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# Projectile height given time

Video transcript

Let's say you and I are playing a game or I'm trying to figure out how high a ball is being thrown in the air How fast would we throwing that ball in the air? And what we do is one of us has a ball and the other one has a stop watch over here So this is my best attempt to make it more like a cat than a stop watch but I think you get the idea And what we do is one of us throw the ball the other one times how long the ball is in the air And what we do is gonna 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 how high it got And there is going to be one assumption I make here frankly that's an assumption we are gonna make in all of these projectile motion type problem is that air resistance is negligible And for something like it, this is a baseball or something like that That's a pretty good approximation So when can I get the exact answer, I encourage you experiment it on your own or even to see what air resistance does to your calculations We gonna assume for this projectile motion in future one at least in the basic Physics playlist We gonna assume air resistance is negligible And what that does for us is 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 takes it to go down If you look at this previous video, we've plot it displacement verse time You see after 2 seconds the ball went from being on the ground or I guess the thrower's hand all the way to its peak height And then the next 2 seconds it took the same amount of time to go back down to the ground which makes sense whatever the initial velocity is, it take half the time to go to zero and it takes the same amount of time to now be accelerated into downward direction back to that same magnitude of velocity but now in the downward direction So let's play around with some numbers here Just so you get a little bit more of concrete sense So let's say I throw a ball in the air And you measure using the stop watch and the ball is in the air for 5s So how do we figure out how fast I threw the ball? Well the first thing we could do is we could say look at the total time in the air was 5 seconds that mean the time, let me write it, that means the change in time to go up during the first half, I guess the ball time in the air 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 m/s in the 2.5 seconds And this is a graph for that example, This is the graph for the previous one, The previous example we knew the initial velocity but in whatever the time is you are going from you initial velocity to be stationery at the top, right with the ball being stationery and then start getting increasing velocity in the downward direction So it takes 2.5 seconds to go from some initial velocity to 0 seconds So we do know what the acceleration of the gravity is We know that the acceleration We know the acceleration of gravity here, we are assuming it's constant or slightly not constant but we are gonna assume it's constant We are just dealing close to the surface of the earth is negative 9.8 m/s*s, so let's think about it This change in velocity, are change in velocity Their change in velocity is the final velocity minus the initial velocity which is the same thing as zero minus the initial velocity which is the negative of the initial velocity That's another way to think about change in velocity We just shown the definition of acceleration change in velocity is equal to acceleration, is equal to acceleration negative 9.8 m/s*s times time or times change in time, our change in time, we are just talking about the first half of the ball's time in the air So the change in time is 2.5 s, times 2.5 s So what is our change in velocity which is also the same thing as negative of our initial velocity Get the calculator out, let me get my calculator, bring it on to the screen, so it is negative 9.8 m/s times 2.5 s Times 2.5 s, it gives us negative 24.5, so this gives us I will write it in new color This gives us negative 24.5 m/s, this seconds cancels out With one of these seconds in the denominator we only have one of the denominator out m/s, and this is the same thing as the negative, as the negative initial velocity Negative initial velocity that's the same thing as change in velocity So you multiply both side by a negative, you will get our initial velocity So that simply we are able to figure out what our velocity is So literally you take the time, the total time in the air divide by two And multiply that by acceleration of gravity and if you take I guess you can take the absolute value of that or take the positive version of that And that gives you your initial velocity So your initial velocity here is literally 24.5 m/s Since it's a positive quantity it is upwards in this example So that's my initial velocity, so we already figure out part of this game The initial velocity that threw upward That's also going to be, we gonna have the same magnitude of velocity The balls about to hit the ground although is gonna be in the other direction So what is the distance or let me make it clear what is the displacement of the ball from its lowest point right when it leaves your hand all the way to the peak, all the way to the peak? We just have to remember, all of these come from very straight forward ideas Change in velocity is equal to acceleration times change in time And the other simple idea is that displacement 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 2, or we assume that acceleration is constant So literally just the arithmetic mean of your initial and final velocity So what is that? That's gonna be 24.5 m/s plus our final velocity In this situation we are just going over to the first 2.5 s So our final velocity is once again 0 m/s We are just talking about when we get to his point over here So our final velocity is just 0 m/s And we just gonna divide that by 2 This will give us the average velocity And we wanna multiply that by 2.5 s, times 2.5 s So we get this part right over here 24.5 divided by 2 When you go with the 0, it is still 24.5 It gives us 12.25 times 2.5, and remember this right over here is in seconds let me write the units down So this is 12.25 m/s times 2.5 seconds And just to remind ourselves We are calculating the displacement over the first 2.5 seconds So this gives us, I get the calculator out once again We have 12.25 times 2.5 seconds gives us 30.625 So this gives us, this gives us, the displacement is 30.625 m The seconds cancel out This is actually a ton, you know, roughly give or take about 90 feet throw into the air, this looks like a nine stories building And I frankly do not have the arm for that But if someone is able to throw the ball for 5 seconds in the air They have thrown 30 meters in the air Hopefully you will find that entertaining In my next video I'll generalize this maybe we can get a little bit of 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 you will see how, at least how I tackle it in the next video