Videos > Visual Proof: a= v^2/r - Calculus, Mathematics
Visual Proof: a= v^2/r - Calculus, Mathematics Video Lecture
FAQs on Visual Proof: a= v^2/r - Calculus, Mathematics Video Lecture
| 1. What is the significance of the equation a = v^2/r in calculus? |  |
Ans. This equation represents the acceleration (a) of an object moving in a circular path with a radius (r) and a velocity (v). It shows that the acceleration is directly proportional to the square of the velocity and inversely proportional to the radius of the circular path.
| 2. How is the equation a = v^2/r derived in calculus? | |
Ans. The equation a = v^2/r can be derived using calculus by considering the position function of an object moving in a circular path. By differentiating the position function twice with respect to time, we obtain the expression for acceleration in terms of velocity and radius.
| 3. Can the equation a = v^2/r be applied to objects moving in non-circular paths? | |
Ans. No, the equation a = v^2/r is specifically valid for objects moving in circular paths. For objects moving in non-circular paths, the relationship between acceleration, velocity, and radius is more complex and cannot be described by this equation alone.
| 4. How does the equation a = v^2/r relate to centripetal force? | |
Ans. The equation a = v^2/r is closely related to centripetal force. In circular motion, centripetal force is responsible for keeping the object moving in a curved path. The magnitude of the centripetal force can be calculated using the equation F = ma = mv^2/r, where m is the mass of the object.
| 5. Is the equation a = v^2/r applicable to all types of motion? | |
Ans. No, the equation a = v^2/r is specific to uniform circular motion. It describes the acceleration of an object moving in a circular path at a constant speed. For other types of motion, such as linear motion or non-uniform circular motion, different equations and principles of calculus need to be applied.
Text Transcript from Video
Welcome back.
In the last couple of videos,
actually the very first video
on centripetal acceleration, I
told you that the necessary
centripetal acceleration, and
we're just talking about the
magnitude, and actually the c
tells you the direction, it's
centripetal so its inward
acceleration, it equals the
velocity squared over
the radius.
I told you that the real
rigorous proof has to be done
with calculus.
But I looked on Wikipedia and
there's actually a pretty neat
proof, although when you read
Wikipedia, it's not so obvious
on what they're saying.
So I thought I would do a video
on it, because this is
cool and you don't need calculus
to understand it.
So let's just do something.
Let's just plot the distance
vector, the velocity vectors
and the acceleration vectors
as something
goes around a circle.
Let me draw two circles.
So let's say this is
my first circle.
Let me draw another circle,
another color just for fun.
OK.
This is my other circle.
This is the center
of the circle.
And this is the center
of the circle.
And so what's the position
vector in any point in time?
Well, the position vector,
you could just
draw it as a radius.
So the position vector at
any given time looks
something like this.
So let's say initially this
is the position vector.
That's the initial
position vector.
Its magnitude is the radius
of the circle.
And this direction is right
here in the positive
x-direction.
And at that point, what is
the velocity vector?
Well, the velocity-- let's
assume we're going
counterclockwise.
I don't know why
I assumed that.
It could go the other way.
Let's say that this is the
velocity vector at that point.
The velocity vector is going to
look something like that.
That is the velocity vector.
That's v.
It's going tangent
to the circle.
Let's plot the velocity vector
as a function of
time on this circle.
So if that's the velocity vector
of that time, I can
draw the velocity vector here.
This the the same vector.
Remember, I'm just saying, at
a particular time, what does
the distance vector look like
and what does the velocity
vector look like?
So at that time, the velocity
vector looks like this.
I'm trying to make it the
same size to show you
it's the same vector.
And I'm doing it in the same
color to show you it's the
same vector.
This is the exact same vector.
This is the actual circle.
Like if you were to draw it,
this is the path of the dot or
whatever is moving around
the circle.
And this, you can just view it
as I'm plotting the velocity
over time, the velocity
vector over time.
