Projectile motion | AP Physics | Khan Academy

- [Instructor] Quick question.

We have a baseball pitching machine.

We use that to shoot a
baseball horizontally,

and at the same time,

we're gonna drop another baseball, okay?

Which one hits the ground first?

Well, my intuition says

the one that is dropped just
goes straight down, right?

So it should hit the ground first.

But let's see what's gonna happen, ready?

Here goes.

What do you find?

Both of them hit the
ground at the same time.

Okay, would the result be different

if we shot it much faster?

Well, let's see.

Here goes, boom.

And again, the same result,

the baseballs hit the
ground at the same time.

But why?

What does it mean and what
can we learn from this?

Let's find out.

A cool way to analyze things

is to first of all look at
a slow motion version of it,

and then we can take snapshots

of the positions of the
baseball periodically,

say every 10th of a second,

and let's see if we can
learn something from that.

So let's look at it one more time.

We'll take snapshots every
10th of a second, let's say.

And let's see what that looks like, ready?

Here goes.

Boom.

Boom.

Boom.

Alright, so now that
we have some snapshots,

what can we learn?

Well, the first cool thing that we can see

is if you look at their
vertical positions,

they're aligned.

That means when the drop
ball comes over here,

the shot ball is over here.

They're vertically at the same positions.

When the ball drop ball is over here,

look, the short ball is over here.

When the drop ball is over here,

the short ball is over here,

and the drop ball reaches,

you know, the ball hits the ground,

the ball that was shot
also hits the ground.

This means that vertically,

they're both traveling together.

In other words, the vertical velocities

of both the balls will always be the same

at any given moment in time.

This is pretty awesome
because I already know

what the vertical velocity looks like

for a ball that is dropped.

We know that the initial velocity is zero,

and then as it falls down,

its velocity increases, it accelerates,

and this acceleration is a constant.

Close to earth the magnitude
of the acceleration

is about 9.8 meters per second square,

which we just call as G.

Okay, A couple of things over here.

First of all, why is there
a Y subscript over here?

Well, that's to remind ourselves

that these are vertical velocities,

and this is the vertical acceleration.

We usually choose,

well, you know, we usually
label the vertical axis as Y.

That's why we use the
subscript Y over here.

But more importantly,

I know the magnitude

of the vertical acceleration

is 9.8 meters per second square.

But what about its sign?

I know this acceleration is downwards,

but is it positive or negative?

Does it depend upon the
direction in which the ball goes?

That used to be confusing for me.

So how do we decide the sign?

Well, the sign purely depends
upon your coordinate system.

It depends on which direction
we choose to be positive.

Now, in this example,

we have chosen the upwards as positive.

Since gravitational acceleration
is always downwards,

that means in this coordinate system,

the sign becomes negative.

So over here, in this particular case,

the acceleration would be,

in the vertical would be minus
9.8 meters per second square.

It has nothing to do with the direction

in which the ball is moving.

Only depends on which direction
we choose positive, okay?

Now, we could have chosen
downwards to be positive.

That is completely our choice.

In that case, the sign
of the acceleration,

the vertical acceleration
would be positive.

So it purely depends upon

which direction we choose to be positive.

And since their vertical
motion is exactly the same,

I can say the same thing about
the shot baseball as well.

Initially its vertical
velocity its also zero,

and then it increases,

it accelerates down at 9.8
meters per second square.

But along with the vertical direction,

it's also moving forward.

So what kind of motion is it
doing in the forward direction?

Well, let's look at it.

If you look at its spacing,

we see the spacing to
be pretty much the same,

which means the distance its covers

in every 10th of a second
is exactly the same.

In other words, it's traveling
with a constant velocity.

So the acceleration in the
forward direction is zero.

Now, for the forward direction,
we can label it as X axis.

And so the acceleration in
the X direction becomes zero.

So whatever velocity with
which it was shot over here,

that velocity stays the same.

Now of course, we're ignoring

the effects of error over here.

It does have an effect

on both the horizontal
and the vertical motion.

But if we ignore it,

then we can pretty much say
that the horizontal velocity

stays exactly the same.

It's a constant.

And now I can combine
both these velocities

and add them up to get the total

velocity of ball at every instant.

So let's do that.

And this is what it will look like.

