An introduction to physics in the context of everyday objects.

Loading...

來自 University of Virginia 的課程

生活中的物理

803 個評分

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 can a ball move upward and still be falling? The answer to that question is

that a falling ball is accelerating downward at the acceleration due to

gravity, regardless of its current velocity.

Recall that a falling ball is one that's experiencing only a single force,

its downward weight. And it is accelerating downward in

response. The ball's acceleration then dictates how

the ball's velocity is changing with time.

After all, that's what acceleration is, the rate at which velocity is changing

with time. What the falling process doesn't

determine is what the ball's velocity was when it started to fall.

That's a completely separate issue. When I drop a ball from rest, I'm

choosing the initial velocity for the ball to be zero.

I let go of it, and for that one moment, I have control over the ball's velocity.

I choose it to be zero. After that, the velocity is out of may

hands, [SOUND] literally.

From the moment I let go of it, acceleration,

the acceleration due to gravity takes over. And the ball's velocity, which

started at zero by my choice, begins to change with time and become more and more

downward. And the ball develops 10, roughly 10

meters per second of downward velocity for every second it has had to fall.

But, I didn't have to choose an initial velocity of zero.

I could choose, for example, to let, let go of the ball with an intial velocity

downward. I can throw it downward.

Like this. In that case, at the moment I let go of

the ball and it became a falling object, it already had a large downward velocity.

In effect, I, I gave it a head start on the falling process. But once it left my

hand, it accelerated according to the rules of a falling object and it picked

up speed in the downward direction at the usual rate of 10 meters per second per

second. Well, for the purist, 9.8 m per second

per second. Well, that leads to another possibility.

What if instead of dropping the ball from rest or throwing the ball downward with a

downward initial velocity, what if I start the ball with an upward

initial velocity? I throw it upward. [SOUND] In that case, it's rising upward.

But it's still accelerating downward, it's still a falling ball.

And so, it continues upward for a while. But as it goes higher and higher, it's

being pulled downward and is accelerated downward, and so it's slowing down.

It eventually comes to a stop, and then it begins to descend faster, and faster,

and faster. Well, to look a little more carefully at

all the properties of a ball that is falling but heading upward, we need some

more room. So, let's head outside and start throwing

a ball around in the open spaces. Before we examine that ball toss

carefully, here's a question about the balls

travels. At the moment that the ball reaches the

peak height on its way up, it comes to a peak and then it comes down.

Right at that peak, what are its velocity and acceleration? At the peak of its

travel, the ball has momentarily come to a stop.

Just before that instant, it was heading upward.

Just after that instant, it'll be heading downward.

So, at that point, it is in transition from heading upward to heading downward,

and it's neither, it's doing neither. It's neither rising or descending.

So, it's motionless. Velocity zero.

But that doesn't mean it's not accelerating.

Like any falling ball, it's still accelerating downward at the full

acceleration due to gravity. So, let's return to the ball that I

tossed straight up. I'll show it to you again to remind you

what it looks like. The basketball took just 2.2 seconds to

complete its journey. It spend half that time heading upward

and half that time heading downward. To help you see how that movement

occurred, I can now begin to play with the video of that basketball toss.

The first thing that I'm going to do is I'm going to show you where the

basketball was at each moment in time prior to the present.

Now, the camera records 30 images per second so I'm going to have the images of

the basketball linger on the screen, each one separated from the next by 1/30th of

a second. Here then, is that same basketball toss

but with all of the previous basketball images still visible on the screen.

That trail of basketball images provides us with enough information to determine

the basketball's velocity and position at each frame of the video.

Now, I'm going to graph those values, velocity and position, as a function of

time. So here, it will be the, the graph of the

basketball's velocity versus time, and its position versus time.

Now, in showing you how they evolve over the course of time,

everything happens pretty fast in real life.

So, I'm going to slow the video down to 1/10th of its normal speed.

Here then, is the same tossed basketball with its velocity and position graphed as

time goes on. The graph of the basketball's velocity

versus time is a straight line. We've seen that straight line before.

When I drop the ball from rest, its velocity versus time is another straight

line. In that case, the velocity started at

zero when I released the ball and then increased steadily in the downward

direction. In this case, the ball started to fall

while it was heading upward with a large upward velocity.

