one good example of a situation where there is uh motion with constant acceleration is something called free fall and free fall is very simply just an object freely falling under the influence only of gravity so if i take my marker let it go it drops and it turns out to drop at a very specific rate and we'll talk about that in a second so free fall is what we'll define as being a motion under the influence only of gravity why is this uh why is this important well it turns out that if an object is freely falling under the influence of only gravity its acceleration has a very specific value its value is what we call g and i'll talk about g in a second but the direction of that motion is always vertically downwards and so remember acceleration just like velocity and position isn't is a vector so we need both a magnitude and a direction the direction is always straight down towards the center of the earth the magnitude it turns out to be this special thing called g so i'm going to write down g here as a value i'm going to take it to be 9.80 meters per second squared now some textbooks might use 9.81 and in fact here in oshawa it's probably closer to 9.81 but averaged over the entire earth it's very close to 9.80 so we're going to take that to be the value just as an aside when i say i'm going to take that to be the value that's the value you should use on your assignments on your tests and so on so that your answers will match mine and not be a little tiny bit off because you use 9.81 so 9.80 is what we're going to use now this is the idea with free fall why is it important it turns out that every object regardless of mass falls at this rate so if i drop my marker or if i drop my bowling ball if i drop my feather they should all fall at the same rate if they're only under the influence of gravity and so this is a really good example here of a bowling ball and a feather being dropped together this is in a huge vacuum chamber and i recommend you take a take a look at the entire video i'll post a link for you later these two objects drop and they drop at the exact same rate they have to do it inside of vacuum because air of course has a big influence on the feather much less so than the bowling ball but in that vacuum they drop at 9.8 meters per second squared precisely okay so that's free fall now it's a couple things we have to mention about free fall and about this letter g so the first thing i want to mention is that the direction is vertically downwards i already said that but what that means is that g here is the magnitude of the acceleration that means it's positive by definition magnitudes of vectors are always positive and so g is a positive value despite the fact that it's vertically downwards and often times i'll write the acceleration as being negative g itself is always positive what else so g is not called gravity right and this is a bit of a pet peeve for me so bear with me g is an acceleration due to gravity right the presence of gravity only or it's called the free fall acceleration you can use either one of those two terms don't just call it gravity though gravity is a force that's a very different thing we'll talk about the force of gravity later on in the course and the last thing i'll mention is that 9.80 as i mentioned was an average over the earth if we're on the moon or mars or somewhere else is not going to be 9.80 that is very dependent on the force of gravity and so different planetary objects will have different amounts of of gravity okay all this said let's try a really quick example i want to do just a quick numerical example so that we can get practice both dealing with free fall but more importantly dealing with our mathematical model of constant acceleration so i'm going to take my marker here i'm going to hold it up 2 meters above the above the ground and i'm going to drop it all right i think you heard it drop there the question is how long does it take for that marker to drop so i'm going to write down that question over there how long does the marker take to fall to hit the floor there that's our question so i want to do this really carefully even though it's a really simple kind of a problem and we'll deal with a lot more sophisticated problems later on but i want to handle this one kind of carefully so the very first thing i want to do is draw a coordinate system so i'm going to draw a little y-axis here i'm going to indicate my origin on that y-axis that's the floor and i'm going to draw my marker up here there we go there's my marker and then i'm going to draw my marker down here there it's hit the floor and and so up here i'm going to call the position of my marker y knot down here i'm going to call it y1 i'm going to suppose its velocity is v naught and it's dropped at a time t naught and down here it's got a velocity of v1 and a time of t1 why am i doing all this because i want to carefully define all of the symbols that i'm going to use to describe the motion of this marker remember our goal here throughout the first part of this course is to describe motion and so i want to do that and i want to be precise in everything this is physics and so precision is really important the last variable that i'm going to use here of course is is the acceleration due to gravity the free fall acceleration and of course that's directed directly downwards and has a numerical value of minus g again note that minus sign g itself is positive that minus sign in my coordinate system indicates that the acceleration is downwards directed okay so this is all of the variables now if i go through these uh i'm going to suppose i drop it from 2 meters so we'll just write in 2 meters there i'm going to drop it from rest i don't know if you saw that but i didn't throw it downwards i just let go of it and it fell and i'm going to suppose that i start my stopwatch right i start timing this at t equals 0. all right so the acceleration is minus g that's minus 9.80 meters per second squared so we've got that number y1 just by definition of where