So let's say a few seconds
later, or a few moments later,
what does the radius
vector look like?
Well, then the radius vector
looks like this.
I'm trying to write as
neatly as possible.
And what does the velocity
vector look like?
Go back to the purple.
The velocity vector once again
is tangent to the circle.
And it'll look something
like that.
It's going to have the
same magnitude,
just different direction.
Actually, let me do it in a
different color, just to show
you what I'm doing.
So let me do it in brown.
So the velocity vector
is going to look
something like this.
It's brown, supposed
to be a brown.
That's the velocity vector.
So after a few seconds, or a
few moments, where is the
velocity vector here?
Well, it's going to
look like this.
Remember, this vector
I'm just plotting
here after a few seconds.
Let me use that line tool.
So it's the same magnitude and
now the velocity is just at a
different angle.
It should be the exact same
angle as what I just drew in
the other circle.
So that's this velocity
vector.
So when we start here, the
velocity is going to change at
the circle.
After a few moments, the
object's rotating around the
circle, so now the velocity
is the same magnitude.
It's just switched directions.
And so what's happening here?
When we plot the velocity vector
over time, it has the
same magnitude, so it'll draw
out a circle, but its
direction changes.
Just as in this case, what takes
us from this point to
this point?
This was the velocity vector
and the velocity vector's
always changing.
But, in general, this is
the change in position.
And what causes the change
in position?
Well, the velocity, or at least
the speed, because the
vector's always changing.
So in this case, what's
changing the velocity?
Well, just like velocity changes
radius, acceleration
changes velocity.
Let me draw an acceleration
vector, and acceleration is
going to be in this direction.
Because if the velocity is
changing from here to here,
the acceleration is going along
the direction of the
change in velocity.
So the acceleration
vector might look
something like that.
It's going to be tangent
to this velocity path.
And that's interesting, too,
because if this is the
acceleration vector when the
velocity is here, so when the
velocity is here, the
acceleration vector's going
directly to the left, so that
means that acceleration
vector's going directly to left,
and so this is also the
acceleration vector, which
coincides with what we learned
about centripetal
acceleration.
The acceleration has to
be going inwards.
And we see that when
we actually plot
the velocity vector.
And if we plot acceleration
vector here, once again it's
going to be going tangent
to this plot of
the velocity vector.
Let me do it in a
different color.
I've already used that color.
I'll do it in yellow.
So then the acceleration
vector is going to look
something like this.
Remember, acceleration
is nothing
but change in velocity.
So at this point, when this is
the velocity vector, the
acceleration vector
will just be this.
Once again, it's just the
opposite direction of the
position vector.
So why am I doing all this?
Well, I'm doing all of
this to set up the
analogy to show you.
As this object completes one
entire path, what's happening
on this circle?
Well, this circle, the velocity
is completing one
entire path, right?
However long it takes to go
around this circle is the same
amount of time it takes to go
from this velocity back to
this velocity.
The magnitude is the same the
whole time, but the direction
is changing.
So if this takes 10 seconds to
go around the circle in real
position, then it takes 10
seconds for the acceleration
to change the direction of this
velocity enough that it
goes back to the original
velocity direction.
So why am I doing all of that?
Well, how long does it take
for the object to do one
rotation around this path?
Well, it's just the distance
divided by the velocity.
So the time to do one revolution
around this path is
the distance-- well,
that's just the
circumference of the circle.
Well, that's 2 pi r divided
by your speed, which is v.
And how much time does it take
to go around this path?
Well, we know it's going be the
same time, and now we're
going to say the time's
going to be the change
in velocity, right?
In this circle, this is the
change in distance, but it's
the change in velocity.
So here we had this much
change in velocity.
I know this might be a
little non-intuitive.
Then we have a little bit more
change in velocity, a little
bit more change in velocity,
a little bit
more change in velocity.
So the total change in velocity
is just going to be
the circumference
of this circle.
And what's the circumference
of this circle?
