Law of energy conservation

suppose I throw this cat from some height and I want to know at what speed it hits the ground don't worry it's a toy cat now how would I solve such a problem well one way would be to think about the forces and accelerations but that could be a little tedious because of the curved path a much faster way is to use the principle of energy conservation so let's see what this principle is so the principle says energy can neither be created nor destroyed but only converted from one form to another you cannot create or destroy energy but you can convert it from one form to another so let's look at an example let's say we take that cat but this time we just drop it from some height what's going to happen to its energy well initially at this point since it has no speed it has no kinetic energy so if this is the kinetic energy meter it's showing zero right now however because it is placed at some height from the ground it has a gravitational potential energy so this is the potential energy meter and let's say it's showing full right now as an example now let's look at what happens to these energies as the cat falls down so as the cat falls down and comes to say this point now the cat has some speed which means the cat has gained kinetic energy right it had zero before but now it has kinetic energy we can ask where did that kinetic energy come from did it get created no it didn't get created you see although kinetic energy has increased notice the cat's potential energy has decreased because it is now closer to the height ooh so we can now say that the potential energy some of its potential energy got converted into kinetic energy so over here initially it was zero now some of that potential energy has been converted into kinetic energy that's where that kinetic energy come came from and so this process continues as the cat keeps falling down it's kinetic increases potential starts decreasing and eventually when the cat is about to hit the ground it will have the maximum speed you know this right so that means it'll have a lot of kinetic energy and it would have lost all its potential energy because it's right now the height is zero that means now all the potential energy got converted into kinetic so now all of that potential has been converted into kinetic energy and so the important thing over here is that as the cat fell down its total energy didn't change at all it was conserved it only converted from one form to another now before we look at what happens to that energy once the cat hits the ground let's see if we can prove this that the total energy didn't change so to do that let's get rid of all the extra stuff let's keep the cat in only two positions let's say the initial height of the cat is h1 and imagine the height after some time is h2 now we want to check that the total energy of the cat here is the same as the total energy of the cat over here right so how do we do that well we can use the work energy theorem to bring energy into the picture the work energy theorem says that the total work done by all the forces on an object that's the net work done that should equal the change in its kinetic energy so k1 would represent the initial kinetic energy and k2 would represent the final kinetic energy of course in our example k1 is 0 but let's just call it as k1 and what is this equation saying this equation is basically saying that the total work done on a body represents how much kinetic energy gets added to it or removed from it ok and we've talked a lot about this equation in a previous video called work energy theorem so if you need more clarity get ready to go back and watch that anyways let's calculate the work done by all the forces on this cat how many forces are acting on it there's only one force right that is the force of gravity of course when things are falling down air also starts pushing on it but we're ignoring that because usually that force is pretty small so since the force is only gravity this would be the work done by gravity how do we care there will be the force of gravity multiplied by the displacement of that object do that force okay the force of gravity is the weight of the cat mg well what's the displacement well the cat went from here to here right how much is that displacement that is just the difference in this height h1 minus h2 right so let's write that down over here so our displacement will be H 1 minus H 2 that is the total work done and that should equal k2 minus k1 let's simplify open up the brackets we'll get mg H 1 minus mg H 2 that equals k2 minus k1 and if you look carefully this is the potential energy this is the potential energy of the cat at H 1 this is the potential energy of the cat as at H 2 right so you've brought potential and kinetic energy into the picture so just like how you gain a symbol K for kinetic let's use a symbol for potential we'll use u capital u don't ask me why but that's the symbol we use for potential so we can call this as u 1 minus u 2 that equals K 2 minus K 1 and if we rearrange this look at what we'll get on the left hand side we'll get u 1 plus K 1 that equals u 2 plus K 2 and what is this equation telling us this equation is saying that the total energy of the cat potential plus kinetic initially equals the total energy of the cat later on and now I'm pretty sure you agree even if I had taken H 2 to be somewhere over here regardless of where I take that cat second position to be the total energy should always be the same all the equations will still work now here's a beautiful thing even if I hadn't dropped the cat straight down but let's say I threw the cat sideward says that the cat went in some curved motion even then this equation is valid why because remember the work done by gravity does not depend on the path it only depends upon the height difference which is still H 1 minus H 2 so we will still end up with