So, it began its fall with a large upward velocity but it finished its fall with a

large downward velocity. And the transition from large upward

velocity to large downward velocity was smooth, steady, and seamless.

The velocity change formed a straight line when plotted against time as its

downward acceleration, the acceleration of a falling object, gradually reduced,

gradually caused its velocity to shift more and more in the downward direction.

The effect of that downward acceleration early on was to slow the rise of the

ball. The ball started out after I threw it

upward with a large upward velocity, so its velocity was upward.

But it's acceleration was downward and that causes what we often refer to as

deceleration. Acceleration opposite the velocity.

So that if you are moving forward but accelerating,

you slow down. The same thing happened for the ball, the

basketball. It was heading upward, velocity was

upward, but its acceleration was downward. and it was a constant downward

acceleration so the velocity in the upper direction gradually decreased, steadily,

steadily, steadily until at one instant in time, one moment, the velocity was

reduced all the way to zero. After that moment, which occurred halfway

through the travels, the ball's downward acceleration cause its velocity to become

more and more downward, it steadily increased in the downward

direction. So, the 1st half of this trip, the upward part of the trip, is the ball

having an upward velocity that decreases steadily towards zero.

The second half of the trip is the ball dropping from great height with the

velocity that's downward and steadily increasing from zero.

Right between these two, these two portions of the motion is that instant in

time where the velocity is neither upward nor downward,

it's zero. The ball is momentarily motionless.

[SOUND] But, it's still accelerating there.

So, it continues its, its transition from velocity upward to velocity downward,

and shortly returns to my hands. The graph of the ball's position versus

time is a smooth arc that curves downward.

It starts rising swiftly at first, then arcs over to, to a, to a flat top.

And then, arcs downward more and more steeply.

That reflects the motion of the ball that starts rising quickly at first, and then

more and more slowly as the downward acceleration of the falling ball saps its

upward velocity. It momentarily stops rising all together

and then begins to descend faster and faster and faster, covering more and more

distance each, each second. And so, we get this rapid rise at first

and then slower and slower then not at all, then slow descent, slow descent,

faster, faster, faster until down it comes at high speed.

[SOUND] The moment at which the basketball reaches peak height is an

interesting moment. Up until that point, the ball had an

upward velocity that was gradually decreasing, but it was still rising

upward to greater and greater height. After that moment of peak height, the

ball's velocity was downward and gradually increasing.

So, the ball was descending away from peak height.

That's why the, the moment of peak height is the moment at which the ball's

velocity stops being upward and hasn't yet been downward.

It's just hit zero exactly, that's the peak moment.

And, the ball's motion is remarkably symmetrical around that moment of peak

height. If you look at where the ball was one

second before the moment of peak height and one second after the moment of peak

height, it's at the same altitude. The ball is the same distance below the

peak on both sides of time, one second before, one second after.

Not only that, but the speed of the ball is the same.

Before peak height, the speed was directly upward.

It was an upward velocity. After peak height, that speed was

directed downward, it was a downward velocity.

But the speeds were the same. So, there's wonderful symmetry around the

peak. Because of that symmetry, the rise and

fall of the ball looks the same if I play it forward as if I play it backward.

You pretty much can't tell the difference between the video of me throwing the ball

up and down., played forward, and played backward.

I'll show you. The bottom line here is that a falling

ball is falling no matter what its velocity is at the start of the fall.

It can start from rest, it can start with me throwing it downward a little bit, or

it can start with me throwing it upward. It doesn't matter.

Once the ball leaves my hands, the only force acting on it is its weight, and it

accelerates downward at the acceleration due to gravity.

All the rest, the initial velocity issues, how I started are details that we

can deal with. So, the formulas that I gave previously

for a ball dropped from rest still, still apply, they're still relevant.

But, we need to spruce them up a little to take into account the possibility that

the ball started to fall with an initial velocity that wasn't zero.

To take a look at that, those slightly revised formulas, let's go back to my

laboratory. To describe the motion of a falling ball

that begins its fall with an initial velocity different from zero, we have to

make some modifications to the equations I gave you to describe the motion of a

falling ball dropped from rest. We have to incorporate the possibility of

an initial velocity that's not zero. So, let's look at the ball's

acceleration, then its velocity, then its position.