The radius of the circle is the
magnitude of the velocity.
So it's 2 pi times the velocity
of the circle.
And so, what is the amount
of time it takes a do one
revolution?
Well, it's the total
change in velocity.
And that's 2 pi times the
magnitude of the velocity
divided by acceleration.
And if this doesn't make
complete sense, just remember,
acceleration is change in
velocity over change in time.
I'm glossing over it a little
bit, because you might say,
oh, well, the net change
in velocity is zero.
But we're actually more
concerned about what's the
total change in the velocity?
So it changes a lot, then it
goes back to its original, so
that the net is still zero.
But this should hopefully give
you a little intuition.
So think about what I said.
But the bottom line is we know
that this expression and this
expression have to be equal,
because in the same amount of
time it takes the object to do
one complete revolution on
this circle, its velocity,
direction, has also done one
complete revolution.
So we can set these two
equal to each other.
So we could say 2 pi r over v
is equal to 2 pi v over a.
Let me switch colors.
We can cross the 2 pi
out on both sides.
And then we could multiply
both sides times v.
You get r is equal to
v squared over a.
Multiply both sides times a,
I'm doing this little bit
circuitously, and you get ar
is equal to v squared.
Divide both sides by r, and
you get a is equal to v
squared over r.
So hopefully, that gives you a
little bit of intuition about
why this works.
And I got this from Wikipedia,
so I want to
give them proper credit.
And hopefully, my explanation
helps clarify what the people
on Wikipedia are talking
about a little bit.
Anyway, I'll also do the
calculus proof, because that's
more just straight up math.
And I'll see you in
the next video.
In the last couple of videos,
actually the very first video
on centripetal acceleration, I
told you that the necessary
centripetal acceleration, and
we're just talking about the
magnitude, and actually the c
tells you the direction, it's
centripetal so its inward
acceleration, it equals the
velocity squared over
the radius.
I told you that the real
rigorous proof has to be done
with calculus.
But I looked on Wikipedia and
there's actually a pretty neat
proof, although when you read
Wikipedia, it's not so obvious
on what they're saying.
So I thought I would do a video
on it, because this is
cool and you don't need calculus
to understand it.
So let's just do something.
Let's just plot the distance
vector, the velocity vectors
and the acceleration vectors
as something
goes around a circle.
Let me draw two circles.
So let's say this is
my first circle.
Let me draw another circle,
another color just for fun.
OK.
This is my other circle.
This is the center
of the circle.
And this is the center
of the circle.
And so what's the position
vector in any point in time?
Well, the position vector,
you could just
draw it as a radius.
So the position vector at
any given time looks
something like this.
So let's say initially this
is the position vector.
That's the initial
position vector.
Its magnitude is the radius
of the circle.
And this direction is right
here in the positive
x-direction.
And at that point, what is
the velocity vector?
Well, the velocity-- let's
assume we're going
counterclockwise.
I don't know why
I assumed that.
It could go the other way.
Let's say that this is the
velocity vector at that point.
The velocity vector is going to
look something like that.
That is the velocity vector.
That's v.
It's going tangent
to the circle.
Let's plot the velocity vector
as a function of
time on this circle.
So if that's the velocity vector
of that time, I can
draw the velocity vector here.
This the the same vector.
Remember, I'm just saying, at
a particular time, what does
the distance vector look like
and what does the velocity
vector look like?
So at that time, the velocity
vector looks like this.
I'm trying to make it the
same size to show you
it's the same vector.
And I'm doing it in the same
color to show you it's the
same vector.
This is the exact same vector.
This is the actual circle.
Like if you were to draw it,
this is the path of the dot or
whatever is moving around
the circle.
And this, you can just view it
as I'm plotting the velocity
over time, the velocity
vector over time.
So let's say a few seconds
later, or a few moments later,
what does the radius
vector look like?
Well, then the radius vector
looks like this.
I'm trying to write as
neatly as possible.
And what does the velocity
vector look like?