the same equation this means that even in such a complicated motion the total energy here should equal the total energy over here and we can use that to solve our original problem so it might be difficult to think in terms of forces and accelerations but if you think in terms of kinetic plus potential it's much easier to solve and we will do that in another video okay anyways since this total energy remains the same we like to give it a name this total value potential plus kinetic is often called the mechanical energy and so if gravity is the only force acting on a body then regardless of what path it takes to go from one point to another it's total mechanical energy should remain the same and so immediately the next question we might have is what happens if other forces start acting on this body will the mechanical energy still remain the same let's find out if we go back to the original cat experiment let's see what happens to the energy once the cat hits the ground so once the cat hits the ground its potential energy is anyway 0 but now it stops that means its kinetic energy also becomes 0 so where does it go does it just disappear well we know that can't happen energy cannot be destroyed but then where did that kinetic energy go now if we look at the spot where the cat just landed we'll find that that spot has become a little hot of course it's not fiery but it's a little hot and heat is also a form of energy whoo that means that kinetic energy didn't get destroyed what happened now is there is that that kinetic energy just got converted into heat energy and so after hitting the ground the mechanical energy didn't get destroyed it just got converted into heat energy and of course you may be wondering ok do I find this patch Hart then no because that heat will then get distributed everywhere but it's there it didn't get destroyed that heat energy is there it just got distributed everywhere so we can't detect it ok now let's ask the question where did this gravitational potential energy come from well if we now go back we will realize that the had jumped and that's how it gained it but before jumping the cat neither had a kinetic energy nor it had potential energy it was at rest on the ground so where did that energy come from well in order to jump your muscles have to move which means some kind of work needs to be done right and there needs to be some energy for that this is the energy present inside all the living beings which allow us to do different kinds of work that energy is often called the chemical energy which means the cat used its chemical energy to jump and gained potential energy gravitational potential energy oh so that means after jumping the chemical energy got converted into the gravitational potential energy so that's where it came from so now we can ask okay where did the cat get the chemical energy from well if we go further back we realized that the cat had just eaten an apple for breakfast maybe and so we realized that that chemical energy was actually inside the Apple after eating it it got it so it came from the Apple then I can eyes okay where did the chemical energy in the Apple came from we realize it came from the apple tree the tree had the chemical energy it was the one that put it inside the Apple okay and then we can ask where did the energy in the tree comes from you might know this it comes from the sunlight you may have heard of this process called photosynthesis in this process the trees actually take up the light energy and convert it into the chemical energy so that energy came from the light alright so now we can ask the question where did that light energy come from we know that light energy comes from the Sun but does the Sun create that energy No then where it come from well it took us some time to realize this but today we know that that energy comes from the atoms you may have heard that everything around us is made of atoms including the Sun and it turns out that this energy was initially present inside the atoms the Sun is just taking the energy from the atoms and converting it into the light and you will learn a lot about that when you study nuclear physics all the fun stuff comes over there but then you might ask okay where did the energy in the atoms come from and that's where things get a little hard to answer we're not sure exactly where the energy in the atoms came from there are theories and everything it leads all the way back to the Big Bang and will not try to think about it over here but as long as we don't go too far back we can understand that all the energy on our planet comes from the sunlight itself isn't it and that's why we love energy because as nature goes through all her intricate changes energy is that one thing that never ever changes it converts from one form to another goes from one body to another but that energy never gets destroyed we can always keep track of it and that also helps us solve problems and so to summarize what did we learn in this video we saw that when cats or any objects are under the influence of gravity their kinetic energy plus potential energy always stays the same that total value never changes and this doesn't depend on what part it takes to go from here to here whether it goes straight down or it goes in a curvy path or whether you would go up it doesn't matter the total energy will stay the same and this total energy is often called the mechanical energy so the mechanical energy of this cat does not change as it's falling however if other forces start acting on this cat like you know when that cat hits the ground then this mechanical energy might get converted to other forms of energy like maybe heat or sometimes chemical energies etc so what we find is that whatever happens the total energy can neither be created nor destroyed it can only be converted from one form to another and that's what we love about energy