hello my name is kaylee and today we aregoing to focus on solving mathematical problems with the equations for kineticand potential energy before we start this lesson make sure you’ve watched thevideo equations for kinetic and potential energy and the video units andphysics now let’s recall our equation forkinetic energy remember it goes E sub K equals one-half MV squared wherem is the mass of the object and V is the velocity of the object here we have ourequation for potential energy remember it goes E sub P is equal to mghwhere m is the mass of the object G is the acceleration due to gravity and H isthe height of the object and remembering our laws of energy conservation thetotal energy of the object never changes and is always equal to potential energyplus kinetic energy as long as friction is not present so we can remember thatthe energy transfers between potential and kinetic and always maintains e totalas the same number alright so using these equations we are going to getstarted on solving some problems all right so our first question is what isthe kinetic energy of a 56 kilogram ball moving with a velocity of 3 meters persecond so first things first let’s write down what we know so they tell us herethe kinetic energy of a 56 kilogram ball so we know the mass is equal to 56kilograms is moving with a velocity of 3 meters per second so V is equal to 3meters per second okay and what are they asking us what is the kinetic energy soremember the equation for kinetic energy isEK is equal to one-half MV squared all right let’s just do a unit check here sowe’re looking for kinetic energy so we know that we want our answer to be injoules right so we’re looking for an energy okay so at the end after we gothrough our equation let’s just make sure that we have joules as a check allright so M we have the mass and V we have the velocity so let’s go ahead andplug these numbers in so one-half times 56 kilograms times 3 meters per secondall of that is squared all right so let’s do the squared part first the Vsquared so we’ll leave 1/2 times 56 kilograms and here we are going tosquare the number and we square our units so whatever we do to the V we haveto do to the number and E units so 3 squared is 9 and then we get meterssquared over seconds squared here and let’s just plug it into our calculatorso we end up with 1/2 times 56 times 9 which is 5 meters squared times secondsquared so remember if I’m multiplying my numbers I also have to multiply myunits and now let’s check so we have a kilogram meter squared over secondsquared and that is if you remember from our unit study equal to a Joule so goodwe’re in good shape Joule is energy that’s what we’regetting here so let’s finish out our math one half 504 is equal to 252 weknow that this is a Joule great so then we can say e K is equal to 252 jouleswe’ll put a box around our answer nice so let’s think about this so the kineticenergy of a 56 kilogram ball so the mass is 56 with a velocity of three metersper second he’s gonna have a kinetic energy that is equal to 252 joules nicejob all right our next question goes a rock has kinetic energy of a hundred andten joules and is traveling with a velocity of four meters per second whatis the mass of the rock okay first things first let’s write down what weknow so told us kinetic energy so e K isequal to a hundred and ten joules and it’s traveling with a velocity so V isequal to four meters per second and the question is what is the mass of the rockokay so let’s think about that we’re looking for mass so m and we know thatwe want our units to be in kilograms so let’s think about what is an equationthat we know that has EK kinetic and mg M for mass and B for velocity sounds alot like our kinetic energy equation remember goes EK equals one-half MVsquared and we’re looking for the mass the M so let’s solve this equation for Mso let’s do that first by multiplying both sides of our equation by two soI’ll put a two on either side so we get two x EK is equal to this 2 cancels outwith this 2 mv squared so now we’re here we need to get this M by itself so let’sdivide both sides now by V squared so we get V squared so we can cancel these outunder our rules from fractions so now we have M is equal to 2 timeskinetic energy divided by the velocity squared and our problem has given us thekinetic energy 110 joules in our velocity which is 4 meters per secondall right so let’s think about this here we plug these numbers and we’re gonnahave 2 times 110 joules divided by 4 meters per second all of those squaredall right so let’s do this squared part first remember that if we are squaringour number we also have to square our units so this is equal to 2 times ahundred and 10 joules all over 4 squared 16 over second squared so let’s do this over here 16 meterssquared per second squared okay so this is a big like triple fraction here rightlooks a little confusing so let’s just do our our unit math right over here tomake it easy on us so we have joules over meter squared over seconds squaredand we can rewrite that as a dual times second squared over meter squares fromour we can flip our fraction and multiply it so now we have joules overseconds squared over meters squared all right and if we remember a dual the longway to write out a Joule is a kilogram meter squared over second squared sothat’s our j times what we have left over a second squared over a metersquared okay so now we can do some fraction math you can cancel out thesecond squared we can cancel out these meters squared and we’re left with akilogram good so then it looks like our units here will equal kilogram which iswhat we need for our mass so let’s just plugthis final number into our calculator so 220 joules divided by 16 meters squaredper second squared comes out to thirteen point seven five joules meters squaredover seconds squared which we did that math over there is equal to thirteenpoint seven five kilograms something right our mass is equal to thirteen let’s put a box around that awesome solet’s think this backwards now so we’re looking for the mass of a rock we have amass down here with our kilogram if we know the kinetic energy is 110 joulesthat we know the velocity is four meters per second then we rearranged ourkinetic energy equation to come up with our equation for a mass we did our unitmath and it came out to thirteen point seven five kilograms nice job alrightour next question now says a thirty five kilogram box is lifted to a height offour meters what is the box’s potential energy okay so first things first let’swrite down what we know they tell us a thirty five kilogram box so we know thatthat is the mass equal to thirty five kilograms and is lifted to a height offour meters so height is H and then what are they asking us what is the