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Kinetic energy derivation

Let's derive the K.E. expression 1/2 m v^2.   Created by Mahesh Shenoy.

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Video transcript

moving things have energy we call this the kinetic energy but the question is how much kinetic energy does a moving thing have for example if we knew the mass of this ball let's say it was 2 kilogram and let's say it was coming in and coming in at 10 meters per second how much kinetic energy does it have and what would that number mean that is what we want to try and find out in this video now there are more than one ways to derive things right so over here we'll take a little different approach compared to the standard textbook derivation we will use a definition of energy to derive things so what is energy energy is the capacity to do work right so what is kinetic energy kinetic energy would be the capacity of moving things to do work so for example if this cricket ball can do a maximum of let's say 500 joules of work when it hits something okay let's say maximum then that's its capacity capacity means maximum and therefore we would say the kinetic energy of this ball is 500 joules on the other hand if you find out that when it hits something it can only do let's say 3 joules of work as an example then that's its maximum capacity then that would be its kinetic energy 3 joules so we have to figure out how much work this ball can do when it hits something maximum work that would be its kinetic energy okay so how do we calculate work well we calculate work as force acting on an object multiplied by the displacement of the object and we've talked a lot about this energy and work in previous videos so if you need more clarity great idea to go back and watch them so let's say this ball goes and hits something let's say it goes and hits my hand let's say I'm trying to catch the ball now you might know whenever you try to catch a ball when you catch it eventually about your hand moves back a little bit so let's say after catching my hand has come over here now can you see the ball has done work on the hand it has pushed on the hand with some force it hits your hand with some force let's call it as force on the hand and notice it has made your hand move it has displaced your hand from here to here and therefore it did work on your hand and by the way since the ball has come to a stop after getting it that means it can't do any more work so it has done the maximum work it could so whatever work it does on your hand by definition that would be the kinetic energy of this pala okay so let me just write that down the kinetic energy of this ball will equal the work done by the ball on your hand so work done on the hand by pushing it and by moving it and now before we substitute and see how to calculate you know what we'll do we'll first write these in general instead of two kilograms let's say the mass of that object is M kilograms M and similarly instead of say 10 meters per second let's write it in general let's say it's coming in with some velocity U so that will get a nice equation all right so we just have to calculate how much this work is and we know how to calculate work it is the force in a displacement so the work done on the hand will be the force on the hand so that is this force on the hand multiplied by the displacement of the hand multiplied by s so what do I do next well I want to get rid of the forces and bring out bring in masses and velocities into the picture right which means I need to connect force and motion how do we do that well there's one equation that connects force and motion which you have stated earlier Newton's second law Newton's second law basically says that the force acting on an object or net force acting on an object equals mass times the acceleration hey I can use this right but there's a problem if I directly substitute F equals MA here I will end up with the mass of the hand because it's a force of this is the on the hand it will be the mass of the hand and the acceleration on the hand hey that's not something that I wanna do I don't know hot mass of the end I don't let the acceleration I'm not interested in it I don't want that that's not what I want so what do I do well think about this when the ball pushes on the hand with some force we know from Newton's third law that the hand pushes back on the ball with an equal and opposite force so the ball also experiences a force I'll call it as force of the ball and this is exactly equal to this and it's in the opposite direction so we can now substitute the force of the hand to be equal to the force of the ball because of Newton's third law it's equal but it also opposite right it's also opposite and how do we write opposite in in in mathematics we write it in negative sign all right into the displacement whose displacement is this by the way this is the displacement of the hand but guess what this is also the displacement of the ball right because balsa gets displaced by the same amount so what we have done now is you have brought in the properties of the ball you now know the force of the ball and the displacement of the ball and that'll be easier to bring in mass and the velocity of the ball so what do I do next next I can apply Newton's second law to this one this is the net force acting on the ball there is no other force acting so I can say the force acting on the ball should equal mass of the ball times the acceleration of the ball and we have displacement oops same color let me use displacement okay fine is this my final answer no this is not my answer because I want to bring in velocities see I know the initial velocity of this ball is you I also know the final velocity is zero all right so final velocity we let's call it it's also zero I want to bring in velocities over here I don't want acceleration I don't want the displacement so can I can I somehow connect acceleration displacement initial velocity and final velocity how do we do that connection hey this brings us back to equations of motion we've seen that in equations of motion we can do these connections so let's quickly recall what are the three equations of motion let's get rid of this over here and let's write down the equations of motion these were the three equations of motion we had now we want to choose an equation which connects the velocities the acceleration and the displacement which of these three equations do that can you try and pause the video and check this for a while alright let's see the first one has velocities but it has time I don't want that there is no displacement also over here I don't want that second one has displacement and initial velocity but it doesn't have final velocity again there is also time which I don't want that's my main culprit I don't want time the third equation that has final velocity initial velocity it also has acceleration displacement it has no time this is our winner so let's use this equation to somehow get rid of a and s and bring in the velocities all right let's see that's right let's let's substitute it and see what we get now before I do it'd be a great idea for you to pause the video and see if you can continue this derivation yourself because we have what we want we can use this to get rid of acceleration and displacement look carefully and somehow bring in the U into the picture we can do that all right so great idea to pause and see if you can continue derivation from here all right let's do this I know that V V is zero so this becomes zero and if you look carefully the a s is over here which I want to get rid of so let's find out what a s is from this equation okay so let's keep a us on this side and let's put everything else on the other side so let's subtract U Square from both sides so I'll get a minus u square over here and that equals to a s and we can divide by you want to get rid of this too right so let's divide by 2 on both sides and so that gets rid of the two and now I know what a s is so a s turns out to be minus u square by 2 let's plug that in over here this will be minus M times S which is minus u square by two and I have gotten rid of everything I wanted and now I have mass and velocity also the negative sign cancels out that's also good and so what we see now is that the kinetic energy of the ball is M u squared divided by two and so in general we say the kinetic energy of any object will be half times mass times its velocity squared okay let's say quickly see if it makes sense to us so the equation is saying that if the velocity increases the kinetic energy will also increase does that make sense well yeah right I mean if this ball were to come and hit you with more speed you would expect it to push you more and move your hand back even more so it would do more work it has more capacity lower more kinetic energy it also says that if the mass of the object increases then also kinetic energy will increase does that make sense well let's see if instead of a cricket ball if a bowling ball were to come and hit your hand with the same speed let's say don't you think it'll now push with much more force and push your hand much farther back so therefore it has a much higher capacity to do er and therefore it will have more kinetic energy so hopefully that makes sense right alright so now that we have the expression let's go ahead and see how much is the kinetic energy of our cricket ball what numbers did we take we said the mass of the cricket ball let's call that as 2 kilograms and we say the the speed of the cricket ball or the velocity was 10 meter per second all right so if we substitute over here we will get mass which is 2 kilograms and I will not substitute the unit's because I already know the units of kinetic energy it's the same as that of the work joules right but of course you can try substituting the units and checking that you will get the same unit as well but I will not do that everything is in standard units so 2 times velocity square velocity is 10 so 10 squared divided by 2 and so 2 and 2 cancel and we end up with 10 squared which is a hundred so in our case the cricket ball has a kinetic energy of hundred joules and what does that mean it means that this cricket ball can do a maximum of hundred joules of work when it goes and hits something not more than that all right so to quickly summarize how did we derive the expression for kinetic energy we said it's the capacity it has to do work and so we calculate how much work it did maximum work it did when it hits something and comes to a stop then we use the formula for work and then we use Newton's laws and some equations of motion to finally arrive at the kinetic energy