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Teachers & SchoolsTime for some fun with 2-D motion. We'll use horizontal speed and distance to find vertical motion, and more. So... a 2-D experiment, but 3-D excitement.

Courses | Physics |

Language | English Language |

the proud country of shmoop sylvania has put up with

our old enemy East ignorance is stand for too long

We plan on launching a barrage of smart rockets to

attack their lack of knowledge and its core thes rockets

will carry payloads of literature algebra history and oh so

much more All right rocket scientists what is rocket science

But physics put in emotion explode emotion one of our

favorite kinds Well but before we just start launching rockets

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

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

to get him there So consider this lesson Rocket science

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

for this lesson We saw these in the last lesson

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

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

in the extraction which we find by multiplying the velocity

and the extraction by the elapsed time we could also

find the change in displacement vice of tracking the initial

displacement from the final displacement like in this equation right

here and these are the only equations will need for

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

any acceleration along the x axis but the only acceleration

will be dealing with will be in the uae direction

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

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

emotion on the y axis first this one it tells

us that the change in displacement in the uae direction

equals the initial velocity in the white direction multiplied by

the time plus one half the acceleration of gravity times

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

a particular emotion takes well in that case we can

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

the square of the final velocity in the wider action

equals the square the initial y velocity plus two times

the acceleration of gravity times the change in displacement along

that why axis So yeah we have a history with

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

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

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

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

mixed up with our wives variables i have to remain

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

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

how to talk about him while hotshot physicists use different

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

talking about motion in the extraction will use the term

range to describe the maximum horizontal distance of projectile travels

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

term maximum height to describe well that maximum height of

our projectile And we might think of a projectile only

in terms of missiles or bullets or whatever but term

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

a hunter kicks a football well that football is now

a projectile If you accidentally knock your fork off the

dinner table the fork is now projectile and if we

toss you a soda that can is a projectile although

it might be an example of an explosion No well

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

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

trip First of all everything will be dealing with here

will still have zero acceleration along the x axis and

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

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

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

of one of our smart rockets All right what is

this trajectory Tell us about the vertical velocity Well for

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

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

of a cliff This rocket is going up up up

been away But as we know from that one time

we were throwing our little cousin in here What goes

up You must come down with a nice pretty problem

like this We can see that the overall motion is

symmetrical You could fold it right in half So when

a projectile begins and ends its vertical motion in the

same position like here where it begins and ends at

the zero point for why then the max height will

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

might see that Height referred to as the change in

height or delta y and the upward velocity that occurs

during the first half of the motion will be equally

matched by that downward motion in the second half And

when we look at the halfway mark again will find

that the y velocity at that precise moment is zero

meters per second just hanging in the air for one

tiny sliver of time Then gravity wins the battle in

the velocity turns downward now believe it or not this

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

if we head to the basketball court so we can

show off our sick moves and our three point range

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

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

show you this graph just like the rocket The best

well traveled in a parabola See how we have our

velocity arrows there at every point on the graph the

horizontal motion has the same velocity which is why all

those arrows are the same size But the vertical arrows

change if you line them up Like looking at the

third basketball from the right and the third basket ball

from the left we see that the arrows are pointing

in different directions but they have the same magnitude Yeah

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

well if the change in displacement in the white direction

is zero it means the whole second half of the

right side will equal zero leaving us with final velocity

equal in the initial velocity at least in terms of

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

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

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

a friendly game A catch near the fortress of learning

you know is friendly again The catch is you can

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

throw with the same velocities both vertical and horizontal and

they throw and catch from the same height Now these

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

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

catching the ball from one point five meters off the

ground and the ball reaches a mac sight of one

hundred one point Five meters How much time does it

take the ball to travel between this superhuman pair Okay

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

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

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

First of all we know that max height is one

hundred one point five meters and the starting height is

one point Five meters when we subtract the initial list

placement from the max when you find a change in

displacement of one hundred meters no what about the initial

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

know what it is at this point We do know

that the initial velocity is the same as the final

velocity when the ball reaches bizarro shmoop her man And

we know that when the balls at its highest point

the velocity in the uae direction is zero meters per

second which is the key right there Now we know

nothing about the horizontal motion velocity distance No clue but

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

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

is figure out how long the ball takes to reach

the max tight or alternatively how long it takes the

ball to fall from the max height Clever each half

of the balls flight will take the same amount of

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

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

got this one for the change in displacement on the

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

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

people and only one equation has time in it and

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

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

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

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

Because that lets us set the initial velocity at zero

zero meters per second which simplifies the equation a whole

lot for us because the first part of the equation

on the right side the initial loss any times time

will equal zero when the initial velocity equals zero which

means that the change in displacement equals one half the

acceleration and gravity Times the square of the time period

Now we just have to isolate a that tea Well

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

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

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

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

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

you why in a second so the acceleration of gravity

will be nine point eight meters per seconds squared And

when we have that we get four point nine meters

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

the equation by that number Leaving us with time squared

equals one hundred meters over four point nine meters per

second squared I was still not done because we need

t not t squared So time equals the square root

of one hundred meters over four point nine years per

second squared Which is why we used a positive value

for the acceleration and gravity because finding a negative square

route leads us into the land of imaginary numbers And

this story is about a superhero and his evil twins

Oh that needs to be you know grounded in reality

when we put the numbers into our trusty calculator we

find that t equals four point five two seconds but

hold on remember this was on ly for the second

half of the trajectory the time for the first half

is the same so we just have to do double

our result to get the total time which means the

ball is in the air for nine point oh four

seconds and were able to figure out the time But

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

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

there Well actually no atleast for the horizontal velocity because

we need one more piece of information which is the

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

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

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

as what we were using before Now we can get

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

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

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

focusing on the second half of the trajectory which once

again lets us use an initial velocity of zero And

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

we looked at the whole trajectory are changing displacement would

equal zero And then our equation would just tell us

that the final velocity equals the initial velocity and you

do like a public chasing its tail which is pretty

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

put numbers into this equation So the final velocity squared

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

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

per seconds squared times the change in vertical displacement which

is a hundred meters So we find the final velocity

squared equals nineteen hundred sixty meters per second And then

we need the square route to get the actual velocity

which comes out forty four point three meters per second

and remember people the final velocity equals the initial velocity

So we killed two birds with one stone here or

a one baseball and not just for the vertical velocity

for horizontal motion We only have one equation worry about

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

velocity times the time Remember the vertical and horizontal motions

are linked by that t there An earlier we figured

out that the time elapsed was nine point oh four

seconds and our distance or displacement for the throw is

five hundred meters In order to isolate the velocity we

need divide both sides by the change in time or

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

velocity equals the distance divided by the time so the

velocity in the ex direction equals five hundred meters divided

by nine point Oh four seconds giving us a horizontal

velocity of fifty five point three meters per second which

is equivalent to about one hundred twenty four miles an

hour which is really fast right Well once again we

were able to get all the answers by looking at

the two perpendicular motions separately which should be everything we

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