ap physics 1 looking at power and energy we're going to define power and understand what its units mean and see how it relates to work and energy we want to understand the rate at which work is done here's a little story uh scenario for helping us think about this imagine it's next year you're out of the house graduated from high school and you're moving into your first apartment it's beautiful fourth floor walk up and you really want to bring your piano with you this piano is quite large it's 350 kilograms you notice that there's very conveniently placed a pulley outside of the window convenient yeah but you're really lazy and you want to get a motor to lift this up you know put a rope over the pulley and winch it up with the motor instead of trying to pull it with your hands you go to the hardware store and you ask to purchase an electric motor and the clerk asks what power motor do you need you realize i need more information so you think back to this very day in ap physics from your senior year of high school where you learned about power and you start by thinking about the energy or the work required to lift the piano up to your window you know the gravitational potential energy would be mgh so about 52 000 joules of work required but you also have to think about time because power is not just how much energy is required or how much work needs to be done it's how fast you are doing that work power is generally defined as energy transformed over time and so this could be the amount of work it took takes to lift the piano with the change in potential energy or the change in energy from one form to another in our case it's how much potential energy the piano needs to gain to get to the top you generally see this written as power is change in energy over time delta e over delta t that's the equation on your ap equation sheet and power is measured in watts given a capital w a watt is a joule per second and this si unit is actually named for someone by the name of james watts credited as inventing the steam engine and we commonly use watts to do a lot of different things like describe the brightness of incandescent light bulbs because the more energy a light bulb uses per second the brighter it is given the same technology incandescent as we've gone to different types of light bulb this is less descriptive but we still talk about watt equivalence you know if you pick up an led light bulb it'll say oh this will be as bright as a 60 watt incandescent light bulb um and that's what we mean 100 joules of electrical energy are being converted into light and heat energy every second with that particular light so if we go back to our piano we have to decide how quickly we want to raise this up and five minutes seems like a reasonable amount of time not so long that you couldn't keep people away from that spot on the sidewalk while you lift it up um but also fast enough so it's not going to draw too much attention while you're doing this and get you in trouble so the window is 15 meters above the ground your piano has a mass of 350 kilograms of which we've already said how much power does it take to lift this piano in five minutes well the power is energy over time we already calculated the mgh the energy required divided by 300 seconds that's five minutes in seconds and it gives us about 175 watts of power you go back to the store you tell the clerk what you want 175 watt motor and they reply that all their motors are rated in horsepower and you think what what do i do now what is this the rating of horsepower well the si unit of power may be the watt but horsepower is an older unit or a unit from non-standardized international system and again we have to go back to james watt to talk about this one um he came up with this basically when marketing his steam engine trying to explain how much work it could replace from animals and this is an interesting quote something about how much work a pony could do and how much more work a horse could do i have no idea where any of this came up he came up with any of this but suffice it to say it was an estimate of comparing a machine to an animal nowadays we just say horsepower is standardized to equal 746 watts reminder that a joule a watt is a joule per second so we can just use this conversion if you do that conversion you find out that the watts cancel out you end up with about 0.23 horsepower so if you get a quarter of a horsepower motor you should be just fine let's do a couple more examples and see how this plays out a 70 kilogram jogger runs along up a long flight of stairs in four seconds vertically they go up 4.5 meters how much power were they outputting well the power is the change in potential energy in this case over time and that gives us about 788 watts and you might ask yourself well how much energy did this require well energy is just power multiplied by time right the rate at which you were transforming energy multiplied by how long you took gives us about 3152 joules now we could have calculated this amount of energy first that's basically the numerator of that fraction but we can also work backwards to find that another way to think about power is that power is the rate at which work is done but work is defined as force times distance times the cosine of the angle and if you think about well if you apply a constant force to an object that delta r is just a distance traveled divided by time and that gives you a speed or an average velocity if you're being careful the cosine of the angle can kind of go away here because generally in this type of problem we're thinking about traveling in a straight line and really that's all we're dealing with an average velocity multiplied by a constant force or an average force is another way to consider thinking about power so let's think about power from an engine or a motor driving a vehicle this is the kind of thing that i think many people have asked about what about you know when wheels are spinning and all this well it's a it can get complicated but if you imagine that a car has 18 horsepowers excuse me 18 horsepower of power output when traveling at a steady 90 kilometers per hour we can think about what was the average force exerted on the car due to air resistance and friction and all the other things that the engine has to overcome to keep it going at a constant rate well we first need to think of this in terms of watts because we need our si units do that conversion and then we remember that power is equal to force times velocity and therefore the average force that is being applied must be the power divided by the speed and that gives us about 537 newtons now you might think that this is actually a small amount of power and a small amount of force but remember this is a steady 90 kilometers per hour a car engine is certainly going to have to output a lot more power to do the acceleration up to that speed or higher um as it goes another example of this is um this is sort of a classic movie reference right james bond driving an aston martin db5 um this car in 1960 whatever it was that he drove this car in the movie was rated by the manufacturer to have 222 horsepower and it had a 0 to 60 mile per hour time that it took to speed up of 8.1 seconds and the mass of the car was about 1775 kilograms so we can sort of compare the predicted power output of the motor of 222 horsepower with the power that was on average while it accelerated from zero to 60 and that's only about 105 horsepower and this isn't really surprising right there's a difference between what an engine or a motor can do at its maximum and what it does in average over a period of time right cars are complicated things there's lots of energy loss or turned into heat certainly the linkages the shifting of gears all sorts of things happening here that can result in this difference so where does that leave us the really big idea here is power is how fast work is done or energy is transformed and you can go from knowing about energy and time to power from power and time to energy