let's say if you hold two balls in the air a 10 gram metal ball and a five gram metal ball and you hold it five feet above the ground so they're at the same height once you release it from rest both balls will be in free fall which one will hit the ground first is it the 10 gram ball or is it the five grand ball both objects will reach the ground at the same time the reason for that they're placed at the same height and are under the influence of the same gravity earth's gravity the acceleration due to gravity is 9.8 meters per second squared which we'll talk more about that later but because everything is the same the height is the same the acceleration is the same if you take away air resistance both objects will hit the ground now let's say if we use an actual demonstration of a brick versus a flat piece of paper so the brick is a lot heavier than the flat piece of paper let's say it's a a one kilogram brick now with air resistance which one will hit the ground first the brick is going to go straight down the paper might go this way that might go that way and then eventually fall to the ground because of air resistance now what's going to happen if you take the same brick but you crumpled up the paper so that it's very very small you take that same paper and crumple it up now the effects of air resistance on this paper will be greatly reduced if you crumple up the paper very tightly and you let it go these two will hit the ground approximately about the same time the greater the surface area of the paper the greater the effect of air resistance air resistance increases with surface area so once you crumple up the paper and then you drop it you'll see that it's about the same time when it reaches the ground with the brick the air resistance is greatly reduced so if you can eliminate air resistance completely then both objects should hit the ground at the exact same time because they're under the influence of the same gravity earth's gravitational field now let's talk more about the acceleration due to gravity so if we have a ball and if we release it from rest it's going to fall down and it's going to accelerate at the rate of 9.8 meters per second square so if you recall acceleration tells you how fast the velocity is changing every second so initially the speed might be zero one second later the velocity is going to be negative 9.8 the speed is just positive 9.8 speed is always positive but velocity can be positive or negative depending on the direction so because the ball is going in the negative y direction the velocity is negative now technically acceleration is negative 9.8 meters per second squared because the object is going to accelerate in the negative y direction gravity pulls things down not up now two seconds later the velocity is going to be negative 19.6 meters per second so every second the velocity is changing the speed changes every every second the speed changes by 9.8 so as the ball drops notice that the velocity is decreasing by 9.8 it's becoming more and more negative the speed however is increasing keep in mind speed is positive so at this point the speed is positive 9.8 it's 19.6 and here it's positive 29.4 so an object in free fall that's under the influence of gravity the speed is increasing by 9.8 every second but the velocity because it's negative is decreasing by 9.8 every second so remember acceleration tells you how fast the velocity is changing every second now before we go over a few free fall problems we need to talk about the equations that you need to solve them so whenever an object is moving with constant speed this is the equation that you need to use d is equal to v t d can be used as distance or displacement just remember distance is a scalar quantity displacement is a vector so displacement can be positive or negative but distance is always positive so anytime an object is moving with constant speed you can use this equation now when an object is moving with constant acceleration you can use any one of these four equations v final is equal to v initial plus a t v final could represent the final speed or final velocity v initial is the initial speed or initial velocity a is the acceleration t is the time so you may see me use the word speed and velocity interchangeably just remember that velocity is speed with direction speed is the scalar quantity it can only be positive velocity can be positive or negative depending on what direction you're going so for an object that's moving to the right the velocity is positive if the object moves to the left the velocity is negative but in free fall situations we're dealing with motion in the y direction when an object goes up the velocity is positive when it goes down the velocity is negative the next formula that you need to know is this one d is equal to v initial t plus one half a t squared and then there's this one d is equal to one half v initial plus v final times t and v final squared is equal to v initial squared plus two a d so there are five formulas that you need to know so these three plus this one and this equation only when an object is moving at constant speed when it's moving with constant acceleration you can use these four equations so let's start with this problem a ball is dropped from rest on a cliff what is the speed of the ball five seconds later so let's say this is the cliff and here is the ball so it falls down but we don't know if it hits the ground at this point we just want to find the speed five seconds later so it's important to make a list of what you have and the variable that you need to find what is the initial speed of the baldness problem notice that the ball is dropped from rest so the initial speed is zero our goal is to find the final speed now we know the time the time is 5 seconds and the acceleration due to gravity is negative 9.8 the acceleration due to gravity is in a negative y direction so that's why it's negative so what equation that was listed earlier has these four variables the equation that we need is this one v final is equal to v initial plus a t the initial is zero the acceleration is negative nine point eight and t is five so negative nine point eight times five is negative 45 i mean not 45 but 49 so now let's go back to the question what is the speed of the ball five seconds later now think about your answer the answer the speed is positive 49 meters per second remember speed cannot be negative so you got to make it positive so this is the answer for part a now part b what is the velocity of the ball at this time the velocity is negative so that's the velocity right there it's negative 49 meters per second because the ball is moving in the negative y direction velocity is negative remember velocity is a vector quantity speed is scalar now let's move on to part c how far does it travel during this time what equation do you think we need to use one equation that can help us define it is this one d is equal to v initial t plus one half a t squared we already have the initial speed we know the time and we have the acceleration so we have enough information to use this equation so v initial is zero zero times t is just going to be zero acceleration is negative nine point eight and t is five so half of negative nine point eight that's negative four point nine if you multiply that by 5 squared that will give you a value of negative 122.5 meters now what does this answer represent that answer is not the distance traveled but rather it's the displacement this is the answer for part d remember displacement is a vector like velocity it can be negative or positive because the object moves in the negative y direction the displacement is negative but to answer part c how far does it travel during this time part c is not