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

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 going to be really tough but instead we're going to 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 going to

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 there's friction because of friction there's some heat generated means

there's 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 transfer to the air as well however sometimes this energy transfer 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're going to be exploiting to 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 earthball system over here is just gravitational

potential energy and we can show that with an indicator like this all of the energy is gravitational potential energy

this bar is completely full and this bar is completely this fire 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 will 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 it I 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 then the last question we could have is will it

be able to roll over this hill again why didn't you pause and think about it all right well as it goes up over

here notice that it comes to 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 remember 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 not 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 because

the same thing will happen the story continues it 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 assuming is conserved it's it's isolated that's what

you're assuming so that's what keep on happening but of course from your daily experience you probably know that things

will not keep keep on moving forever it will 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 that

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 will keep losing its energy total energy keeps decreasing and thermal because of you know go out as thermal

energy over here and maybe by the time it reaches over here it would have lost all its kinetic energy now when the ball

comes over here can you can you think about what the you know what would the indicators look like again well it would

have no kinetic energy over here it will have a little bit of potential energy so look a lot of energy is pretty much gone

it don't even make it till here now as it turn as it goes back it'll keep further losing energy and eventually it

would have 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 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 you 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 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 or 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 goinging closer so you'll 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 such so much more straightforward so much more elegant but we're 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 kum's

repulsion they will slow down then eventually they will 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 till 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 will

fuse them if the R value is not small enough then it will not be within the nuclear range and it'll not be able to

fuse so this is an actually really important problems 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 initial condition I'll just call this as I and this might consider 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 uh 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 they are you can

imagine they're very far apart to begin with so far apart they're hardly interacting you can imagine they're

infinitely distance apart and you might recall when charges are infinitely distance apart we say that that in

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 K1 + K2 let's

call that what is the final potential energy 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 kums constant K

* q1 Q2 divided R we know the values of K1 and K2 we know what the colums 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'll get r equal K q1 Q2 q1 Q2 ided by K1 + K2 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^ s and if I just plug in all the

values we get and if I simplify these numbers I'll get about 4.61 * 10 to the power I have a 9 then I

have a - 38 then I have a-3 in the numerator which goes in denominator from

denominator which goes into numerator as + 13 giving me -6 so I'll get this as my value for R

now turns out that this is small enough for 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 released 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