An introduction to physics in the context of everyday objects.

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來自 University of Virginia 的課程

生活中的物理

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An introduction to physics in the context of everyday objects.

從本節課中

Falling Balls

Professor Bloomfield examines the physics concepts of gravity, weight, constant acceleration, and projectile motion working with falling balls.

- Louis A. BloomfieldProfessor of Physics

How does a falling ball move after it is dropped? This question asks for a careful

analysis of the ball's velocity and position as it falls.

In a nutshell, the ball's velocity increases steadily in the downward

direction. But its position shifts down further with

each passing second. [SOUND] Now, to help you understand these

positions and velocities, let me suppose initially that gravity has vanished, and

let's look and see what happens. To a falling ball in the absence of

gravity. In a live class I can make that

supposition, but only in my head. In video, I can show you that

possibility. Now, this is a class about how the real

world works. It's a science class.

It's not about science fiction. So whenever I do this sort of thing, when I

show you how things would happen if I changed the rules slightly, I'll let you

know. So, off goes gravity.

Lets see what happens to a ball that's released from rest in the absence of

gravity. Gravity has been switched off.

Now, the acceleration of the gravity is 0.

And lets see what happens when we drop a ball from rest.

Ready, get set, go. There's no acceleration due to gravity,

so the ball's velocity starts at 0, which is when I release it, and it stays at 0.

The ball's inertial, and it does nothing. It just remains there in space.

Gravity is still switched off. And I'm going to drop the ball from rest

again. But this time, I'm going to plot the

ball's velocity versus time. Now velocity is a vector quantity, and

plotting a vector quantity is difficult. After all it has an amount and a

direction. So what I'm actually going to plot.

Is the vertical component of the ball's velocity.

That is, the portion of the ball's velocity that lies along the vertical

direction, and therefore that affects the ball's altitude or height above the

ground. So if it's moving downward That

contributes a downward component of velocity.

If it's moving upward, that is an upward component of velocity? Ready? Again, no

gravity. Here we go.

Ready? Get set. There you have it.

The balls velocity starts at 0, and as time passes, the velocity stays at 0.

It's inertial. The acceleration due to gravity is 0, in

the absence of gravity. And so, the ball's velocity is constant.

Well, this is getting tedious. We need some action.

Gravity's still switched off, though. So, if we want some action, we want some

motion, I've gotta do it myself. So I'm going to drop the ball again, but

this time I'm going to give it a push before I let go of it.

I'm going to make sure that it has a downward velocity from the moment I let

go. What it does with that velocity is up to

it, but I'm going to give it that starting velocity in the downward

direction. And as it moves I'm going to plot 2

things. I'm going to, first I'm going to plot the

vertical component of the ball's velocity, as before, but I'm also

going to plot the vertical component of the ball's position.

What's the vertical component of position? Well, it's the altitude of the

ball. Relative to some starting point, some 0,

and I'm going to make the 0 the point at which I let go of the ball, so if the

ball moves downward relative to the point where I let go of it, that's downward, a

position that's down, below where it started.

Those'll be the negative values for my component, vertical component of position

graph. If it moves upward, which it won't this

time, those'll be the positive values of my vertical component of position graph.

So here we go. I'm going to release the ball, not from

rest, but with an initial downward component to its velocity, and we'll

watch the ball fall in this gravity-free environment.

Are you ready? Get set. Go.

[SOUND] It was inertial. The ball coasted downward. After all, it

has 0 net force acting on it, here in the world of no gravity.

And so whatever velocity it started with it retained.So as time passed,the

velocity didn't changed. The position of the ball did changed

however. The ball used its velocity to cover the

distance. And with each passing second, it went

lower and lower, lower and lower,until it finally.

Until finally it drifted out of view. Without gravity the ball becomes inertial

after you let go of it. It's experiencing no external forces, and

so it travels at constant velocity, in accordance with Newton's first law of

motion. Whichever way it was heading when you let

go, it'll keep heading in that direction, and there's nothing special.

Above downward anymore. Finally, its time for some gravity, but

just a little. If I turn on the full earth gravity, the ball drops so fast

that I can't show you what's going on. So I'm going to start with just a little

bit of gravity, 1/100th. Of the full earth gravity.

That means that the full acceleration of the gravity here in my special video

world is going to be 0.098 meters per second.

That's 1/100th ofthe real-world value. What you'll see then is the ball

accelerate downward. It will go faster and faster as time

passes. At the same time, its position will

change. It will cover distance in the downward

direction. I'm going to drop it from rest, make

things simple, and you'll watch it accelerate downward and travel downward.

In response to a very weak version of gravity.

Are you ready? Here we go. Ready, get set, go.

How about that? It started very slowly. In fact, from the moment I let go of it,

it was at rest. And then it went faster and faster and

faster as time passed. and it used that downward velocity to travel more and more

with each passing second. So in the first second, it didn't go very

far because it was traveling very slowly on average.

