Impact velocity from given height

what I want to do in this video is answer an age-old question or at least an interesting question to me and the question is let's say I have a Ledge here I have a ledge or cliff or maybe this is a building of some kind and let's say it has height H so let's say it has a height of H right over here and what I'm curious about is if I were to either let's say that this is me over here so this is me if I were to either jump myself and that's not recommended for very large H's or if I were to throw something maybe a rock off of this ledge how fast would that either myself or would that rock be going when it gets right before it hits right before it hits the ground and like all of the other videos were doing on projectile motion right now we're going to ignore air resistance then for small HS and for small velocities that's actually reasonable or if the object is very aerodynamic and has and it's kind of dense then the air resistance will not matter less if it's me kind of belly-flopping from a high altitude then the air resistance will start to matter a lot but for the sake of simplicity we're going to assume we're going to assume no air or we're not going to take into effect the the effects of air resistance or we could assume that we're doing this on on an earth-like planet that has no atmosphere however you want to do it so let's just think about the problem and just so you know some of you might think that's that's not realistic but this so actually would be realistic for a small H if you were to jump off of a off of the roof of a one-story building air resistance will not be a major component in determining your speed if it was to be a much larger building than all of a sudden matters and I don't recommend you do any of these things those are all very dangerous things much better to do it with a rock so that's actually the example we're going to be considering so let's just think about this a little bit we want to figure out we want to figure out so at the top of right when the thing gets dropped right when the rock gets dropped you have an initial velocity you have an initial velocity of zero and once again we're going to use the convention here that positive velocity means upwards negative or a positive vector means up negative vector means down so we're going to have an initial velocity over here of zero and then at the bottom at the bottom we're going to have some final velocity we're going to have some final velocity some final we're going to have some final velocity here that is going to be a negative number so it's going to have some negative value over here so this is going to be negative this is going to be a negative number right over there and we know that the acceleration of gravity for an object of on freefall an object in freefall near the surface of the earth we know and we're going to assume that it's constant so our constant acceleration is going to be negative 9.8 meters per second squared so what we're going to do is give it an H and given that their initial velocity is zero and then our acceleration is negative 9.8 meters per squared we want to figure out what our final velocity is going to be right before we hit the ground we're going to assume that this is this H is given in meters right over here and we'll get an answer in meters per second for that final velocity so let's see how we can figure it out so we know we know some basic things and the whole point of these is to really show you that you can always derive these more interesting questions from very basic things that we know so we know that displacement is equal to displacement is equal to average velocity average velocity times change in time times change in time and we know that average velocity average velocity if we assume acceleration is constant which we are doing average velocity is the final velocity plus the initial velocity plus the initial velocity over to and that our change in time our time or our the amount of elapsed time that goes by this is our change in velocity so elapsed time is the same thing I'll write it over here is our change in velocity divided by divided by our acceleration and just to make sure you understand this it just come straight from the idea that acceleration or let me write that change in velocity change in velocity is just acceleration times time or I should say acceleration times change times change in time so if you divide both sides of these this equation by acceleration you get this right over here so that is what our displacement remember I want an expression for displacement in terms of the things we know and the one thing that we want to find out well for this example right over here for this example right over here we know a couple of things well actually let me let me take it step by step we know that our initial velocity is zero our initial velocity is zero so this first expression for the example we're doing the average velocity is going to be our final velocity our final velocity divided by two since our initial velocity is zero our change in velocity change in velocity is the same thing change in velocity is the same thing as final velocity minus initial velocity minus initial velocity and once again we know that the initial velocity is zero here so our change in velocity is the same thing as our final velocity so once again this will be times instead of writing change in velocity here we can just write our final velocity because we're starting at zero initial velocity is zero so times our final velocity divided by our acceleration divided by our acceleration final velocity same thing is change in the velocity because the initial velocity was zero and all of this is going to be all of this is going to be our displacement and now it looks like we have things in we have everything written in things we know so if we multiply both sides of this expression or both sides of this equation by two by two times our acceleration by two times our acceleration on that side and we multiply the left-hand side by do the same colors two times our acceleration two times our acceleration on the left hand side we get two times our acceleration times times our displacement times our displacement is going to be equal to on the right hand side the two cancels out with the two the acceleration cancels acceleration it will be equal to the velocity our final velocity squared it will be equal to our final velocity our final velocity squared final velocity times final velocity and so we can just solve or for final velocity here so we know what our we know our acceleration is nine point is negative 9.8 meters per second squared so let me write this over here so this is negative 9.8 so we have 2 times negative 9.8 so I can just let me just multiply that out so that's negative nineteen point six meters per second squared times meters per second squared and then our what's our displacement going to be what's the displacement over the course of dropping this rock off of this ledge or off of this roof so you might be tempted to say that our displacement is H but remember these are vector quantities so you want to make sure you get the direction right from where the rock started to where it ends what's it doing it's going to go it's going to go a distance of H but it's going to go a distance of H downwards and our convention is down is negative so in this example our displacement our displacement from when it leaves your hand to when it hits the ground the displacement is he going to be equal to negative H it's going to travel a distance of H but it's going to travel that distance downwards and that's why this vector notion is very important here in our convention is very important here so our displacement over here is going to be it's going to be negative H and negative H meters negative H meters so this is this is the variable and this is the shorthand for meters so when you multiply these two things out lucky for us these negatives cancel out and you get nineteen point six nineteen point six H nineteen point six H meter squared per second squared meter squared per second squared is equal to our final velocity squared is equal to our final velocity squared and notice when you square something you lose the sign information if our final velocity was positive you square it you still get a positive value if it was negative and you square it you still get a value but remember this example we're going to be moving downwards so we want the negative version of this so to really figure out our final velocity we take the essentially the negative square root of both sides of this equation because if you just do so if we were to take the square root of both sides of this so you take the square root of that side you take the square root of that side you will get and I'll flip them around you'll get your the velocity is equal to your final velocity we could say is equal to the square root of nineteen point six H and if you can even take the square root of the meter squared per second squared treat them treat them almost like variables even though their units and then outside of the radical sign you will get a meters meters per second and the thing I want to be careful here is if we just take the principal root here the principal root here is the positive square root but we know that our velocity is going to be downwards here because that is our convention so we want to have we want to make sure we get the negative square root so let's try it out with some numbers we've essentially solved what we set out to solve at the beginning of this video how fast would we be falling as a function of the height but let's try it out with some things let's say that the height is I don't know let's say the height is let's say the height is five meters five meters which would be probably jumping off of a or throwing a rock off of a one-story maybe a commercial one-story building it's about that's about five meters would be about fifteen feet so yeah but by the roof of a commercial building give or take and so let's turn it on and so what do we get if we put five meters in here we get nineteen point six nineteen point six times five times five gives us 98 so almost 100 almost 100 and then we want to take the square root of that it's going to be almost ten so the square root of 98 it gives us nine point roughly nine point nine and we want the negative square root of that so in that situation when the height is five meters so if you jump off of a one-story commercial building you're going right at the bottom or if you throw a rock right at the bottom right before hits the ground it will have a velocity it will have a velocity of negative nine point nine meters per second so negative nine point nine meters per second I'll leave it up to you as an exercise to figure out how fast this is in either km/h or miles per hour and you'll see it's pretty fast it's like you know it's not something you'd want to do and this is just off of a one-story building but you can really figure this out you can use this for really any height as long as we're reasonably close to the surface of the earth and you ignore the effects of air resistance at really high Heights especially if the object is not that aerodynamic then air resistance will start to matter a lot