Understanding Energy Conservation: The Dynamics of a Ball on a Ramp

- [Lecturer] We place a ball on this ramp and we want to now figure out what happens to the speed of the ball as it goes forward.

If you try to do this using forces and accelerations, it's gonna be really tough. But instead, we are gonna use

energy conservation in this video and tackle not only this, but other interesting problems as well. So let's begin.

To use energy conservation, we need to first decide what our system is going to be. What's a system you ask?

Well, a system is basically a group of things that we are interested in. For example, in this particular case,

I am gonna consider the ball and the earth as my system. Now, everything outside of it, like the ramp, the air, all of that, is my surrounding, okay?

Now, usually, when a system interacts with the surrounding, it can exchange energy with it. For example, as the ball falls down,

as it goes down it, there's friction. Because of friction, there is some heat generated. It means there is thermal energy.

Where did the energy come from? It must have come from here as the ball went down, right? Which means there was energy transferred over here.

Similarly, because of air resistance, it can heat up the air as well, and therefore, there will be energy transferred

to the air as well. However, sometimes this energy transferred is minimal and we can ignore it.

Other times, it's not. So in this particular case, let's ignore it. It's not very practical, but it'll be super useful

to get an intuition. So whenever a system is not transferring any energy to its surrounding,

we say it's an isolated system because it's isolated from the surrounding, energy-wise. And in that case, the total energy of the system

stays the same. And that's what we will be exploiting to figure out what happens to the ball as it goes forward.

Its total energy should stay the same. So let's keep track of total energy. At this particular point, the kinetic energy is zero

because we just dropped the ball, but it has some gravitational potential energy because there is some height.

Therefore, all of the energy, total energy of my earth ball system over here is just gravitational potential energy.

And we can show that with an indicator like this. All of the energies, gravitational potential energy, this bar is completely full

and this bar is completely... This bar is empty because there is no kinetic energy. Now what happens as it goes down.

As it goes down, it comes closer to the earth, the height reduces, so the potential energy will reduce, which means kinetic energy increases,

therefore the speed increases. So it'll speed up as it goes down. And then what happens when it comes to this point?

Notice, this is the lowest point in this entire track, which means for this entire track, it'll have the lowest potential energy.

Usually, we like to call the lowest potential energy in our scenario as zero whenever we are closest to, whenever we are close to earth.

So we can just say that this is zero gravitational potential energy, and therefore that means we have the maximum kinetic energy,

and you'll have the maximum speed. If we knew the mass and the height over here, we can plug in and we can find the potential energy here.

And then as a result, we can find the kinetic energy here and then we can find the speed. That's how, using energy conservation,

we can find the speed. So what do you think will this thing look like when it's over here?

Can you imagine the potential and the kinetic energy and the speed at this particular location? Okay, so let's see.

When you are over here, I don't care about the fact that it goes up and down. I don't have to worry about it.

All that matters to me is when it goes from here to here, it is going higher. It's at a more height.

More height means more potential energy, which means less kinetic energy, so less speed than over here.

Boom. And the last question we could have is, will it be able to roll over this hill?

Again, why don't you pause and think about it. All right, well, as it goes up over here, notice that at this particular height,

it should have the same potential energy as before because the potential energy only depends upon the arrangement,

and when we are close to the earth, it only depends upon the height. And since it has the same potential energy as before,

the kinetic energy must be zero because the total energy cannot change. Which means at this point, all the kinetic energy

has been converted to potential, so it has no more kinetic energy left, it has no more speed left, so it cannot go over.

So your ball will now never be able to go over the initial height, and therefore now the ball will just go roll back and keep going back

exactly in reverse, 'cause the same thing will happen. The story continues.

And go all the way back, stop over here. And the same thing will continue forever and ever. Because the total energy of my system, we are assuming,

is conserved. It's isolated, that's what we're assuming. So that's what keep on happening.

But, of course, from your daily experience, you'll probably know that things will not keep on moving forever.

It'll eventually come to a stop. Why does that happen? Well, that's because our system is not really isolated.

It is exchanging energy with its surrounding. For example, as the ball falls down over here, there is friction because of which there is heat generated,

meaning there's thermal energy over here. Where is the energy coming from? It's coming from this total energy of the system.

So let's take an example. So as you go from here to here, let's say there's a lot of thermal energy generated,

because of that, the total energy will keep reducing, total energy of our system keeps reducing, because it's going into this, you know, this ramp.

And so by the time it comes over here, the total energy is less than the total energy over here. That means the kinetic energy over here should be less than

what the potential energy was over here. So it might look somewhat like this, meaning the speed over here would be

less than what we predicted in the ideal case. Now the same thing would continue. As this thing goes on over here,

it'll keep losing its energy, total energy keeps decreasing and you know, go out as thermal energy over here. And maybe by the time it reaches over here,

it would've lost all its kinetic energy. Now when the ball comes over here, can you think about, you know,

what would the indicators look like again? Well, it would have no kinetic energy over here. It'll have a little bit of potential energy.

So look, a lot of energy is pretty much gone. It won't even make it till here. Now as it goes back, it'll keep further losing energy

and eventually it would've lost all its energy. It'll settle somewhere over here. And again, if we knew how much thermal energy

was lost over here, then I can subtract it from here and I can know how much is the remaining potential energy.