Well, the ball's acceleration doesn't change because of any initial velocity.

The ball's acceleration has nothing to do with its initial velocity.

It's simply the acceleration due to gravity.

It's a falling ball. And whether the ball is going up, or

down, or sideways, or any which way, as long as the only force acting on it is

its weight, it's accelerating downward at the acceleration due to gravity.

End of story. So, that was easy.

Acceleration's unchanged. Okay, that brings us to velocity.

Now, velocity does change. In the old relationship, we had for a

ball dropped from rest that the velocity at, at any given time is, is simply the

acceleration due to gravity times the time over which the ball has been

falling. That product of the acceleration due to

gravity times time at time zero, at the moment the, the fall starts, that becomes

zero. So, the velocity at time zero is zero.

Well, if we allow for the possibility of some initial velocity that isn't zero,

well, we have to add it in. So, at time zero, when the ball hans't

had any time to fall and threfore hasn't undergone any acceleration due to

falling, the velocity of the ball is the velocity it started at.

So, we simply add it in. So, the, the, the formula that gives us

the velocity of a falling ball that started with an initial velocity is the

ball's initial velocity plus the acceleration due to gravity times time.

That's all there is to it. Finally, that then brings us to the

ball's position. This one's a more difficult calculation.

First off, we should really allow for the ball to start its fall at a position

other than zero. And, if we do that, we add in the

starting position. So, the position at any given later time

is equal to the starting position, that's, that's the beginning, plus

additional substance. And, what is the additional stuff we have

to add in there? Well, it's the, once again, it's the average velocity of the

ball over the course of its, its fall. From the moment we, we let it start

falling until now, the moment in question.

So, it's that average velocity times the time between the start of the fall and

now. So, it's the same product as before, but

now the average velocity is more complicated because the velocity didn't

start at zero. It started at something else.

And If we work through that carefully, and this is, this is simply a algebra

work. It's not very, not very difficult but

it's enough, I'll leave it out of this, out of this video.

The final result is that the position of the ball during its fall at, at any given

time where, where time zero is the moment the fall began,

that position is equal to the initial position of the ball, where you let go of

it, plus the ball's initial velocity times time.

Plus, 1/2 of the acceleration due to gravity times time squared.

So, the three terms that are present in that relationship that give you the

ultimate, the position of the ball overall.

The first term adjusts for the fact that you might want to start the fall, the

ball's fall, at a position other than zero.

The second term accounts for the fact that the ball might start with an initial

velocity other than zero. And the third term recognizes that a

falling ball is accelerating downward, the acceleration due to gravity.

And it experiences this evolution of position that goes in proportion to the

square of the time, that is time to the second power.

So, whether you, you care about these quantitative relationships that, that,

that actually tell you specifically where the ball is in space, how fast its

moving, and, and what its acceleration is, or when you simply want to watch the

ball fall, its motion is still very simple.

The ball starts with its initial velocity, and once it's falling, it, it

accelerates downward. So, its velocity is changing in the

downward direction. If it's heading upward at a given moment,

well, it's, it's becoming less and less upward as time goes on.

If it's heading downward, it's becoming more and more downward as time goes on.

So, so it makes use of this velocity then to cover distance, and so you see these

rises and falls, or simply falls, they're all dictated by this constant downward

acceleration, the downward acceleration of a falling ball.

It's time to ask the question I asked you to think about during the introduction to

this episode. Suppose I throw a ball straight up.

During the time that the ball is above my hand and heading upward, is there a force

pushing the ball upward? In answering this question, neglect any effects due to

the air. The ball moves upward not because of any

force pushing it upward, but because of its own inertia.

The ball left my hand with an upward velocity.

And even though it begins to fall the moment I stop supporting it,

it takes time for that ball's downward acceleration to change the ball's upward

velocity into a downward velocity. By now, you should have a pretty good

idea how balls move if you drop them from rest or toss them straight up.

But, most ball sports involve motions of falling balls that are not vertical.

If the only thing you could do in baseball, or basketball, or football, or

soccer, or volleyball was make the ball go straight up and down,

those would be pretty dull sports. They still involve falling balls, but

those balls have room to move. And, in the next part of this episode,

we'll take a look at motions that aren't strictly vertical.