The 2nd second, it traveled farther. The 3rd second, farther still, and by the

time 5 seconds have passed, it had pretty much drifted out of view.

So, that's life with weak gravity. To show you full gravity, the whole earth's

gravity, to return to the real world. I need more height.

I can't work in this little, little, laboratory.

We gotta go use the entire Physics Building at the University of Virginia.

And that's what we're going to do. The earth's gravity is strong enough to

make things happen fast. And that's why I need more height to work with.

I'm going to drop this bowling ball out

of the 3rd floor window of the Physics building and let the ball fall all the

way past the basement. But even with that amount of height to work with, the fall's

going to be over in a little more than 1.2 seconds. So we'll do it initially at

full speed. And then because this is video, I'll slow

down the video and begin to mark it up, so that you can see how a falling ball

moves. Going down.

Here's a bowling ball. Ready. Set.

Go. I told you that fall would be quick.

It's hard to even see the bowling ball as it plummets.

To make it easier for you to follow the bowling ball during its descent, I'm

going to highlight it with a red dot. So here's the same fall, but with a red

dot marking the position of the bowling ball as it falls.

Ready, set, go. Because I'm trying to explain how a falling ball moves after is

dropped I need to be able to show you how the ball's position and its velocity

change with time. But recall that while it takes only a

single glimpse to absorb the ball's position, it takes 2 glimpses to

determine the ball's velocity. And if you go as far as looking for

acceleration, it takes 3 glimpses. It would be helpful therefore, if we had

more than the 1 glimpse of the ball's position visible simultaneously.

Since this is a video, I can do that. The camera records 30 frames per second,

that's 30 glimpses of the ball's position every second.

What I can do is cause the whole video to remember all the previous glimpses of the

ball up until the current moment. That means that every 30th of a second,

we'll have a glimpse of the ball. So, here's that same falling bowling

ball, marked by a red dot, and all the previous red dots will linger on the

screen so you can see the, the evolution of the ball's position. And from that.

Take a loot at its velocity and acceleration.

Ready, set, go. That trail of red dots tells us a great deal about the bowling

ball's movement after I dropped it. At first, the bowling ball was moving

downward very slowly, and it remained close to my hands.

It had a small downward velocity, so its position was shifting downward slowly.

But that slow descent didn't last. After about a second of falling.

The ball had accumulated a much larger downward velocity.

After all, it's accelerating downward rapidly, at the acceleration due to

gravity. So, there near the bottom of the fall,

the ball had a large downward velocity, and so its position was shifting downward

rapidly. To help you observe the motion I just

described, I need to slow the video down. So here's the same fall again at 1/10th

of normal speed, slow motion, and I'm going to mark out the ball's position

every 5th of a second. So 5 times a second.

I'm going to draw a line, to indicate where the ball is.

[SOUND]

Let's choose as the 0 of position, the point from which I dropped the bowling

ball, that's 0. I can then measure the ball's position

every 1/5th in a second in indicating on the video.

I can also measure the ball's velocity every 5th of a second, and indicated as

well. But to do that, to make the measurement

of velocity, I have to compare two positions at different times.

After all. Velocity is the rate at which position is

changing with time. I need to look at the change in position

to observe velocity. So, here is the same fall, once again in

slow motion at 1/10th of full normal speed, with approximate values for the

ball's position and it's velocity. Indicated every 5th of a second.

[SOUND]. As you can see, the falling bowling

ball's velocity is increasing in the downward direction by about 2 meters per

second every 5th of a second. It's a steady increase in downward

direction, so this corresponds to a steady acceleration downward.

Over the course of an entire second, the bowling ball, ball's velocity increases

by about 10 meters per second in the downward direction.

That's an acceleration of 10 meters per second.

Per second, or equivalently, about 10m/sec^2,

in the downward direction. That's not a coincidence.

This is a falling ball, and falling balls accelerate downward, at about 10m/sec^2.

The acceleration due to gravity. So this is a falling ball, its velocity's

increasing steadily in a downward direction at a rate of 10 meters per

second per second or 9.8 meters per second per second if you like, physics

works. Well, you may find it helpful if I graph

that falling bowling ball's position and velocity, each as a function of time.

So here, here in this next version of the same video, I will give you a plot of the

bowling ball's velocity versus time. And the bowling ball's position versus

time. Here we go ahead, slow motion, a tenth of

normal speed, the bowling ball falling out of the window.

[SOUND]. The ball's steadily increasing downard

velocity is the hallmark of constant downward acceleration.

After all, this is a falling ball, and falling balls are always accelerating

downward at the acceleration due to gravity.

That steady downward increase in the ball's velocity causes the graph of the

ball's position to arc Downward, for some thoughts on the shape of that arc, let's

return to my laboratory. The bowling ball survived it's fall just

fine, but the ground outside the Physics Building has a pretty good dent in it.

So where do these motion curves come from? That is for falling ball drop from

rest, its velocity plotted versus time gives you a straight line and its

position plotted versus time gives you a curve that bends downward.

