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.

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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 ball's horizontal motion affect its fall? The trivial answer to

that question is, that it doesn't. But a more complete answer is that the ball's

horizontal coasting motion has no effect on its vertical falling motion.

The ball is doing two things at once. It's falling vertically, while it's

coasting horizontally. The space we live in is 3-dimensional,

and so motion occurs in 3 dimensions. Up until now, I focused only on the

vertical dimension, the balls altitude or it's height above

the ground. In more general cases, the ball can move

horizontally as well as vertically. And if I throw it up at an angle, it does

two things at once. It falls vertically.

It goes up and down like a falling object, but at the same time it coasts

horizontally at a steady pace. Geometry tells us that in three

dimensions, we can describe a ball's position, or any other vector quantity

for that matter, in terms of coordinates along three separate directions.

Now, we have some flexibility in choosing those directions, but the best choices

are three directions that are perpendicular to one another.

So, for example, I have this box. Boxes are made in such a shape that if

you put sticks along three sides, you have three mutually perpendicular

directions pointed out. So, here's one direction, a second

direction that is at right angles to the first, and a third direction that is also

at right angles to both the previous directions.

So, this is a system that is three perpendicular and mutually perpendicular

directions. And any of these is a pretty good choice

of coordinates, of coordinate directions, along which to describe any vector

quantity you like. When describing the motion of a falling

ball, there is one particularly simple choice of coordinate directions. In that

choice, one of the coordinate directions points straight up.

The second points horizontally along the path that the ball is taking.

I call that direction the down-field direction because if you're playing

American football or soccer, then you're trying to make progress along with that

down feel direction. The third coordinate points to one side

or the other, and it actually doesn't matter in this situation.

So, here are the three simplest coordinates.

One of them points straight up, one of them points horizontally along the

downward direction, and the third points to the side.

Well, with that description then, we can look at how a ball falls.

If I drop it from rest or toss it straight up, then all of its motion is

along that vertical coordinate. But if I throw it up and at an angle, it

moves both along the vertical coordinate direction and along the downfield

direction. [SOUND] Now, for something remarkable.

A falling ball's motion separates perfectly into two parts,

a vertical falling motion, and a downfield coasting motion.

The ball's vertical motion, that is the component of its motion that lies along

the vertical coordinate direction, is that of a falling ball.

And the ball's down field motion, that is the component of its motion that lies

along the down field coordinate direction, is that of a coasting ball.

And the ball is doing both of these things simultaneously and they have no

effect on one another. To give you an idea of why this works,

before I set out and show you that it works, let's look at how the ball

accelerates. The ball is being pulled downward by

gravity and so it's acceleration is straight down, perfectly along the

vertical coordinate direction. The acceleration of the ball is entirely

along that vertical core direction, and so the component of the ball's

acceleration along that direction is the entire acceleration.

The, therefore, the ball accelerates along the vertical coordinate direction

perfectly. It's a, you know, it's a falling ball

along that direction. Everything relating to falling is taking

place along that coordinate direction. On the other hand, there is no

gravitational force component pointing along the down field direction.

That down field direction is horizontal and gravity is a vertical effect.

So, the ball has no acceleration due to gravity along the down field coordinate

direction. Gravity has no effect on motion along the

down field coordinate direction, and so the ball simply does what it was doing in

accordance with Newton's first law. It's unaffected and therefore it coasts,

it travels. It, it's, the, the component of its

velocity along the downfield coordinate direction is constant. So, the ball's

motion vertically is that of a falling object.

Its motion along the downfield coordinate direction is that of a coasting object.

Well, let me show you this. I need more room so we're going to go up

to the third floor again and throw things out the window.

Now, I want to make the, the ball head horizontally fast so that you can see

that downfield motion. And because a bowling ball has a pretty

big mass, I can't throw it sideways very fast.

that's an inertia issue, right? So, I'm going to stick with a basketball.

So, here we go. I'm going to throw a basketball out the 3rd floor window of

the Physics Building at, at UVA and I'm going to get it going horizontally as

fast as I can so that you'll see that down field coasting motion at the same

time. The ball experiences the vertical falling

motion. So, here we go.

So here goes a basketball. I'm going to throw it as horizontally as

possible. So, it'll start with no vertical

component to its velocity. Ready, get set, go.

[SOUND]

The basketball's fall and its subsequent bounces took a few seconds but it's still

hard to see what happened in real time. So, I'm going to use the fact that this

is video to show you that same arcing descent, but with the images of the

basketball lingering on the screen. So, recall this camera takes 30 frames

per second, so the images that you'll see in a moment are separated in time by

1/30th of a second. So, here again is that, is the, the ball

thrown horizontally out the window and arcing downward in the arc of a falling

ball. That trail of basketball images allows us

to determine the basketball's position, velocity, and even acceleration at each

moment during its fall. Now, I'm going to concentrate first on

the vertical motion of the ball. That is, the component of its motion that

lies along the vertical coordinate direction.

