 Determine the power required for a 1,150kg car to climb a 100m long uphill road with a slope of 30° from horizontal in 12 seconds under 3 different situations. In the first situation we are considering a constant velocity across the entire process. In the second situation we are considering the car going from rest at the bottom of the hill to 30m per second at the top of the hill. And in our third situation we are considering a situation where the car starts at 35m per second and ends at the top of the hill at 5m per second. In our analysis we are told explicitly to disregard friction, air drag and rolling resistance. So we have a hill, the hypotenuse of that hill is 100m and its slope is 30° from horizontal. The car is starting the process at the bottom of the hill and ending the process at the top of the hill. I'm going to define my system as the car itself and I'm going to assume that the mass of the car doesn't change throughout the process, therefore I have a closed system. Furthermore I recognize that I'm going to treat this as a transient process because time matters and I'm going to establish the process as starting at state 1 which is a state point in time and space and ending at a state point that I'm calling state 2 because I'm particularly clever when it comes to naming my state points. In my list of assumptions so far I have closed system. Remember from our discussion of open versus closed systems and transient versus study processes that the default state of the universe is an open system undergoing a transient process. So when you're treating something as an open system that's not really an assumption that you're making you just aren't making any assumptions about simplifications regarding the system. Similarly when you are treating something as being transient it's not really an assumption that you're making. The assumption that you make is simplifying reality to being a steady state process or simplifying reality to behaving as a closed system and not having mass across the boundary. Does that distinction make sense? So I'm running down closed system as an assumption because we are modifying the default state but I'm not listing transient as an assumption because I'm not modifying the default state. In situation A I'm saying v2 is equal to v1. In situation B I'm saying v1 is 0 and v2 is 30 meters per second and in situation C I'm saying v1 is 35 meters per second and v2 is 5 meters per second. So I will first set up an analysis in the abstract case and then we will simplify it for A, B, and C individually. And that analysis as you can probably imagine is going to start with an energy balance on the car. Like with all energy balances we start with delta E is equal to E in minus E out. For our purposes here we aren't distinguishing between different subcategories of energy entering nor exiting. We are just treating it as energy entering the system and increasing its kinetic and potential energy. So the E in doesn't have to be broken apart. It's just the category that we are solving for. And at that we aren't even solving for the magnitude of energy entering. We are trying to determine the power required. So instead of energy we are going to want to write this in terms of in terms of rate of energy entering E dot in. Therefore we are treating E dot in which is E in over the duration if we look at the entire process all at once. And writing E in as E dot in multiplied by duration. And then we are neglecting energy exiting because we don't have enough information to determine it anyway. And like in the previous example we are saying that the energy change of the system which could be delta U, delta KE and or delta PE is not going to include internal energy. So we are neglecting any internal energy changes of the car during this process. So I will list that as an assumption as well. We can't neglect kinetic energy, we can't neglect potential energy. And by the way it's important to note that we are distinguishing that assumption as the change in internal energy is zero. It's not that we are saying there is no internal energy, we are just saying whatever it is it isn't changing. Similarly in the previous example when we assumed that the change in potential energy was zero we are not saying there is no potential energy, we are just saying the change is zero. That's going to be important for part A where the change in kinetic energy is zero. It's not that it isn't moving, that there is no kinetic energy, it's just that whatever the kinetic energy is at the beginning it's the same at the end. Anyway. So we have the change in kinetic energy of the car plus the change in potential energy of the car is equal to the rate of energy entering the car times the duration during which that rate of energy enters. And that is what we are using as our tool for all three analyses. If we solve this for ii.in what we're saying is the average entering energy rate must be delta ke plus delta pe divided by duration. So the power required to accomplish this process which is the rate of energy entering the system is the change in kinetic and potential energy divided by the duration of the process. I know the duration is 12 seconds in all three cases. All I have to do now is determine the kinetic energy change for all three parts and the potential energy change for all three parts. Let's begin with part A where we are climbing the hill at a constant velocity. So the cruise control is locked, velocity isn't changing, whatever it is at one it still is at two. So ii.in which is delta ke plus delta pe divided by 12 seconds is going to simplify down to just the change in potential energy divided by the duration. Again, it's not that there is no kinetic energy it's just that it doesn't change. So the total change in potential energy would be the mass at state two times the gravitational acceleration at state two times the height at state two minus the mass at state one times gravity at state one times the height at state one. All divided by the duration and because I've assumed it's a closed system that the mass doesn't change the mass can be factored out and I think it would be reasonable to assume we have standard gravitational acceleration so I will call that 9.81 meters per second squared then I can factor out both mass and gravity and write this as mass times gravity times delta h or delta z if you prefer divided by duration that's something we can compute we know the mass is 1150 kilograms we know gravity is 9.81 meters per second squared the change in elevation here is not 100 meters it is 100 meters multiplied by sine of 30 degrees and then we are dividing by 12 seconds and it doesn't specify what unit it wants for power so because everything else is in metric we will express an answer in the metric unit system the relevant unit for power would be watts or more likely for cars kilowatts so I'm going to shoot for a kilowatt as my destination and as a general rule it's best to start at your destination and work backwards for these unit conversions I'm going to start with a kilowatt recognize that it is 1000 watts and then I will break apart my secondary dimensions into primary dimensions so I will write a watt as a jewel per second because that is how a watt is defined if you don't remember that off the