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Shmoop! Projectile Prediction: Galileo, trigonometry, and an experiment.

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[mumbling]

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[mumbling]

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So, a long time ago there was this guy named, Galileo Galilei and his parents

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were cruel. He was a super-smart Italian guy, who figured out a way to measure

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gravity. Now that may not sound too impressive today, but he did all this

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science, in the 1500s. He didn't have any fancy computers, or a smart phone to [Galileo in different rooms experimenting]

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record stuff, or YouTube to watch videos of dogs and monkeys being best friends.

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Nope, he had to figure out a simple way to measure gravity. So what he did was

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set up a ramp, roll the ball down it and timed how long it took. And he did this

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experiment hundreds of times. Because again no YouTube, what else was he going

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to do all day. What did he find? Well for one thing, he found that each time he did

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the experiment, when the ball was halfway through the trip, in terms of time. It was

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only one quarter of the way through it, in terms of distance. So the ball covered [Galileo rolling experiment]

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three quarters of the distance, in the last half of the roll. He also learned

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that the angle of the incline, directly correlated with the speed of the ball at

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the bottom of the ramp. In fact the acceleration equalled the

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force of gravity, times the sine of the angle of inclination.

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Oh yeah, consider this a warning, in this lesson we'll be getting our trigonometry

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on. Galileo also found that the ball, would continue to travel horizontally, at

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the same speed, as when it left the ramp and that speed would stay constant until

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something stopped it. Like hitting a wall, or falling on the Galileo's foot. Well

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we're gonna be doing an experiment of our own, in just a minute, that's kind of [atom talking]

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similar. But let's make sure we're clear on the math, first. In our other lessons

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we've used the acceleration of gravity, as 9.8 meters per second, squared, as our

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only form of acceleration. However since we'll be dealing with an

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incline, we can't use gravity, because we're not dealing with freefall anymore.

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So the first thing we have to do, is to calculate the correct acceleration, using [atom talking]

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that equation, we just mentioned. This one rod chair. Well once we have that

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acceleration, we're able to find the final velocity.

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Remember this equation, that one. It tells us that the square, of the final velocity,

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equals the square, of the initial velocity, plus two times the acceleration,

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times the change in displacement. In this case the change in displacement will be

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the length of the ramp. Oh and all of this motion is in the horizontal [ramp with equations]

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direction. Which is why we've got all these X's. Once we have that final

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velocity, it becomes the speed of the ball as it leaves the ramp and then? Well

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then we can predict the future. Not at tomorrow's winning lottery numbers kind

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of prediction. More of a here's where a marble will land, when it rolls off the

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table, kind of prediction. Okay well let's get our lab set up. First we need

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equipment. We need some small dense ball like a marble, or maybe something metal.

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As long as it's not bouncy, we should be just fine. Next up, a table and we mean an

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actual table this time, not a data table. Some kind of smooth surface that we can [ball rolling on table]

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roll the ball off of. Yep it could be a counter top, or the top of the dresser.

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Next up a measuring tape, or a ruler, or a meter stick. Well we want to be using

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metric measurements, but a worse comes to worse, we can always convert. And

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if you're making a conversion, just know, that one inch equals 2.54 centimeters.

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Then we need something to make our ramp. Now if you already have some sort of

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ramp like thing, like maybe an old triangular wooden block you used to play

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with, or a piece of Hot Wheels track, well then feel free to use that. We're [man being snob in empty room]

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not gonna be ramps snobs. Just make sure that angle isn't too steep. Nothing more

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than 30 degrees. Otherwise see if you have some heavy cardstock, or some

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lightweight cardboard, something like that. We can DIY our own ramp out of that

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stuff, and pen, paper, scissors and tape. Oh and also we might want to use a plumb

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bob and no it's not a guy named Bob, who can unclog your bathtub. A plumb bob is a

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weight, that hangs straight down from a string. This weight will let us find exactly

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where the edge of the table is on the ground. We just hang our plumb bob from[atom doing experiment]

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the end of the table, like this. It takes some guesswork out of determining where

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the freefall will start. And last, but never least, we need a calculator. An

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actual calculator, a calculator app, something on a webpage, whatever, okay. Now

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we need to put everything together. Make sure the table is level, set your marble

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down and see if it rolls to one side, or another.

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If it does, put some paper, or something under one of the tables legs. Help set it

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straight. If you need to make your own ramp from the cardstock, or cardboard,

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well and go ahead and do that now. We're gonna leave this feat of engineering to

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you though, all on your own. Just figure out some way to make a stiff ramp, that's

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pretty shallow. This isn't a scary waterslide we're building, we want just a[man on resort water slide]

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fairly gentle roll. So here's the plan we're gonna set up our ramp at one end

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of the table. We'll let the ball roll down it. Then on the other side, when the

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ball falls off the table. We're gonna mark where we think it will land. So how

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do we figure out that landing spot? Well first it might help to draw a little

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diagram of what we're working with. The ramp, the table and the floor for

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starters. Then measure the length and height of the table, go ahead and write

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those measurements down on the diagram and we need to measure the ramp to, the

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length, height and hypotenuse. And yeah write those measurements down, we don't [measurements of experiment]

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want to forget them. While we're doing all these measurements, figure out how

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tall you are. Has nothing to do with the experiment, it's just you know good to

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keep track. All right well with the measurements of the ramp, we can

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calculate the angle of the incline. Remember sohcahtoa, no it's not an

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ancient druid chant. It's a way to remember trig functions. We'll just look

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at the SOH part. That tells us that the sign of an angle, equals the opposite [equations on chalkboard]

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side, over the hypotenuse. Which would be helpful if we knew the angle already and

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knew the length of one of the sides. Yeah, then we could find the length of

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whichever side we didn't know. Well in this case we know the length of both

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sides. We don't know the angle, which means we need to break out the inverse

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function of sign. Well ladies and gentlemen, please welcome back to the

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stage, the arc sign. Ya, the arc sign is kind of the opposite of the sign. So if

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sign x, equals y, arc sign y, equals x. Now make sure your calculator is set for [calculator preforming functions]

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degrees and for the inverse of functions. Then find the arc sign of the length, of

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the opposite side, divided by the length of the hypotenuse. Because we know

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the lengths of each side, we can use any of the inverse functions, arc cosine, arc

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tangent, pick your poison. And slap that number up on the diagram too. Okay almost

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time to look into our crystal ball. But we have to calculate our velocity first.

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Step one, acceleration. Which equals gravity times, the sign of the angle of [formulas on chalkboard]

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incline. The gravity is always, 9.8 m/s^2 because, we're on earth. Let's

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say that we happen to have, a perfect 30-degree angle of incline. When we put

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that number in, we find that our acceleration equals 4.9 m/s squared. And

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then we need that final velocity. First let's figure out the horizontal velocity.

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The square of the final velocity, will equal the square of the initial velocity,

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plus 2 times the acceleration, times the change in displacement. Our initial [atom talking and chalkboard equations]

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velocity will be 0. So that makes things a little easier and let's say the ramp

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is 20 centimeters long. We want our velocity to be in terms of meters per

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second though, so we'll call it 0.2 meters. So we double our acceleration,

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making that 9.8 meters per second squared and we multiply that

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acceleration by, 0.2 meters. Which means that the square of the final velocity

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equals 1.96 meters per second. And when we find that square root, to solve for [formulas on chalkboard]

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the velocity. We get 1.4 meters per second. Well now we have to figure out

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how long it'll take this ball, to fall to the ground, after it rolls off the table.

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Which means it's time for another equation. We're sure you remember which

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one to use. Which is good because we don't. Oh yeah, now it's coming back to us.

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We'll use this one, the final displacement, equals the initial

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displacement, plus the initial velocity, times the elapsed time, plus 1/2

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acceleration, times the square of the time. Our final displacement will be, the [equations on chalkboard]

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height of the table and our initial displacement will be, 0. Our initial

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velocity will be 0, just standing there, because you know, right now we're just

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looking at this vertical velocity. Forget about all that horizontal junk, we were

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looking at before. Well don't actually forget it, we'll need it in a minute. With

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those two values, equaling zero, we're left with this, the height of the table,

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equals 1/2 the acceleration. In this case gravity, times the square of the time and

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that time, is what we need to solve for. Time to rearrange the furniture, in this [atom talking in classroom]

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equation. Well we'll start by multiplying both sides by 2, then we'll divide both

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sides by the acceleration and don't forget to find the square root of each

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side also, so we can get all the way down to plane

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T. So the square root of two times the displacement, divided by the acceleration,

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equals the time. If we say that the table is one meter tall and plug in the

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numbers, we'd find that the time equals 0.45 seconds. A little longer than the

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blink of an eye. So keep those eyes peeled, we don't want to miss anything. [atom talking with red background]

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All right now we're ready to make a prediction. We have our horizontal

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velocity and we know how long the ball will be in flight. Multiply those two

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numbers and we'll have the horizontal distance, aka the range. Go ahead and

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write that value down as the predicted distance. And measure out that distance

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from the edge of the table, putting that plumb bob to use, if necessary. Now tape

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your paper down, so it's centered. You know, where you expect the ball to land.

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Go ahead and draw a line across the paper, of that predicted distance, good.[atom setting up experiment]

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Experiment assembly is officially complete, time to get the ball rolling. Go

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ahead and place your marble at the top of the ramp and let that bad boy get

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going and hustle over to see where it lands. Make an X, at that landing spot.

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Then measure the shortest distance from that spot, to the line we drew earlier.

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That distance, if there is one, is our experimental error. Feel free to run the

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experiment a few more times. Go ahead, don't be shy, more data is always good.

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Besides it took a lot to set all this stuff up, so like let's amortize it, a [atom talking with blue background]

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little bit people. All right, yeah. Okay, all done? Did we get it right? If not

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well and we had some experimental error. Well what what might have gone wrong? Was

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the table not as level as we thought? Did the ball hit a stray cheerio as it

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rolled? Did our sister start talking, creating a sudden gust of wind, that blew

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the marble off course. And what was our actual horizontal velocity? We can

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calculate that vector, by finding our actual change in horizontal displacement

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and the time, to figure out how fast the ball was going when it fell off the

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table. And how about this question, which would be more accurate? Calculating the [atom running experiment]

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expected time the ball takes to fall to the floor, or using a stopwatch to

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measure it. Well thanks to our good buddy Galileo, or the big double G, as we like to call

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him. We know just how strong gravity is. Without that knowledge, we wouldn't have

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been able to do this experiment at all. Now some of Galileo's later work got him

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in trouble. In fact his insistence that the earth isn't the center of the

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universe, got him placed under house arrest by the Catholic Church. But I

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promise, stick with me and the Spanish Inquisition won't come knockin at your

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door.[Galileo wondering halls]

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