# Physics: Practicing Perfect Projectiles

Time 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.

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### Transcript

00:26

the proud country of shmoop sylvania has put up with

00:29

our old enemy East ignorance is stand for too long

00:34

We plan on launching a barrage of smart rockets to

00:36

attack their lack of knowledge and its core thes rockets

00:39

will carry payloads of literature algebra history and oh so

00:43

much more All right rocket scientists what is rocket science

00:47

But physics put in emotion explode emotion one of our

00:52

favorite kinds Well but before we just start launching rockets

00:55

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

00:58

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

01:00

to get him there So consider this lesson Rocket science

01:03

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

01:06

for this lesson We saw these in the last lesson

01:08

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

01:10

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

01:14

in the extraction which we find by multiplying the velocity

01:17

and the extraction by the elapsed time we could also

01:20

find the change in displacement vice of tracking the initial

01:22

displacement from the final displacement like in this equation right

01:26

here and these are the only equations will need for

01:29

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

01:31

any acceleration along the x axis but the only acceleration

01:35

will be dealing with will be in the uae direction

01:37

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

01:40

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

01:43

emotion on the y axis first this one it tells

01:46

us that the change in displacement in the uae direction

01:50

equals the initial velocity in the white direction multiplied by

01:52

the time plus one half the acceleration of gravity times

01:55

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

01:58

a particular emotion takes well in that case we can

02:00

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

02:03

the square of the final velocity in the wider action

02:05

equals the square the initial y velocity plus two times

02:09

the acceleration of gravity times the change in displacement along

02:12

that why axis So yeah we have a history with

02:15

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

02:17

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

02:20

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

02:23

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

02:26

mixed up with our wives variables i have to remain

02:28

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

02:32

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

02:34

how to talk about him while hotshot physicists use different

02:38

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

02:41

talking about motion in the extraction will use the term

02:44

range to describe the maximum horizontal distance of projectile travels

02:49

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

02:51

term maximum height to describe well that maximum height of

02:55

our projectile And we might think of a projectile only

02:58

in terms of missiles or bullets or whatever but term

03:01

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

03:04

a hunter kicks a football well that football is now

03:07

a projectile If you accidentally knock your fork off the

03:10

dinner table the fork is now projectile and if we

03:13

toss you a soda that can is a projectile although

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it might be an example of an explosion No well

03:19

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

03:22

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

03:28

will still have zero acceleration along the x axis and

03:32

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

03:35

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

03:39

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

03:41

of one of our smart rockets All right what is

03:43

this trajectory Tell us about the vertical velocity Well for

03:47

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

03:53

of a cliff This rocket is going up up up

03:55

been away But as we know from that one time

03:59

we were throwing our little cousin in here What goes

04:02

up You must come down with a nice pretty problem

04:05

like this We can see that the overall motion is

04:07

symmetrical You could fold it right in half So when

04:10

a projectile begins and ends its vertical motion in the

04:13

same position like here where it begins and ends at

04:17

the zero point for why then the max height will

04:20

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

04:23

might see that Height referred to as the change in

04:26

height or delta y and the upward velocity that occurs

04:30

during the first half of the motion will be equally

04:32

matched by that downward motion in the second half And

04:35

when we look at the halfway mark again will find

04:38

that the y velocity at that precise moment is zero

04:42

meters per second just hanging in the air for one

04:44

tiny sliver of time Then gravity wins the battle in

04:47

the velocity turns downward now believe it or not this

04:50

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

04:53

04:55

show off our sick moves and our three point range

04:58

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

05:02

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

05:07

well traveled in a parabola See how we have our

05:10

velocity arrows there at every point on the graph the

05:13

horizontal motion has the same velocity which is why all

05:17

those arrows are the same size But the vertical arrows

05:20

change if you line them up Like looking at the

05:23

05:26

from the left we see that the arrows are pointing

05:29

in different directions but they have the same magnitude Yeah

05:33

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

05:37

well if the change in displacement in the white direction

05:40

is zero it means the whole second half of the

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right side will equal zero leaving us with final velocity