Where does that come from? To answer those questions, we need to look at how

the falling ball dropped from rest, how it's acceleration depends on time, how

it's velocity depends on time, and lastly, how it's position depends on

time. The first of those.

How the following ball's drop from rest. Acceleration depends on time is simple.

Its a falling ball, its acceleration is constant.

Its the acceleration due to gravity, approximately 10m/sec^2 straight down,

and we represent that by the little letter g.

So. The falling ball's acceleration is just

equal to little g, nothing else. Time is out of the picture.

Second issue, the velocity of that following ball dropped from rest as a

function of time. Well now time does matter because the

following balls. Velocity starts at 0, but because it's

accelerating downward, the velocity gradually accumulates more and more

downward, aspect to it. It goes faster and faster.

After 1 second, the falling ball, dropped from rest, is heading downward at, ten

meters per second, approximately. After 2 seconds, it's a 20m/sec,

approximately, and so on. All right? What's the formulaic

relationship, then, between the velocity and the time? It turns out that the

velocity of the falling ball dropped from rest is simply the acceleration, which is

little g, times time. Nothing else.

So that is the formula for a straight line.

When you plot The falling ball dropped from rest at velocity versus time there

in a straight line. The slope of that line is little g, the

acceleration due to gravity. All right? That brings us to position.

The falling ball dropped from rest; position is more complicated, because to

determine how far the ball has moved from where we started We have to know the

average velocity of the ball over the time we're, we're considering.

From start, from the drop moment, to, to now, and we have to know it's average

velocity over that period, and we also know, have to know, the length that

period because the, the, the ball will move using its, its average velocity to

make progress, to go somewhere. To change its position and the, the new

position of the ball will be its average velocity times the time over which it's

used at average velocity, namely the time between the drop and now, the moment in

question. That passes, that begs a new question.

Okay, so what's the average velocity? Of this ball droped from rest.

In many cases in physics, calculating an average velocity is difficult.This is a

very simple case where you can do it pretty easily, the average of a ball it

drop from rest is simply. The average of its starting velocity

times 0 and its ending velocity, the moment in question.

Just take those 2 values and average them because they'll be times early on when

the ball was traveling more like the starting velocity, they'll be times later

on when the ball is traveling more like the final velocity and it all averages

out. The middle, the middle point, in terms of

velocity, is, the average. And so, we know, what the velocity of a

falling ball is, as, as a function of time.

It's, it's g * time. So, the velocity at the start of the drop

is 0. The velocity at the end of the drop is,

g*time, when, not necessarily the end of the drop, but the moment in question, the

moment we're paying attention to. So the average of 0 in g*time is

g*time/2, halfway between the 2.

So that is the average velocity of a falling ball dropped from rest.

G*time/2, the falling ball uses that average velocity to make progress and it

has time,amount of time to work with so we multiply g*time/2, the average

velocity, * time, we get. The new position of the falling ball

dropped from rest. It's g*times^2/2 and that is the, the

formulatic relationship between the position of this falling ball dropped

from rest and the time it's had to fall. So there's the whole story.

The, the relationship between acceleration and time, is simple, it's a

constant. The relationship between the falling

balls velocity, and time, is, is, is a straight line.

It, it is, that the, the falling balls velocity, is proportional to time; g *

time, time. And finally, the falling ball dropped

from rest position is proportional to time squared.

It's g/2 times time squared. And that kind of a formulaic relationship

between position and time gives you an arc, if you plotted against time.

time squared. If you plot time squared or anything

proportional to time squared against time, it gives you an arc.

And the shape of that arc is parabolic, so this is the mathematicians would

identify that and go, oh, that's a special kind of arc.

It's a parabola and parabolas show up all the time in falling objects, and will see

more of them later on in this episode. During the first second of its fall the

stone is moving down relatively slowly on average and so it doesn't travel very

far from you hands, but during the second second of its fall The stone is moving

downward much faster on average, and so it covers far more distance during that

second second than during the first second.

That's why, after only 1 second of falling, the stone is much closer to your

hand than it is to the water. So we've seen that when you drop a ball

from rest, all of its motion occurs along that vertical coordinate direction.

And that motion is relatively simple. The ball's acceleration, is that of a

falling object. It accelerates downward at the

acceleration due to gravity. The ball's velocity, is also pretty

simple. It starts at 0, because we're dropping it

from rest, and then it increases in the downward direction In proportion, to the

time, over which the ball has been falling.

Lastly that brings us to position. The balls position, we can define as

starting at 0, and then that position, increases in the downward direction, in

proportion to the time it's been falling squared.

That's because As the balls moves faster and faster, it covers more and more

distance with each passing second. So, it, it, it, its velocity increases in

proportion to time, its position increases in proportion to time ^ 2.

Well, this is the simplest case of falling.

Falling from rest. In the next video we'll take a look at

what happens if you have a ball that's not falling from rest. Its actually

falling, [SOUND], from a start in the upward direction.