Go back to the purple.
The velocity vector once again
is tangent to the circle.
And it'll look something
like that.
It's going to have the
same magnitude,
just different direction.
Actually, let me do it in a
different color, just to show
you what I'm doing.
So let me do it in brown.
So the velocity vector
is going to look
something like this.
It's brown, supposed
to be a brown.
That's the velocity vector.
So after a few seconds, or a
few moments, where is the
velocity vector here?
Well, it's going to
look like this.
Remember, this vector
I'm just plotting
here after a few seconds.
Let me use that line tool.
So it's the same magnitude and
now the velocity is just at a
different angle.
It should be the exact same
angle as what I just drew in
the other circle.
So that's this velocity
vector.
So when we start here, the
velocity is going to change at
the circle.
After a few moments, the
object's rotating around the
circle, so now the velocity
is the same magnitude.
It's just switched directions.
And so what's happening here?
When we plot the velocity vector
over time, it has the
same magnitude, so it'll draw
out a circle, but its
direction changes.
Just as in this case, what takes
us from this point to
this point?
This was the velocity vector
and the velocity vector's
always changing.
But, in general, this is
the change in position.
And what causes the change
in position?
Well, the velocity, or at least
the speed, because the
vector's always changing.
So in this case, what's
changing the velocity?
Well, just like velocity changes
radius, acceleration
changes velocity.
Let me draw an acceleration
vector, and acceleration is
going to be in this direction.
Because if the velocity is
changing from here to here,
the acceleration is going along
the direction of the
change in velocity.
So the acceleration
vector might look
something like that.
It's going to be tangent
to this velocity path.
And that's interesting, too,
because if this is the
acceleration vector when the
velocity is here, so when the
velocity is here, the
acceleration vector's going
directly to the left, so that
means that acceleration
vector's going directly to left,
and so this is also the
acceleration vector, which
coincides with what we learned
about centripetal
acceleration.
The acceleration has to
be going inwards.
And we see that when
we actually plot
the velocity vector.
And if we plot acceleration
vector here, once again it's
going to be going tangent
to this plot of
the velocity vector.
Let me do it in a
different color.
I've already used that color.
I'll do it in yellow.
So then the acceleration
vector is going to look
something like this.
Remember, acceleration
is nothing
but change in velocity.
So at this point, when this is
the velocity vector, the
acceleration vector
will just be this.
Once again, it's just the
opposite direction of the
position vector.
So why am I doing all this?
Well, I'm doing all of
this to set up the
analogy to show you.
As this object completes one
entire path, what's happening
on this circle?
Well, this circle, the velocity
is completing one
entire path, right?
However long it takes to go
around this circle is the same
amount of time it takes to go
from this velocity back to
this velocity.
The magnitude is the same the
whole time, but the direction
is changing.
So if this takes 10 seconds to
go around the circle in real
position, then it takes 10
seconds for the acceleration
to change the direction of this
velocity enough that it
goes back to the original
velocity direction.
So why am I doing all of that?
Well, how long does it take
for the object to do one
rotation around this path?
Well, it's just the distance
divided by the velocity.
So the time to do one revolution
around this path is
the distance-- well,
that's just the
circumference of the circle.
Well, that's 2 pi r divided
by your speed, which is v.
And how much time does it take
to go around this path?
Well, we know it's going be the
same time, and now we're
going to say the time's
going to be the change
in velocity, right?
In this circle, this is the
change in distance, but it's
the change in velocity.
So here we had this much
change in velocity.
I know this might be a
little non-intuitive.
Then we have a little bit more
change in velocity, a little
bit more change in velocity,
a little bit
more change in velocity.
So the total change in velocity
is just going to be
the circumference
of this circle.
And what's the circumference
of this circle?
The radius of the circle is the
magnitude of the velocity.
So it's 2 pi times the velocity
of the circle.
And so, what is the amount
of time it takes a do one
revolution?
Well, it's the total
change in velocity.