05:47

equal in the initial velocity at least in terms of

05:49

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

05:52

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

05:55

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

05:58

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

06:16

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

06:45

placement from the max when you find a change in

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displacement of one hundred meters no what about the initial

06:50

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

06:58

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

07:07

second which is the key right there Now we know

07:10

nothing about the horizontal motion velocity distance No clue but

07:15

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

07:17

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

07:25

ball to fall from the max height Clever each half

07:29

of the balls flight will take the same amount of

07:30

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

07:34

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

07:38

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

07:45

people and only one equation has time in it and

07:48

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

07:53

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

08:00

Because that lets us set the initial velocity at zero

08:03

zero meters per second which simplifies the equation a whole

08:07

lot for us because the first part of the equation

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on the right side the initial loss any times time

08:11

will equal zero when the initial velocity equals zero which

08:14

means that the change in displacement equals one half the

08:18

acceleration and gravity Times the square of the time period

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Now we just have to isolate a that tea Well

08:25

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

08:27

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

08:30

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

08:32

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

08:36

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

08:40

you why in a second so the acceleration of gravity

08:42

will be nine point eight meters per seconds squared And

08:45

when we have that we get four point nine meters

08:48

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

08:50

the equation by that number Leaving us with time squared

08:53

equals one hundred meters over four point nine meters per

08:55

second squared I was still not done because we need

08:58

t not t squared So time equals the square root

09:02

of one hundred meters over four point nine years per

09:04

second squared Which is why we used a positive value

09:07

for the acceleration and gravity because finding a negative square

09:10

route leads us into the land of imaginary numbers And

09:13

this story is about a superhero and his evil twins

09:17

Oh that needs to be you know grounded in reality

09:20

when we put the numbers into our trusty calculator we

09:22

find that t equals four point five two seconds but

09:25

hold on remember this was on ly for the second

09:28

half of the trajectory the time for the first half

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is the same so we just have to do double

09:34

our result to get the total time which means the

09:37

ball is in the air for nine point oh four

09:39

seconds and were able to figure out the time But

09:41

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

09:44

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

09:47

there Well actually no atleast for the horizontal velocity because

09:52

we need one more piece of information which is the

09:54

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

09:58

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

10:01

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

10:05

as what we were using before Now we can get

10:07

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

10:11

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

10:13

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

10:15

focusing on the second half of the trajectory which once

10:18

again lets us use an initial velocity of zero And

10:21

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

10:24

we looked at the whole trajectory are changing displacement would

10:27

equal zero And then our equation would just tell us

10:30

that the final velocity equals the initial velocity and you

10:33

do like a public chasing its tail which is pretty

10:36

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

10:39

put numbers into this equation So the final velocity squared

10:42

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

10:45

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

10:48

per seconds squared times the change in vertical displacement which

10:51

is a hundred meters So we find the final velocity

10:54

squared equals nineteen hundred sixty meters per second And then

10:58

we need the square route to get the actual velocity

11:00

which comes out forty four point three meters per second

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and remember people the final velocity equals the initial velocity

11:07

So we killed two birds with one stone here or

11:10

a one baseball and not just for the vertical velocity

11:14

for horizontal motion We only have one equation worry about

11:17

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

11:20

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

11:30

seconds and our distance or displacement for the throw is

11:33

five hundred meters In order to isolate the velocity we

11:36

need divide both sides by the change in time or

11:39

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

11:42

velocity equals the distance divided by the time so the

11:45

velocity in the ex direction equals five hundred meters divided

11:48

by nine point Oh four seconds giving us a horizontal

11:51

velocity of fifty five point three meters per second which

11:55

is equivalent to about one hundred twenty four miles an

11:57

hour which is really fast right Well once again we

12:00

were able to get all the answers by looking at

12:02

the two perpendicular motions separately which should be everything we

12:06

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