Kinematic formulas and projectile motion
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Average Velocity for Constant Acceleration
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Acceleration of Aircraft Carrier Takeoff
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Deriving Displacement as a Function of Time, Acceleration and Initial Velocity
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Plotting Projectile Displacement, Acceleration, and Velocity
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Projectile Height Given Time
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Deriving Max Projectile Displacement Given Time
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Impact Velocity From Given Height
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Viewing g as the value of Earth's Gravitational Field Near the Surface
Deriving Max Projectile Displacement Given Time Deriving a formula for maximum projectile displacement as a function of elapsed time
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- I just want to follow up on the last video
- Where we threw balls up in the air and saw how long they stayed up in the air
- and we used that to figure out how fast we inicialy threw the ball
- and how high they went in the air
- and in the last video we did it with specific numbers
- and 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
- on some type of a field and we don't necessarily have, have any,
- any paper around
- so lets say that the ball is in the air for 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
- we'll do that the 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 get's released
- and it goes and its, and it's going as kind of it's
- maximum upward velocity it goes slower and slower and slower
- all the way until it's stationary for just a moment then starts going down
- again, remember the acceleration is costed downwards
- this entire time so what is the final velocity if we just
- consider half of this time well the time is zero so
- it's going to be zero minus our
- inicial velocity, minus our inicial velocity
- when it was taking off
- thats 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
- now negative nine point eight meters per second square
- or the acceleration due to gravity
- when an object is in freefall
- to be technicaly correct times,
- times, times the time we are going up
- so times delta T up witch 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 divided be two is going to be
- four point nine meters per second square
- we still have our negative out in front
- times our delta T, times our delta T, and 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 the negative and we get
- our initial velocity, is just going to be equal to four point nine, four point nine
- meters per second square times the total time, the total time
- that we are in the air, or you could say, the or you could say
- it's the, it's going to be nine point eight meters per second square times half
- of the time that we're in the air, either of those will get you the same
- calculation, so lets figure out our total distance,
- or the distance that we travel in that first in the time of, so that will give us the peek distance, remember that distance
- or should i say displacement in this situation, displacement is equal to
- average velocity, average velocity times the change in time, the change in time that we care
- about is the time of, so that is our delta T over two, our total time
- our total time divided by two, this is our, this is our time of
- time of, and whats our average velocity? well the average velocity
- if we assume constant acceleration is your initial velocity plus your final velocity
- over two, its 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 thing is
- this thing over here, our final velocity, remember we're just talking about
- the first half of the time the ball's in the air so it's final velocity
- is zero we're talking when it gets to this peak point over here
- its 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 square, times delta T
- times delta T over, over two, so this right here this is our average velocity
- velocity average so lets stick that back over here
- so our maximum displacement
- our maximum displacement is going to be our average velocity, so that is going to be
- four point nine meters per second square times delta T, times delta T, all of that over two
- and then we multiply that again times the total times the time of,
- so times delta T over two again, this is the same thing, and 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 four point nine
- divided by four, four point nine divided by four is, what is it, it's one point
- one point two, one point two, two, five I believe, let me just get my calculator out
- I don't want to do that in my head, get this far and make a careless mistake
- four point nine divided by four is one 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, witch is a pretty, witch is a pretty
- straight forward, witch is a pretty straight forward calculation
- so this is, this is our max displacement, kind of how far do we, how high are we getting
- displaced, right when, right when that ball is stationary, or is, is , is no net velocity
- just for a moment and starts decelerating downwards, so we can use that is a ball is in the air
- for five seconds we can verify our computation from the last video
- it would, our max velocity is one point two two five times five squared
- which is twenty five, will give us thirty point six two five, that's what we got in the last video
- if the balls in the air for, i don't know two point three seconds, so its one point two, two, five
- times two point three squared, then that means that it went
- six point four, eight meters in the air, so anyway I just wanted to give you a, a simply 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
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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