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# Deriving max projectile displacement given time

## Video transcript

just want to follow up on the last video where we threw balls in the air and saw how long they stayed up in the air we use that to figure out how fast we initially threw the ball and how high they went in the air and the last video we did it with specific numbers in this video I just want to see if we can derive some interesting formulas so we can do the computations really fast in our brains while we're playing this game out on some type of a field and we don't necessarily have have any paper around so let's say that the ball is in the air for a delta T delta T is equal to time in the air time in the air then we know that the time up is going to be half that which is the same thing as the time down the time up is going to be equal to delta T do that same color is going to be equal to the time in the air divided by two so how what was our initial velocity well all we have to do is remind ourselves that the change in velocity the change in velocity which is the same thing as the final velocity minus the initial velocity so the final velocity remember we're just talking about half of the path of this ball so the time that it gets released and it goes as going as kind of its maximum upward velocity and go slower and slower slower all the way until it's stationary for just a moment and then starts going down again remember the acceleration is constant downwards this entire time so what is the final velocity if we just consider half of the time well it's the time it's zero so it's going to be zero minus our initial velocity minus our initial velocity when it was taking off that's our change in velocity this is our change in velocity this is our change in velocity is going to be equal to the acceleration of gravity the acceleration of gravity not negative 9.8 meters per second squared or the acceleration due to gravity when an object is in free fall to be technically correct x times the time that we are going up so times delta T up which is the same thing I won't even write it delta T up is the same thing as our total time in the air our total time in the air divided by two and so we get we get negative the initial velocity is equal to this thing when you divided by two is going to be four point nine meters per second squared we still have our negative out front times our delta T times our delta T you remember this is our total time in the air not just the time up this is our total time in the air and then we multiply both sides times a negative we get our initial velocity is just going to be equal to four point nine four point nine meters per second squared times the total time the total time that we are in the air or you could say that or you could say it's the it's going to be nine point eight meters per second squared times half of the time that we're in the air either of those would get you the same calculation so let's figure out our total distance or the distance that we travel in that first in the time up so that will give us our peak distance remember that distance or I should say displacement in this situation displacement is equal to average velocity average velocity times change in time the change in time that we care about is the time up so that is our delta T over two our total time our total time divided by two this is our this is our time up time up and what's our average velocity well the average velocity if we assume constant acceleration is your initial velocity plus your final velocity over to it's really just the mean of the two things well we know what our initial velocity is our initial velocity is this thing over here so this is this thing over here our final velocity remember we're just talking about the first half of the time the balls in the air so our it's final velocity is zero we're talking when it gets to this this peak point right over here that's from two videos ago that peak point right over there so our average velocity is just going to be our average velocity is just going to be this stuff divided by two so it's going to be four point nine meters per second squared times delta T times delta T over over - so this right here this is our average velocity velocity average so let's stick that back over here so our maximum displacement our maximum displacement is going to be our average velocity so that is 4.9 m/s squared times delta T times delta T all of that over 2 and then we multiply it again times the total times the time up so times delta T over 2 again this is the same thing these are the same thing and then we can simplify it our maximum displacement is equal to four point nine meters per second squared times delta T squared times delta T squared all of that over all of that over four and then we can just divide 4.9 divided by four 4.9 divided by four is was it it's one point one point two one point two two five I think let me just get the calculator out I want to do that in my head get this far make a careless mistake 4.9 divided by 4 is 1 point two two five so this is so our maximum displacement is going to be one point two two five times our total time in the air total time in the air squared which is pretty which is a pretty straightforward which is a pretty straightforward calculation so this is this is our max displacement kind of how far do we how high are we get displaced right when right when that ball is stationary or is has no net velocity just for a moment it starts decelerating downwards so we can use that if a ball is in the air for five seconds we can verify our computation from the last video it our maximum displacement 1.2 - 5 times 5 squared which is 25 will give us 30 point 6 - 5 that's what we got in the last video if the ball is in the air for I don't know two point three seconds so it's one point two to five times two point three squared then that means it went six point four eight meters in the air so anyway I just wanted to give you a a simple expression that gives you that gives you the maximum displacement from the ground assuming air resistance is negligible as a function of the total time in the air and I don't know I find that pretty pretty fun and it's a it's a neat game to play