And I'm going to show you the same video, slowed down to 1/10 of, of normal speed

so that you can see what's happening, and I'm going to mark the ball's vertical

component of position and vertical component of velocity every fifth of a

second as it plummets. So, here again, same basketball fall, but

with slowed down and with the, the vertical components of velocity and

position there for you to see. As you can see, the vertical component of

the basketball's motion is that of a falling object.

There's no difference between the vertical component of motion of this

basketball, which I threw sideways to start with, and the bowling ball that I

dropped previously from rest out the same window.

In both cases, the ball is accelerating downward at the acceleration due to

gravity. Its vertical component of velocity is

increasing in the downward direction by about 10 meters per second every second.

And the ball is covering more and more distance with the passing time.

So, the vertical motion of a falling ball is that of falling regardless of what its

horizontal motion is. So now, I'm going to show you the same video again at

tenth, at 1/10th normal speed. But instead of worrying about the

vertical component of motion, I'm going to worry about the downfield component of

motion. That is, the portion of the motion that's

occurring along the down field coordinate direction, which will be towards your

left. So, here's the same video in slow motion,

with the, the horizontal component of position marked off every fifth of a

second. As you can see, the downfield component

of the ball's motion is that of a coasting object.

The ball moves steadily downfield, equal distances in equal times.

So, the ho, the down field component of motion is that of an inertial object.

The ball's experiencing no force along that down field coordinate direction, and

so it's not accelerating along that down field coordinate direction.

It's simply coasting horizontally. So, the ball's doing two things at once.

It's falling vertically, it's coasting horizontally, and that gives you the arc

that you saw as the ball descended out this window starting horizontally then

arcing downward, until finally it hit the ground.

Well, that was a basketball falling out the window.

The story will be identical if I throw a bowling ball out the window.

And, of course, it's fun to throw bowling balls out the window.

So, you wanted to see it, I'm sure. Here we go.

Ready? [LAUGH] Here goes the bowling ball.

[SOUND] Didn't bounce much. Well, that was fun.

I think I'd better go repair the dent. Now, bowling doesn't have much to do with

falling balls, but basketball does. Still, tossing basketballs out the window

isn't how the sport is played. Instead, it's played on a court by people

throwing the ball, either between themselves or toward the basket.

And once the ball is free from anybodies hands, it's a falling ball.

So, basketball is all about falling balls.

To see that, let's go over to the gym and watch some people shooting hoops.

[SOUND] Once the basketball leaves the hands of the shooter, it's experiencing

only one force, its weight straight down. And so it's a falling ball.

And the arc that it travels in is the arc of a falling object.

[SOUND] Every time a player takes his shot and the ball leaves his hands, the

ball begins to fall and it travels in the arc of a falling object.

I could take any one of these shots, show you the trail of basketball images, and

then show you that, that trail is the arc of a falling object.

Here are a couple of those curves. I will show you the arc that's formed by

the basketball images, and then I'll show you both the vertical motion, and the

horizontal motion. So you can see if the vertical motion is

that of falling, then the horizontal motion is that of coasting.

Now, depending on the perspective we have on a particular shot, some of the

distances get compressed as the ball gets farther away from us.

And so, the, the, the, the evenness of the horizontal motion maybe not perfectly

visible. Nonetheless, this, this basic behavior of

falling vertically and coasting horizontally is everywhere in the game of

basketball. Here is a beautiful three point shot

right through the basket. [SOUND] And here is that same shot again

With all the previous basketball images lingering on the screen.

And here is a still photograph of that same shot showing you all the basketball

images. And I've marked it up so that you can see

the ball's vertical component of motion and it's down field component of motion.

The ball's vertical motion is that of a falling object.

The ball rises quickly at first, then more and more slowly. It's momentarily

neither rising nor descending, and then it descends more and more quickly as it

approaches the basket. The ball's horizontal motion, the motion

along the downfield coordinate direction, is that of a coasting object.

The ball is moving steadily downfield at a you, uniform pace.

And, it looks like the, the vertical yellow lines are getting closer and

closer together only because the ball is getting farther and farther from us. And

because of our perspective on this shot then, the lines that are far away from us

appear closer together because the space out there is compressed.

It's, it's just off in the distance. So, this really is the arc of a falling

ball thrown up at an angle and basketball has lots of these arcs,

every single shot is like this. Here's another nice shot through the

basket. And here is that same shot with a trail

of basketball images lingering on the screen.

Lastly, here is a still photograph the same shot marked up to show you that the

basketball is falling vertically and coasting horizontally.

It's time for a question. I have an orange ball and a black ball,

and I'm going to roll them off the table side by side and let them hit the floor.

The black ball weighs twice as much as the orange ball, but the orange ball will

be moving to your right twice as fast when the two of them leave the table.

The question is, which ball will hit the ground first, and which ball will hit the

ground farthest from the table? Here we go.

Ready? Get set. Go. Both balls left the table with the same vertical component of

velocity, namely zero. In effect, they began their fall from

rest. They dropped together, it hit the floor

at the same time. But the orange ball was traveling

sideways faster. It had a greater down field component of

velocity than the black ball, and so it used its time to travel farther from the

table, it hit the ground farther from the table.

