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Current time:0:00Total duration:8:51

let's say you and I are playing a game where we're trying to figure out how high a ball is being thrown in the air or how fast that we're throwing that ball in the air and what we do is one of us has a ball and the other one has a stopwatch over here so this is my best attempt to it looks more like a cat but a stopwatch but I think you get the idea and what we do is one of us throws the ball and the other one times how long the ball is in the air and then what we do is we're going to use that time in the air to figure out how fast the ball was thrown straight up and how long it was in the air or Starion and how high it got and there's going to be one assumption I make here and frankly this is assumption that we're going to make in all of these projectile motion type problems is that air resistance is negligible air resistance is negligible and for something like if this is a baseball or something like that that's a pretty good approximation so we're not going to get the exact answer and I could encourage you to experiment on your own to see or even to see what air resistance does relative to your calculations we're going to assume for this projectile motion and really all of the future ones or at least in the basic plate physics playlist we're going to assume that air resistance is negligible and what that does for us is that we can assume that the time up that the time for the ball to go up to its peak height is the same thing as the time that it takes to go down if you look at this previous video where we plotted displacement versus time you see after two seconds the ball went from being on the ground or I guess the throwers hand all the way to its peak height and then the next two seconds it took that same amount of time to go back down to the ground which makes sense whatever the initial velocity is it takes half the time to go to zero and it takes that same amount of time to now be accelerated in the downward direction back to that same magnitude of velocity but now in the downward direction so let's let's play around with some numbers here just so we get a little bit more of a concrete sense so let's say I throw a ball in the air and you measure using the stopwatch that the ball is in the air for five seconds so how do we figure out how fast I threw the ball well the first thing we can do is we could say look if the the total time in the air was five seconds that means that the time let me well that means that the change in time to go up the during the the first half of I guess the balls time of the year is going to be 2.5 seconds and which tells us that over this 2.5 seconds we went from our initial velocity whatever it was we went from our initial velocity to our final velocity which is a velocity of 0 meters per second in the two and a half seconds and this isn't the graph for that example this is the graph for the previous one I the previous example where we knew the initial velocity but in whatever that time is you're going from your initial velocity to being stationary at the top right right when the ball is stationary and then starts getting increasing velocity in the downward in the downward direction so it takes 2.5 seconds to go from some initial velocity to zero seconds so we do know what the acceleration of gravity is we know that the acceleration we know that the acceleration of gravity here we're assuming it's constant although it's slightly not constant but we're going to assume it's constant we're just dealing close to the surface of the earth is negative 9.8 meters meters per second squared so let's think about it this our change in velocity our change in velocity that our change in velocity is the final velocity minus the initial velocity which which is the same thing as zero minus the initial velocity which is the negative of the initial velocity and what's another way to think about change in velocity well just from the definition of acceleration change in velocity is equal to acceleration is equal to acceleration negative 9.8 meters per second squared times time or times change in time our change in time we're just talking about the first half of the ball's time in the air so our change in time is 2.5 seconds times 2.5 seconds so what is our change in velocity which is also the same thing as the negative of our initial velocity get the calculator out let me get my calculator bring it onto the screen so it is it is negative 9.8 meters per second times 2.5 seconds times 2.5 seconds it gives us negative 24 point 5 negative so this gives us let me write in a new color this gives us negative 24 point five meters per second this second cancels out with one of these seconds in the denominator so we only have one in the denominator now so it's meters per second and this is the same thing as our negative as the negative initial velocity negative initial velocity that's the same thing as our change in velocity and so you multiply both sides by a negative u we get our initial velocity so that's simply that simply we were able to figure out what our velocity was so literally you take the time take the total time in the air take it divide it by 2 and then multiply that by the acceleration of gravity and if and if you take s you can take the absolute value of that or you take the positive version of that and then that gives you that gives you your initial velocity so your initial velocity here is literally twenty four point five meters per second and since it's a positive quantity it is upwards in this example so that's my initial velocity so we already figured out part of this game the whole the initial velocity that I threw it upwards and that's also going to be that we're gonna have the same magnitude of velocity when the ball is about to hit the ground although it's going to be in the other direction so what is the distance or let me make it clear what is the displacement of the ball from from its lowest point right when it leaves your hand all the way to the peak all the way to the peak well we just have to remember and once again all this comes from very straight very straightforward ideas change in velocity is equal to acceleration times change in time and then the other simple idea is that displacement displacement is equal to is equal to average velocity average velocity times change in time now what is our average velocity our average velocity is your initial velocity plus your final velocity divided by two or if we assume acceleration is constant so it's literally just the earth edek mean of your initial and final velocities so what is that that's going to be twenty four point five meters per second plus what's our final velocity in this situation room we're just going over the first two point five seconds so our final velocity is once again zero meters per second we're just talking about when we get to this point right over here so our final velocity is just zero meters per second and we're just going to divide that by two this will give us the average velocity and then we want to multiply that times two point we want to multiply that times two point five times two point five seconds so we get this part right over here twenty four point five divided by two we can ignore the zero that's still is twenty four point five that gives us twelve point two five times two point five times two point five and remember this right over here is in seconds let me write the unit's down so this is 12.25 meters per second times two point five seconds times two point five seconds and just to remind ourselves we are cat we're calculating the displacement over the first two and a half seconds so this gives us I'll get the calculator out once again we have twelve point two five times two point five seconds gives us 30 point six two five so this gives us this gives us so our displacement is thirty point six two five thirty point six two five meters these seconds cancel out meters so someone this is actually a ton this is you know roughly give or take about 90 feet thrown in the air so this would be like a nine story building and I frankly do not have the arm for that but if someone is able to throw the ball for five seconds in the air they have thrown it thirty meters in the air well hopefully you found that entertaining in the next video I'll generalize this maybe we can get a little bit of a formula so maybe you can generalize it so regardless of the measurement of time you can get the displacement in the air or even better try to derive it yourself and we'll see how at least how I tackle it in the next video