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No practicing perfect projectiles more fun with two t emotion

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I put next one together being the rocket's trajectory I

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can't forget the equation court Wait forget that i'm getting

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more complex No All right here we go Ladies and

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gentlemen we regret to inform you for it War now

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the proud country of shmoop sylvania has put up with

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our old enemy East ignorance is stand for too long

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We plan on launching a barrage of smart rockets to

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attack their lack of knowledge and its core thes rockets

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will carry payloads of literature algebra history and oh so

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much more All right rocket scientists what is rocket science

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But physics put in emotion explode emotion one of our

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favorite kinds Well but before we just start launching rockets

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willy nilly it'll help if we can figure out how

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far they're going to go and how long it'll take

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to get him there So consider this lesson Rocket science

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one oh one here the equations we're going to need

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for this lesson We saw these in the last lesson

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but yeah let's go over it again just to make

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sure we remember him We've got the one for displacement

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in the extraction which we find by multiplying the velocity

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and the extraction by the elapsed time we could also

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find the change in displacement vice of tracking the initial

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displacement from the final displacement like in this equation right

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here and these are the only equations will need for

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motion along the x axis because we're not gonna have

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any acceleration along the x axis but the only acceleration

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will be dealing with will be in the uae direction

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Yeah and that'll be gravity doing it's a you know

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gravity thing So we'll be using to equations for an

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emotion on the y axis first this one it tells

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us that the change in displacement in the uae direction

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equals the initial velocity in the white direction multiplied by

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the time plus one half the acceleration of gravity times

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time squared And if we don't know how much time

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a particular emotion takes well in that case we can

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put this equation to use what's it telling us while

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the square of the final velocity in the wider action

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equals the square the initial y velocity plus two times

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the acceleration of gravity times the change in displacement along

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that why axis So yeah we have a history with

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these three equations Phone we go way back but we're

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doing something new with them Today we'll be using more

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than one to help us find whatever solution that we're

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looking for but remember we can't get our ex variables

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mixed up with our wives variables i have to remain

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separate it's that time that links them together and while

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we're keeping our x and y separate let's talk about

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how to talk about him while hotshot physicists use different

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terms when it comes to these different motions when we're

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talking about motion in the extraction will use the term

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range to describe the maximum horizontal distance of projectile travels

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but when we're looking at vertical motion will use the

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term maximum height to describe well that maximum height of

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our projectile And we might think of a projectile only

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in terms of missiles or bullets or whatever but term

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doesn't have to refer to things that go bang when

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a hunter kicks a football well that football is now

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a projectile If you accidentally knock your fork off the

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dinner table the fork is now projectile and if we

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toss you a soda that can is a projectile although

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it might be an example of an explosion No well

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now we're dealing with something as complicated as rocket science

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we're gonna have to expand our arsenal of handy physics

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trip First of all everything will be dealing with here

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will still have zero acceleration along the x axis and

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it'll also have the acceleration of gravity in the y

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axis There won't be any other accelerations to wrap our

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minds around Let's take a look at the full trajectory

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of one of our smart rockets All right what is

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this trajectory Tell us about the vertical velocity Well for

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one thing we know there's an initial vertical velocity This

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isn't the case where a car drives off the side

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of a cliff This rocket is going up up up

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been away But as we know from that one time

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we were throwing our little cousin in here What goes

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up You must come down with a nice pretty problem

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like this We can see that the overall motion is

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symmetrical You could fold it right in half So when

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a projectile begins and ends its vertical motion in the

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same position like here where it begins and ends at

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the zero point for why then the max height will

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occur halfway through whatever time period we're looking at You

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might see that Height referred to as the change in

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height or delta y and the upward velocity that occurs

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during the first half of the motion will be equally

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matched by that downward motion in the second half And

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when we look at the halfway mark again will find

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that the y velocity at that precise moment is zero

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meters per second just hanging in the air for one

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tiny sliver of time Then gravity wins the battle in

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the velocity turns downward now believe it or not this

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type of motion isn't restricted on ly two rockets What

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if we head to the basketball court so we can

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show off our sick moves and our three point range

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Okay that was an air ball which is perfect it's

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what we're trying to do Really Because then we can

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show you this graph just like the rocket The best

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well traveled in a parabola See how we have our

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velocity arrows there at every point on the graph the

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horizontal motion has the same velocity which is why all

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those arrows are the same size But the vertical arrows

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change if you line them up Like looking at the

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third basketball from the right and the third basket ball

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from the left we see that the arrows are pointing

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in different directions but they have the same magnitude Yeah

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remember this equation where we're finding the final velocity Yeah

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well if the change in displacement in the white direction

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is zero it means the whole second half of the

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right side will equal zero leaving us with final velocity

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equal in the initial velocity at least in terms of

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magnitude moving in a direction So let's put this stuff

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in action Let's say that shmoop er man in bizarro

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shmoop her man that trooper man's evil twin We're having

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a friendly game A catch near the fortress of learning

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you know is friendly again The catch is you can

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get with your evil twin Well since they're twins they

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throw with the same velocities both vertical and horizontal and

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they throw and catch from the same height Now these