box’spotential energy okay if we can recall or check your cheat sheet for potentialenergy is equal to M G H and quickly now let’s go over remember little G is theacceleration due to gravity and it is a constant weassume that this question means that we are on earth so then remembering thatlittle G is that constant we can say little G remember it’s 9.8 meters persecond squared I remember meters per second squaredbecause it is the acceleration due to gravity on earth okay all right so thenit looks like we have a mass we have a height and we know our constant G so wehave all those let’s make sure that our units are gonna work out so we’relooking for an energy so remembering an e it means we want units of joules okayand then if we’re gonna be multiplying a mass times an acceleration times theheight let’s say the mass is gonna be kilograms acceleration is meters persecond squared and height is another meter and so allof this is kilogram meters squared we’re multiplying those together and overseconds squared remember that is the definition of a Joule so good thisshould all work out so let’s plug our numbers in so we know our mass is 35kilograms and G is our constant 9.8 meters per second squared and our heightis 4 meters all right so I’m gonna go ahead and plug all of that now into mycalculator so 35 kilograms times 9.8 meters per squared times before metersgives me 1,000 372 kilogram multiplying my units also meter per second squaredtimes a meter which is equal to one thousand three hundred seventy-twokilogram squaring our M’s all over second squared remembering that here isthe definition of a Joule so we can say EPR potential energy is equal to 1372 joules perfect all right so now we know a box box will havea potential energy of 1372 joules if it has a mass of 35 kilograms and has leftit to a height of 4 meters off the groundnice work all right and here we are the last question goes a 60 kilogram cart istraveling up a ramp with a velocity of 2 meters per second when the car is 10meters off the ground what is the total energy of the cart so there’s a littlemore going on in this one so let’s start with just writing down what we know sothey tell us 60 kilograms so we know the mass is equal to 60 kilograms and it’stravelling up a ramp with a velocity so V is equal to 2 meters per second andwhen the car is 10 meters off the ground so off the ground means height H isequal to 10 meters and then what are they asking us what is the total energyof the cart okay so remembering total energy e totalis equal to all the energies added together so let’s think about thisquestion here so we know that there is a cart and it has a mass so that’s goodand we know that has a velocity so what energy checklist has mass and velocitythat would be kinetic energy so we can write EK + what else when the car is 10meters off the ground so there’s a height and it’s the cart so there’s amass so now what energy checklist has mass and height that would be potentialenergy so we can write plus Ep okay so now we know our total energy is gonnaequal the kinetic energy of the cart plus the potentialyou have a card so let’s write out what these equations are so e total equalsone-half MV squared plus potential energy is mgh remembering like fromour previous question G is our constant the acceleration due to gravity and R onearth so it’s 9.8 meters per second squaredremember meters per second squared because it is an acceleration okay solet’s think about this one-half M we know the massyes that’s given times V squared velocity yes we have that and then forpotential energy mass yes we know it’s given G is a constant we know that andwe were given the height H yes so you can go ahead and we can plug our numbersin here we have equal to one-half 60 kilograms times two meters per secondall of that is squared plus 60 kilograms remembers the mass of the cart so it’sthe same for potential and kinetic energy G 9.8 meters per second squaredand lastly the height 10 meters all righty now so let’s start doing somemath with the kinetic energy equation so we have 1/2 60 kilogramsremember if we Square this number we also have to square our units so we get4 meters squared over second squared we carry our math 4 units plus let’s see inhere 60 times 9.8 times 10 so I can do 60 times 10 in my calculator real fastit’s gonna be six hundred kilogram times meter remember again I’m carrying mymath to my units and I still have to multiply that by 9.8 meters per secondsquared okay now I’m gonna go ahead I’m gonna puthigher kinetic energy equation into my calculator so I end up getting this isequal to 120 kilogram meter squared over seconds squared because I can remember Ihave to multiply my units when I’m multiplying my numbers plus 600 times9.8 comes out to be five thousand eight hundred and eighty kilogram meterssometimes a meter per second squared and these units I could simplify even moreso I could multiply this so that it is also kilogram meter squared per secondsquared so I have these units and these units which are the sameI have kilogram meter squared over second squaredand remembering from our unit’s lesson this is the definition of a Joule andwe’re looking for total energy so we know an energy unit has to be a Joulewhich makes sense because we’re adding an energy plus an energy so it shouldthere should be a Joule plus a Joule is gonna equal a Joule so now let’s do someaddition here 120 kilogram meter squared per second squared plus five thousandeight hundred and eighty kilogram meter squared per second squaredhe’s going to equal six thousand kilogram meter squared per secondsquared and I know that this is the definitionof a Joule so I can say e total is equal to six thousand joules alright so let’sgo back and think this through so we knew that the cart had kinetic energyand potential energy because we went through the question and we found thechecklist mass and velocity and mass and height perfect and we’re able to plugthose numbers in we knew that our units would need to be a Joule here andwe we know since we’re adding an energy plus an energy we should be getting aand we got kilogram meter squared per second squared like this and this onealso for potential energy with kilogram meter squared per second squaredwe knew that was the definition of a Joule who we were able to do our math tohave a total of a cart that that has a mass of 60 kilograms a velocity of 2meters per second squared and it’s 10 meters off the ground we’ll have a totalenergy of 6,000 joules so nice work we just wrapped up our solving problemsusing our mechanical energy equations for kinetic and potential energy greatjob make sure you play the game the solving problems solving energy problemsgame and to practice this work on your own and remember to always be clever