my origin is is equal to zero so that's uh that's the ground and then i don't know what v1 is but i don't care i don't know what t1 is but that one i do care about i want to figure out how long it took to fall to the ground this right here is what we would call the visualization step of the problem solving process and we'll go into a lot more detail about this when we do examples full problem examples this one here though i just want to do really quickly and so this is all of the information we need in order to answer this problem to actually answer the problem of course we need to turn to our mathematical model so we're going to model this marker dropping as being under constant acceleration in free fall so i know the acceleration and if we turn to our to our mathematical model for constant acceleration the position equation we wrote down as this so i'm going to write it out here in full just so that we can see it and then we're going to talk about all of these different pieces there we go so there is my constant acceleration position equation now it's got a bunch of s's in it it's got a bunch of eyes and f's in it and i need to change that the reason why it looks like this is because physicists try to be as general as possible and so we want to write down this equation for every situation possible now i have a marker falling and so i need to adapt this equation specifically for that and so i need to then rearrange this equation or rewrite it in terms of these variables so rather than sf my final position is actually y1 that's what i'm calling my final position i'm using y here instead of s because s is meant to be general and i have vertical motion which is more specific so y is the correct one to use and then i've got s i here i called my initial position y naught so i will place it replace it with that my initial velocity i just called v naught so i'll just throw in v naught here delta t i'll sort of write as t one minus t not just to again be more specific and then finally i'll throw in my acceleration here as well and so this is one half a and then t one minus t naught squared okay so again this equation going to that equation that's going from a very general statement that's applicable to all situations to something that's applicable just to this situation here now i'm going to simplify this equation a little bit here y1 for example is zero right i said it was zero here so get rid of that v naught i said i dropped it from rest so that whole term is going to go away and lastly t naught i also said with zero so a lot of these were my own decisions of where i put my origin where i put my time origin and so a lot of these things go away and so my equation here reduces down to very simply this equation here uh d1 squared there we go so now that i have it down into this form i'm just going to rearrange for the thing i'm looking for which is t1 and so if i put my y naught on the other side and then multiply by 2 and divide by a and then take the square root those are all the things i have to do and so what i end up is is t 1 equals the square root of 2 y naught divided by minus a there we go notice i wrote down that minus a there um like that because it's going to be a negative number that negative times a negative is going to end up being positive and so this will end up being 2 y not divided by g and then if we plug in all of the numbers we end up getting a time of 0.61 seconds all right so that kind of answers our first really simple problem here it takes 0.61 seconds to drop to the floor let's test and see if that sounds reasonable so this is about 2 meters uh and uh and let's see definitely less than a second i'm not sure how much less but that sounds like a pretty reasonable number so i think we've done a good job here trying to understand free fall and doing a nice simple example where we've got constant acceleration as our last sort of specific kind of motion we want to talk about let's talk about motion that happens on an incline right so a good example of this is say just a box sliding down a ramp something like this the ramp is inclined at some angle we'll call it theta so something like this as a situation where we call that inclined motion now one thing that we want to be clear about here and this is something we'll change later on in the course but for now we're going to make the motion frictionless in other words it's a very slippery ramp a very slippery slippery box and it's going to slide down without friction in other words it's under the influence of gravity not so much under the influence of friction and so in that case we can write down the acceleration as well and so i'll write down here a s now it turns out that the acceleration is going to be less than 9.80 which makes sense right and in fact as you lower that angle the acceleration should get less and less so it turns out that we can write it down as g sine theta where that theta is the angle of incline here g of course as always is 9.8 positive number but notice that i have a plus or minus sign there and the reason for that is because it depends on the direction of the ramp if the wrap is going down like this the acceleration is going to be that way as well on the other hand if we've got a ramp like this and we've got a box and let's say it's moving upwards in that case the acceleration is actually down the ramp and so usually we'll put our x-axis this way and our x-axis this way and so that means the acceleration here is going to be negative anyway the moral sort of of the story is that take a look at your situation and figure out which direction the acceleration is going to be if it's in the negative x direction then it should pick up the negative sign as usual that's it for for incline motion if you've got something sliding down a ramp you just have to change your acceleration to g sine theta and then you can handle that case as well this is still motion with constant acceleration and so we turn to our usual one-dimensional kinematic equations to solve that one as well