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aren't normal people tossing a baseball around so they're putting

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some oomph into these things Let's say they're throwing in

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catching the ball from one point five meters off the

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ground and the ball reaches a mac sight of one

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hundred one point Five meters How much time does it

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take the ball to travel between this superhuman pair Okay

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well first of all let's figure out what we know

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what we don't know and what we want to know

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Well let's look at the motion in the uae direction

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First of all we know that max height is one

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hundred one point five meters and the starting height is

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one point Five meters when we subtract the initial list

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placement from the max when you find a change in

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displacement of one hundred meters no what about the initial

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y velocity It must be pretty high but we don't

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know what it is at this point We do know

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that the initial velocity is the same as the final

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velocity when the ball reaches bizarro shmoop her man And

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we know that when the balls at its highest point

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the velocity in the uae direction is zero meters per

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second which is the key right there Now we know

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nothing about the horizontal motion velocity distance No clue but

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we're only trying to find the time here so we

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don't need all that stuff What we need to do

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is figure out how long the ball takes to reach

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the max tight or alternatively how long it takes the

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ball to fall from the max height Clever each half

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of the balls flight will take the same amount of

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time So what equation will we use Well we've got

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two to choose from for vertical motion Well first we've

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got this one for the change in displacement on the

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y axis and then we've got this one to find

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the final velocity when we're trying to find the time

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people and only one equation has time in it and

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everything so it looks like we'll be choosing bachelor number

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one We'll use the change in height of one hundred

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meters and we'll find the time it takes for the

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ball to go from its peak to pizarro's glove Why

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Because that lets us set the initial velocity at zero

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zero meters per second which simplifies the equation a whole

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lot for us because the first part of the equation

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on the right side the initial loss any times time

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will equal zero when the initial velocity equals zero which

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means that the change in displacement equals one half the

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acceleration and gravity Times the square of the time period

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Now we just have to isolate a that tea Well

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in most cases we've set the acceleration of gravity as

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a negative number but remember it's totally upto us and

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in this case we're actually going to use the positive

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version Why Because we're all about the power of positivity

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and keeping the acceleration positive will really help We'll show

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you why in a second so the acceleration of gravity

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will be nine point eight meters per seconds squared And

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when we have that we get four point nine meters

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per second squared And now let's divide both sides of

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the equation by that number Leaving us with time squared

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equals one hundred meters over four point nine meters per

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second squared I was still not done because we need

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t not t squared So time equals the square root

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of one hundred meters over four point nine years per

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second squared Which is why we used a positive value

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for the acceleration and gravity because finding a negative square

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route leads us into the land of imaginary numbers And

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this story is about a superhero and his evil twins

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Oh that needs to be you know grounded in reality

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when we put the numbers into our trusty calculator we

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find that t equals four point five two seconds but

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hold on remember this was on ly for the second

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half of the trajectory the time for the first half

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is the same so we just have to do double

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our result to get the total time which means the

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ball is in the air for nine point oh four

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seconds and were able to figure out the time But

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what about the velocities Both horizontal and vertical Well there's

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no way we'd actually be able to calculate that is

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there Well actually no atleast for the horizontal velocity because

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we need one more piece of information which is the

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distance or range between the two super twins So we'll

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say it's well half a kilometre better known asked five

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hundred meters and we'll keep the vertical values the same

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as what we were using before Now we can get

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there so let's tackle the vertical velocity first this time

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we'll use that other equation we were looking at and

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we'll use the same trick we did last time by

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focusing on the second half of the trajectory which once

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again lets us use an initial velocity of zero And

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to be frank it's not really a trick Because if

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we looked at the whole trajectory are changing displacement would

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equal zero And then our equation would just tell us

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that the final velocity equals the initial velocity and you

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do like a public chasing its tail which is pretty

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cute but not helpful in doing physics All right let's

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put numbers into this equation So the final velocity squared

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equals two times the acceleration of gravity and we'll use

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the positive number again So that's nine point eight meters

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per seconds squared times the change in vertical displacement which

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is a hundred meters So we find the final velocity

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squared equals nineteen hundred sixty meters per second And then

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we need the square route to get the actual velocity

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which comes out forty four point three meters per second

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and remember people the final velocity equals the initial velocity

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So we killed two birds with one stone here or

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a one baseball and not just for the vertical velocity

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for horizontal motion We only have one equation worry about

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and that's this One where the changing displacement equals of

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velocity times the time Remember the vertical and horizontal motions

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are linked by that t there An earlier we figured

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out that the time elapsed was nine point oh four

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seconds and our distance or displacement for the throw is

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five hundred meters In order to isolate the velocity we

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need divide both sides by the change in time or

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delta t Once we've done that we find that the

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velocity equals the distance divided by the time so the

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velocity in the ex direction equals five hundred meters divided

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by nine point Oh four seconds giving us a horizontal

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velocity of fifty five point three meters per second which

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is equivalent to about one hundred twenty four miles an

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hour which is really fast right Well once again we

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were able to get all the answers by looking at

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the two perpendicular motions separately which should be everything we

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need to launch our smart rockets and wipe east ignorant 00:12:10.6 --> [endTime] to stand off the map

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