And from that, I can predict what this height is going to be. Amazing, isn't it?

But we can do so much more with energy conservations. For example, let's consider the orbit of the mercury around the sun.

If you consider mercury and the sun as our system, then pretty much this system is not interacting with anything else.

All the other orbits are... Other planets are very far away, this is vacuum, so we can assume our system to be isolated.

So the total energy must be conserved. So now let's use this to predict the speeds over here, okay? If we look at this position, notice, in this entire orbit,

we are the closest to the sun over here. When you are closer to a planet or a star, you have the lower potential energy.

Since you are the closest, you have the least gravitational potential energy. So you have the maximum kinetic energy.

So I know that should have the maximum velocity over here. What happens when I go from here to here?

Well, notice, I have now slightly farther away from the sun. Potential energy has increased. The kinetic energy should reduce,

the speed should be slightly lower than over here. What happens when I go from here to here? At this location, notice, I'm at the maximum distance

from the sun in this entire orbit. So I have the maximum potential energy in the entire orbit. I have the minimum kinetic energy of the entire orbit.

So I should have the minimum speed over here. And then the same thing continues in the opposite direction. The speed should of course increase

because the kinetic energy increases, the potential energy reduces because I'm going closer. So it have a higher kinetic energy, higher speed.

And therefore, look, as it goes from here to here, the speed reduces and then the speed increases. We had seen this before, right?

We had analyzed this before using forces, and we had to think about the directions and all of that. But look, just using kinetic and potential energy

is so much more straightforward, so much more elegant, but we are not done with energy conservation. Let's look at another example.

This time we want to smash two nuclei into each other, okay? So you have two positive nuclei coming towards each other. As they come towards each other,

because of the Coulomb's repulsion, they will slow down. Then eventually, they'll come to a stop within some distance.

Our goal is to figure out what this stopping distance is. Let's say we are given the initial kinetic energy of both these nuclei.

Now you may be wondering why should we care about, you know, until what distance it come and stops. The reason we care about this

is because if it's close enough, if this r value is small enough, then the nuclear force will take over and it'll fuse them.

If the r value is not small enough, then it'll not be within the nuclear range, and it'll not be able to fuse.

So this is actually really important problem for things like nuclear fusion reaction calculations and all of that.

So again, how do we do this? How do we figure out what this closest approach is going to be?

Well, we do the same thing as before. We consider the nuclei as a part of our system, and then notice there's nothing else in the surrounding.

So our system is completely isolated. So the total energy of our system is conserved. So if I consider this as my initial condition,

I'll just call this as i, and this is why I consider this as my final condition, final case.

The total initial energy, the total initial energy should equal the total final energy.

What's total initial energy? That is the initial potential energy. This time the potential energy is electric,

so we'll call this electric potential energy, plus the initial kinetic energy, that should equal to final potential energy

plus final kinetic energy. All right, so again, what do we know over here. The initial potential energy,

you can imagine they're very far apart to begin with. So far apart, they're hardly interacting. You can imagine they're infinitely distanced apart.

And you might recall, when charges are infinitely distance apart, we say that in each potential energy is zero.

So this potential energy is zero. Finally, I know when they come very close to each other, the closest they should stop,

their kinetic energy finally becomes zero. So we just have to equate this. So the total initial kinetic energy is given to us.

So it's just K one plus K two, let's call that. What is the final potential energy? Well, how do we figure out the potential energy?

Well, again, you can look it up. We don't have to remember it, but it turns out the potential energy at any distance

is given as the Coulomb's constant k times q1 q2 divided by r. We know the values of K one and K two.

We know what the Coulomb's constant is and the charge is, we can just plug in and calculate what r is.

So why don't you pause and try yourself again. All right, so if I rearrange this to get r, I get r equals K q1 q2,

q1 q2, divided by K one plus K two. And q1 and q2 are the charges over here.

And since both are just having one proton each, the charge over here is e and e, so you get e and e, e squared.

And if I just plug in all the values, we get... And if I simplify these numbers, I'll get about 4.61

times 10 to the power... I have a 9, and I have a minus 38. Then I have a minus 30 in the numerator,

which goes into denominator. From denominator which goes into numerator plus 13, giving me minus 16.

So I'll get this as my value for r. Now it turns out that this is small enough for a nuclear fusion to happen.

And so just with a pen and paper, we can actually predict whether the nuclei will fuse together, isn't that beautiful?

And this idea is even useful in doing physics inside the atoms. For example, if you consider a hydrogen atom,

you might be familiar with this model where you have the proton at the center and you sort of have the electron cloud.

Now if you provide more energy to it, that energy electron takes up that energy and jumps into a higher orbital, we say.

But since it can't stay there for long, it jumps back, and when it does, the difference in that energy is emitted as a photon.

And this energy release depends upon the energy levels of the atoms, which means it's a signature of a particular atom.

So by looking at the photons released by excited atoms, we can even figure out which atoms we are looking at. This is basically how we analyze

the light coming from the distant stars and the suns and the atmospheres of exoplanets, for example. And we can figure out, or we can make an estimate

of what material makes it up. So the idea of conservation of energy has far, far reaching consequences.