The arcs of falling balls are part of nearly every ball sport,

including football, basketball, baseball and soccer.

Effects due to the air modify those arcs somewhat, and we'll deal with those air

issues in later episodes. But even if we continue to ignore air, we

can make some interesting observations about how ball sports work.

Whenever you throw or kick or hit a ball, there's usually a limit to how fast you

can make that ball move. For example, I can only throw a baseball

so fast. Let's suppose that you throw the, a ball

as fast as you can, what path does that ball take? Well, that depends on the

direction in which you throw it. I've set that upper speed for the ball

but I haven't selected yet its velocity. Remember, velocity is a vector quantity

and it has a direction to it. So, when you throw the ball, you're

choosing not only its speed, and we'll choose the maximum.

But you're choosing the direction of that ball's velocity when it leaves your hand.

Well, when you choose the ball's velocity,

and particularly the direction of its velocity,

you're choosing both the vertical component of the ball's velocity and its

down field component of velocity. If you throw the ball straight up, you're

putting all of that speed into the vertical motion and giving the ball its

maximum vertical component of velocity. At the same time, though, you're giving

the ball zero down field component of velocity.

So, straight up, it distributes all of the speed in the vertical direction.

On the other hand, you can throw it straight horizontally and give the ball,

put all the ball's speed into the down field component of velocity,

not into the vertical. And everything in between.

Well, that choice of angle, and we'll measure angle relative to horizontal. So,

this is zero degrees on up to 90 degrees. That choice of angle at which to throw

the ball has a big effect on the ball's path.

It, it determines both how long the ball stays above the ground and how far it

travels down field during its time aloft. Let's start by throwing the ball straight

up. That is, 90 degrees above the horizontal.

Here is a plot of the ball's position, both its vertical component of position

and its downfield component of position at times, at equal times during its

travels. By throwing the ball straight up, you're

putting all of the ball's initial speed into its vertical component of velocity

and none into the downfield component of velocity.

That ball has the maximum vertical upward speed, and so it rises to its, the

greatest height it can go to and takes as long as possible to return to the ground.

But, of course, it makes no progress down field at all because you haven't given it

any down flow component of velocity. It hits you on the head, which isn't

typically very useful in most ball sports.

Let's try something else. Let's lower the angle at which we throw

the ball to about 70 degrees. In this case, we're putting the ball

speed mostly into its vertical component of velocity, but still somewhat into its

down field component of velocity. At this point then, the ball doesn't stay

above the ground as long. It simply doesn't have as much upward

component of velocity, but it uses the time that it's above the ground to make

progress down field. And it lands somewhere down field, this is good, you

didn't get hit in the head again. Alright, let's go to a lower angle. Let's

go down to 45 degrees. 45 degrees is special because at that angle, you're

distributing the ball's initial velocity equally.

So that the vertical component of velocity and the down field component of velocity

are the same. They're different direction but they're

equal in amount. So, the ball has a perfect balance

between the vertical motion which keeps it above the ground, and the down field

motion which causes it to travel in the direction you want it to go.

And with this perfect balance, at 45 degrees and in the absence of air, air

changes this somewhat. The ball stays loft long enough and

travels down field fast enough to, to travel as far as you can make it go from

where you threw it. So, if you throw it from this height, it

will pass through this height again as far as possible down field.

Alright, let's lower the angle still further from 45 down to about 20 degrees.

At this point you're putting most of the ball's initial velocity along the down

field direction so that the down field component of velocity is, is, is quite

large. There isn't all that much vertical

component of velocity left. The ball doesn't stay above the ground

very long, but it uses what little time it has to travel very fast down field.

Finally, if we go all the way down to, to, to horizontal, the ball in principle

doesn't stay above the ground at all. I mean, depends on the fact you throw the

ball from up here and the ball, the ground is down there.

But if you threw the ball from ground level horizontally, you hit the ground

immediately. So, a totally level throw is, is often

not very valuable if you're, if you're low.

But, for baseball for example, it's not bad because you are far enough above the

ground that the ball can can stay above the ground by the time it get's to home

plate. All of these possible paths are

potentially useful in ball sports. For example, if you're going for maximum

distance with a football, you probably want to throw it about 45 degrees above

the horizontal because that gives you the best balance between vertical component

of velocity and down field component of velocity to maximize its flight distance

down field. But sometimes, maximizing that distance

isn't your goal. For example, a punter kicking the ball to

the other side may want to keep the ball out of the hands of the opponents as long

as possible while still achieving some amount of downfield distance. And so, a

punter will often kick the ball above 45 degrees, more like 70 or 80 degrees, so

that the ball has a longer time above the ground, even if that costs some amount of

downfield distance. On the other hand, if there are players all

over the field in football and you want to throw the ball to someone as quickly

as possible, and they're not all that far from you,

distance isn't the goal, it's speed down field.

In which case you want to throw the ball below 45 degrees,

more like 20 degrees or maybe even 10 degrees, to put as much of the speed in

the down field component of velocity as possible to get it there fast.

So, in all the sports choosing those angles matters,

it gives different flights and many of them are useful.