looking for the displacement it's looking for the distance traveled the distance is simply positive 122.5 whenever an object moves in one direction if it doesn't change direction if it moves straight the distance and displacement they have the same value they have the same magnitude but the signs may be different depending on what direction it's going but anytime you have an object that's moving in a straight line the distance and displacement have the same numerical value just the signs might be different so that's it for this problem number two a ball is thrown downward at an initial speed of 15 meters per second from the top of a cliff what is the speed and velocity of the ball eight seconds later so this problem is similar to the last problem only one key difference the ball is not released from rest rather it's drove down with an initial speed and that initial speed is 15 meters per second but let's use the initial velocity when dealing with this equation so we're going to use a negative 15 meters per second because it's going in the negative y direction our goal is to find the speed and velocity eight seconds later so we need to find the final velocity the time is 8 seconds and the acceleration is still negative 9.8 so let's use the same equation to find the final velocity so the initial velocity is negative 15 plus the acceleration of negative 9.8 multiplied by the time of 8 seconds negative 9.8 times 8 that's negative 78.4 so negative 15 plus negative 78.4 that's negative 93.4 meters per second so this is the final velocity eight seconds later the speed eight seconds later is simply positive 93.4 that's it you just gotta change the sign just remember speed is always positive now let's find the displacement and also the distance that it travels so let's use the same equation as the first example so d is equal to v initial t plus one half a t squared so v initial this time is negative 15 and t is eight the acceleration is negative nine point eight and we're going to plug in eight for t again so negative fifteen times eight that's negative 120 half of negative 9.8 is negative 4.9 so negative 4.9 times 8 squared that's negative 313.6 so when adding these two together you should get negative 433.6 meters so this is the displacement of the ball during these eight seconds now the distance that it travels is positive 433.6 meters so that's the distance that it travels all you need to do is just make the answer positive number three a stone is dropped from the top of the building and hits the ground five seconds later how tall is the building so let's start with a picture so let's say that's the building and the stone is dropped from the top of a building our goal is to find the height of the building the height of the building is basically the distance that the ball travels until it hits the ground so if we could find the distance that it travels we could find the height of the building now what equation should we use in order to find out which equation to use we need to make a list of everything that we have now it didn't say the stone is thrown down so therefore if the stone is dropped from the top of a building we know it's dropped from rest which means the initial speed is zero we don't know what the final speed is but we do have the time and we know the acceleration for any object in free fall any object that's fallen under the influence of gravity this acceleration will always be the same value in the y direction so our goal is to find d in the y direction which is the same as the height so once again we could use this equation d is equal to v initial t plus one half a t squared but because v initial is zero we don't need this portion of the equation so therefore the height of the building which we can replace with the displacement the height of the building is going to be just one half a t squared for those of you who just want a simple formula uh to find this answer i'm going to use d in this example a is negative 9.8 t is 5. so negative 4.9 which is half of 9.8 times 5 squared that's negative 122.5 meters so keep in mind that's the displacement of the ball so what that means is that the ball travels 122.5 meters down before it hits the ground so therefore the height of the building is the same but you don't need to describe the height of the building using a negative number all you need to say is is that um the building is 122.5 meters tall you don't have to say negative 122.5 that's not going to make any sense so this is the height of the building so now that you know how to do the last problem go ahead and try this one a stone is thrown downward from the top of a cliff at 24 meters per second and hits the ground seven seconds later how tall is the cliff so to no longer release from rest but someone just throws it down so therefore we know that this is an initial speed which is 24 but we're going to use the initial velocity which is negative 24 meters per second we have the time it takes for it to hit the ground that's seven seconds the acceleration in the y direction is still negative 9.8 meters per second squared our goal is to find the height of the cliff so basically we just need to find the displacement of the ball so let's use the same formula d is equal to v initial t plus one half a t squared this time we need this portion of the equation so v initial is negative 24 t is seven a is the same and now let's go ahead and find the answer negative 24 times 7 that's negative 168 and negative 4.9 times 7 squared that's about negative 240.1 so the displacement of the ball is negative 408.1 meters so what this means is that the ball travels 408.1 meters in a negative y direction that's why it's negative but the distance traveled which is the height of the cliff that's positive 408.1 meters and that's the answer a rock is released from rest on a 700 meter building how long does it take to hit the ground what is the speed and velocity of the ball just before it hits the ground well let's start with a picture so this time we're given the height of the building and initially we need to find out how long does it take to hit the ground so we got to find the time let's make a list of what we have the rock is released from rest so the initial speed is zero we have the acceleration in the y direction that's negative 9.8 we're looking for the time but we do have the displacement the displacement in the y direction is not positive 700 but it's a negative 700 because the ball is moving in the negative y direction so let's use this equation d is equal to v initial t plus one half a t squared now d is negative seven hundred v initial fortunately zero so we can avoid using the quadratic equation the acceleration is negative 9.8 so now all we got to do is find t so half of 9.8 is 4.9 now to isolate t squared let's divide both sides by negative 4.9 so negative 700 divided by negative 4.9 is 142.86 and it's positive so now let's take the square root of both sides so t i'm gonna write it right here is 11.95 seconds so that's how long it takes for the rock to hit the ground now what about part b how can we find the speed and velocity of the ball just before it hits the ground so we're looking for vf in this case we have the initial speed we have the acceleration and we have the time so therefore we can use this familiar equation v final is equal to v initial plus a t so v initial is zero the acceleration is negative 9.8 and we now have the time which is 11.95 seconds negative 9.8 times 11.95 that's negative 117.1 meters per second so this is the velocity of the wall just before it hits the ground the speed of the ball before it hits the ground it's the same number but positive it's positive 117.1 meters per second so these are the two answers to part b you