And that's 2 pi times the
magnitude of the velocity
divided by acceleration.
And if this doesn't make
complete sense, just remember,
acceleration is change in
velocity over change in time.
I'm glossing over it a little
bit, because you might say,
oh, well, the net change
in velocity is zero.
But we're actually more
concerned about what's the
total change in the velocity?
So it changes a lot, then it
goes back to its original, so
that the net is still zero.
But this should hopefully give
you a little intuition.
So think about what I said.
But the bottom line is we know
that this expression and this
expression have to be equal,
because in the same amount of
time it takes the object to do
one complete revolution on
this circle, its velocity,
direction, has also done one
complete revolution.
So we can set these two
equal to each other.
So we could say 2 pi r over v
is equal to 2 pi v over a.
Let me switch colors.
We can cross the 2 pi
out on both sides.
And then we could multiply
both sides times v.
You get r is equal to
v squared over a.
Multiply both sides times a,
I'm doing this little bit
circuitously, and you get ar
is equal to v squared.
Divide both sides by r, and
you get a is equal to v
squared over r.
So hopefully, that gives you a
little bit of intuition about
why this works.
And I got this from Wikipedia,
so I want to
give them proper credit.
And hopefully, my explanation
helps clarify what the people
on Wikipedia are talking
about a little bit.
Anyway, I'll also do the
calculus proof, because that's
more just straight up math.
And I'll see you in
the next video.
About this Video
Sep 10, 2025
Last updated
Visual Proof: a= v^2/r - Calculus, Mathematics for 2025 is part of preparation. The notes and questions for Visual Proof: a= v^2/r - Calculus, Mathematics have been
prepared according to the exam syllabus. Information about Visual Proof: a= v^2/r - Calculus, Mathematics covers all important topics for
2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Visual Proof: a= v^2/r - Calculus, Mathematics.
Introduction of Visual Proof: a= v^2/r - Calculus, Mathematics in English is available as part of our for & Visual Proof: a= v^2/r - Calculus, Mathematics in Hindi
for course.
Download more important topics related with notes, lectures and mock test series for Exam by signing up for free.
Description
Video Lecture & Questions for Visual Proof: a= v^2/r - Calculus, Mathematics Video Lecture full syllabus preparation | Free video exam.
Information about Visual Proof: a= v^2/r - Calculus, Mathematics
Here you can find the meaning of Visual Proof: a= v^2/r - Calculus, Mathematics defined & explained in the simplest way possible. Besides explaining types of
Visual Proof: a= v^2/r - Calculus, Mathematics theory, EduRev gives you an ample number of questions to practice Visual Proof: a= v^2/r - Calculus, Mathematics tests, examples and also practice
tests.
Visual Proof: a= v^2/r - Calculus, Mathematics is the part of for 2025 exam preparation. The content of Visual Proof: a= v^2/r - Calculus, Mathematics has been
prepared for learning according to the exam syllabus. Visual Proof: a= v^2/r - Calculus, Mathematics covers all important topics for
2025 Exam. Find important questions, notes, tests & features of Visual Proof: a= v^2/r - Calculus, Mathematics here.
How EduRev helps you in preparation?
EduRev provides you with complete coverage and for 2025 Visual Proof: a= v^2/r - Calculus, Mathematics as a part of our plan for
syllabus to prepare for exam. Our subject experts have curated special courses, tests & mock test
series. All previous year questions (PYQs) & topic wise tests like tests for Visual Proof: a= v^2/r - Calculus, Mathematics are provided with detailed
solutions. Download more important topics, notes, lectures and mock test series for Exam by signing up for free.
Study Visual Proof: a= v^2/r - Calculus, Mathematics on the App
Students of can study Visual Proof: a= v^2/r - Calculus, Mathematics alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Visual Proof: a= v^2/r - Calculus, Mathematics,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Visual Proof: a= v^2/r - Calculus, Mathematics is prepared as per the latest syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup