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Physics: Projectile Prediction: Galileo, Trigonometry, and an Experiment 15 Views


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Description:

It's experiment time. We'll be rolling a marble down a ramp... and we'll see what it tells us about gravity, acceleration, and velocity.

Language:
English Language
Subjects:

Transcript

00:00

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

00:09

[mumbling]

00:18

[mumbling]

00:31

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

00:38

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

00:42

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

00:46

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

00:52

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

00:56

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

01:01

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

01:06

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

01:10

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

01:15

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

01:20

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

01:25

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

01:29

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

01:34

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

01:37

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

01:42

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

01:46

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

01:51

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

01:56

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

02:02

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

02:05

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

02:09

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

02:16

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

02:19

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

02:24

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

02:28

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

02:32

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

02:36

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

02:40

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

02:44

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

02:50

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

02:54

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

02:57

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

03:02

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

03:06

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

03:10

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

03:14

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

03:18

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

03:23

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

03:27

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

03:31

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

03:37

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

03:40

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

03:45

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

03:49

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

03:53

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

03:57

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

04:01

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

04:06

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

04:10

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

04:14

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

04:20

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

04:25

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

04:28

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

04:32

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

04:36

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

04:42

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

04:46

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

04:49

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

04:52

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

04:55

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

04:58

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

05:01

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

05:05

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

05:10

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

05:13

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

05:18

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

05:21

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

05:25

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

05:28

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

05:32

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

05:36

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

05:39

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

05:43

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

05:45

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

05:50

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

05:54

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

06:00

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

06:05

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

06:09

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

06:12

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

06:16

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

06:20

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

06:25

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

06:31

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

06:36

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

06:40

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

06:45

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

06:50

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

06:55

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

07:00

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

07:06

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

07:12

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

07:17

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

07:21

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

07:26

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

07:32

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

07:36

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

07:41

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

07:46

making that 9.8 meters per second squared and we multiply that

07:50

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

07:54

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

08:00

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

08:05

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

08:08

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

08:12

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

08:16

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

08:20

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

08:25

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

08:30

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

08:34

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

08:37

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

08:41

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

08:45

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

08:49

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

08:55

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

08:59

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

09:04

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

09:07

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

09:11

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

09:15

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

09:19

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

09:25

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

09:28

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

09:31

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

09:36

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

09:40

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

09:44

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

09:48

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

09:52

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

09:56

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

10:00

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

10:03

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

10:08

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

10:11

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

10:17

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

10:21

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

10:26

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

10:30

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

10:34

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

10:39

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

10:44

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

10:48

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

10:51

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

10:56

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

11:01

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

11:05

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

11:10

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

11:15

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

11:19

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

11:24

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

11:28

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

11:32

door.[Galileo wondering halls]

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