top of your head it is listed on your conversion factor sheet right here and and then I recognize that a jewel is defined as a newton times a meter which is also on my definition sheet and a newton is a kilogram meter per second squared which surprise surprise is on your conversion sheet so we've started with 1000 watts that's a kilowatt and we've broken it all the way apart into its primary components what cancels what jewel cancels jewels newtons cancel newtons kilograms are going to cancel kilograms meters and meters cancel meters and meters second squared cancels seconds and I guess let me try that again second squared cancels second squared and the singular second cancels the singular second that leaves me with kilowatts as my answer so I'm going to pop up the calculator zoom out just a scosh and let's calculate a number here we go 1 1 5 0 times 9.81 times 100 times sine of 30 and I'm already in degrees but I will add the degree simple just for good measure and then I'm dividing by 12 times 1000 that will give me an answer in kilowatts and I determine it takes 47 kilowatts on average to accomplish this process so I'm going to call that e.in the rate of energy entering on average that would accomplish this process that power is going to increase for part b wherein in addition to accomplishing this change in potential energy I'm also accomplishing an increase in kinetic energy the process begins at rest and ends at 30 meters per second so I will start part b the same equation I used for part a before I simplified it and I will write this as ke2 minus ke1 plus pe2 minus pe1 all divided by 12 seconds and total kinetic energy can be written as one half times mass times velocity squared so ke2 would be one half times m2 times v2 squared ke1 would be one half times m1 times v1 squared and then pe2 would be mass 2 times gravity 2 times h2 minus pe1 would be mass 1 times gravity 1 times h1 all divided by 12 seconds per duration so again closed system I can factor out the mass gravity I can factor out gravity I could actually go even further and pull mass all the way out but I don't want to do that for a reason that'll make more sense in a second here so one half times mass times v2 squared minus v1 squared plus mass times gravity times h2 minus h1 divided by delta t so for convenience here I can split my denominator and write this as one half times mass times v2 squared minus v1 squared divided by delta t plus mass times gravity times h2 minus h1 divided by delta t and the cool thing about this is that this term is what we just calculated it's 47 kilowatts so all we have to do to answer this question is figure out how much it would take to increase the kinetic energy in this process and add it to the change in potential energy that we already know furthermore for part b v1 is zero because the process begins at rest so this really becomes mass times v2 squared divided by two times duration plus 47 kilowatts my mass was 1150 kilograms and I'm divided by two multiplying by the velocity at state two which was 30 meters per second which I will write as 30 squared meters squared per second squared divided by 12 seconds and again my goal is to get to kilowatts so I'm going to break out kilowatt into 1000 watts again and a watt into a joule per second then a joule into a newton meter again and a newton into a kilogram meter per second squared again newton cancels newton joule cancels joule watts cancels watts second squared cancels second squared kilogram cancels kilograms meter squared cancels meters and meters and I wrote the seconds in the wrong spot a little bit too quick there a watt is a joule per second not a joule times a second okay then seconds cancel seconds and I'm left with kilowatts and then to that I'm adding 47 kilowatts so if I pop up my calculator again and I calculate 1150 that is 1150 times 30 squared divided by two times 12 times 1000 I get 345 eighths thank you calculator 43.125 kilowatts which when I add that to 47.0063 I get 90.1313 so to accomplish this process the engine has to require on average 90.13 kilowatts 43 of those kilowatts are going into increasing the kinetic energy of the car about 47 are going into increasing the potential energy of the car then for part c I'm going to open a new page here and I'm going to copy over my equation and my approach is going to be largely the same the biggest difference here are my actual velocities themselves so in part c the car begins at a velocity of 35 meters per second and ends at a velocity of five meters per second so we begin in the same way we have the change in kinetic and potential energy divided by duration and I'm grouping those together factoring out mass and gravity and we pick up here where we got rid of v1 last time the difference here now is that I am talking about a change in velocity terms squared so I'm going to write this as one half times 1150 kilograms then I'm going to divide by 12 seconds here for good measure and then I'm taking v2 which remember was five squared minus v1 which was 35 squared meters squared per second squared and then the same unit conversion soup we begin with kilowatts and a kilowatt is a thousand watts and a watt is a joule per second and a joule is a newton times a meter and a newton is a kilogram meter per second squared and then once I calculate that quantity I add 47 kilowatts to it so watt cancels watt joule cancels joule newton cancels newton meters and meters cancels meter squared second squared cancels second squared and seconds cancels seconds and kilograms cancels kilograms leaving me with kilowatts so heck of it now is a bad time to go to sleep it's time to wake up we got math to do I'm taking 1150 multiplied by the quantity five squared minus 35 squared and then I'm dividing by two times 12 times a thousand and I am missing a parentheses and for that first part I am getting negative 115 seconds which is negative 57.5 and then I add to that 47 and I get a result of negative 10 so while I'm writing that down take a minute to ponder what that means so a common mistake for thermal one students would be to ignore the minus they have a tendency to ignore a negative sign if they don't expect it and that's a really bad habit to get into the negative signs are very important especially when we are talking about energy changes like this the fact that the change in kinetic energy was negative implies that we are getting energy out of that decrease in kinetic energy that change in kinetic energy is going to supply the energy for the increase in potential energy what that means is if we start at 35 meters per second and we don't touch the accelerator at all we will reach the top of the hill at a velocity higher than five meters per second because that negative delta was a larger magnitude than the amount of potential energy change required that means we have to supply breaking power the engine would have to work to decrease the kinetic energy in order to only reach five meters per second at the top of the hill instead of something higher than that you could also think of this as we are getting 10 and a half kilowatts out of this situation if we had an electric vehicle and we were able to recover that breaking energy we could get potentially 10 and a half kilowatts of the course of this process