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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.
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
Full Transcript
- 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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