kilometers per hour. So if you think
about its velocity, the magnitude of it's
velocity is constant, it's 100 kilometers per hour. But what is happening to the
direction of the velocity? Remember, velocity
is a vector quantity. It has magnitude and direction. So up here, when it's
starting to enter the curve, it's going in this direction. And you tend to show
vectors by arrows like this. And what you do is,
the arrow's going in the direction of the
velocity, in this case, and normally you would draw
the length of the arrow shows what is the velocity. The magnitude of the
velocity, I should say. So it's velocity's constant. So the length of this arrow
will always be constant. But as we see, it's
direction changes. When it's halfway
through the turn, it's not going in
that same direction. It is now going in a
different direction, and when it comes to
the bottom of the turn, it's going in a very
different direction. And the direction keeps changing
as long as it is turning. And I'm not going to
go into the math here. We're going to wait for the
math on this a little bit later. But remember, acceleration is
a change in velocity over time. Acceleration is equal to a
change in velocity over time, or we could say over
a change in time. And although the velocity's
magnitude is constant here, it's direction is changing. If there was no
acceleration on it, it's magnitude and the
direction of it's velocity would be constant,
and the car would just keep going in that direction. So somehow, the
car's direction is changing inward over
and over and over again. And so this is just kind
of a little bit of a trick question, something
for you to think about, we're going to discuss
the math in more detail in future videos. But what's happening here is the
cars actually are accelerating. And they're actually
accelerating inwards, and that's what's
changing inwards. And when I say inwards,
they're being accelerated towards the center of
the curve, and that's what's allowing their
direction to actually change.