So now I can even visualize
the path of the ball.

This is the path of the shot ball.

As it moves forward,

its forward velocity is a constant,

but its downward velocity increases,

accelerates at 9.8
meters per second square.

Okay, now here's a quick question for us.

What changes if we shoot the ball slower?

How would the path be different?

Can you pause the video
and think about it now?

Alright, let's see.

First of all,

because it's still shot horizontally,

its initial vertical
velocity is still zero.

That means vertically,

it still has the same motion.

Its velocity would still be the same

after every 10th of a second,

and it would hit the
ground in the same time

as the drop ball does.

But horizontally, since it's shot slower,

what happens to its velocity vector?

Horizontal velocity would be smaller,

and therefore, in every 10th of a second,

it would now cover smaller distance.

So if you put that together,

this is what the new snapshots
should look like, right?

And let's confirm that.

Let's now look at the animation again,

and we can see it's exactly
the same as we predicted.

Isn't that incredible?

Which means the new path looks like this.

So look, the velocity

with which we are shooting it horizontally

does not affect the
vertical motion at all.

It's independent of that.

If you shoot it slower,
well, it'll fall closer.

If you shoot it faster,
it'll fall farther away.

But again, vertically look,

its motion will be exactly similar

to the one that is dropped.

Okay, so far we analyze
what happens to a ball

when it's shot horizontally.

But what if it's shot at an angle?

Like for example, when
the batter hits the ball,

it's initial velocity is at an angle.

Now, how do we analyze the motion?

Well, now what we can do

is we can look at this
initial velocity vector

and resolve it into horizontal

and vertical velocities.

So we can say it now has

this much vertical initial velocity.

And this is its horizontal
initial velocity.

And again, we can now analyze
the motion separately.

We can say, hey, vertically,

it's kind of like throwing a ball up.

Again, we know in the vertical

the acceleration is 9.8
meters per second square.

And just like before it's sign

purely depends upon which direction

we choose to be positive.

But when we throw the ball up,

what happens to its velocity?

Well, as it goes up,

because the acceleration is downwards

and velocity is upwards, it slows down.

So as it goes up, it's slows down,

slows down, slows down, it stops,

and then it speeds up,
speeds up, speeds up,

and it falls down.

So let's visualize this.

Let's only first consider
the vertical motion.

Then we'll take snapshots
every 10th of a second.

You ready?

Here it goes.

Look, it's slowing
down, it's slowing down,

slowing down, stops,

and then starts speeding up,

starts accelerating.

Makes sense, right?

During the upward motion
the velocity is upwards,

but acceleration is downwards

that's why it slows down.

During the downward motion

both the velocity and acceleration

are in the same direction

that's why it speeds up.

But we now know that along
with the vertical motion

it'll also be traveling
forward at a constant velocity.

Which means all these snapshots

will be spaced apart horizontally,

equally spaced apart.

So just like before,

if we labeled the horizontal axis as X,

the acceleration in the
X direction is zero.

So let's visualize that
now and there we have it.

Isn't that amazing?

So we have the vertical velocities.

And what about the horizontal velocity?

It's gonna be a constant.

So whatever was the
horizontal velocity over here,

the same would be everywhere.

And again, we can find the total
velocity by combining them.

And as a result, we can get
the path of the baseball.

So look at what we have done.

We have analyzed the motion of an object

that's thrown at a random
velocity at a random angle.

Such objects, which are only
under the influence of gravity,

we call them projectile.

And this motion is called free fall.

Even though during the
upward motion it's going up,

we still call it as a free fall motion.

Now, in contrast, if you
consider the motion of a rocket

for example, that's not a free fall,

it's not a projectile

because it's not only under
the influence of gravity,

there are other forces like thrust,

for example, are acting on it.

Now, of course, a stone or
a baseball thrown in air

will also have air resistances.

So they're not truly projectiles.

But if we ignore air resistance,

then yeah, these are all projectiles

and they'll be under free fall.

But what's cool over here is
if we ignore air resistance

and we treat this
baseball as a projectile,

then it doesn't matter at
what angle you shoot it

in the three dimension,

its motion is always stuck to a plane.

It's moving forward,

and at the same time, it's
also moving vertically.

And therefore, projectile motion

is always a two dimensional motion.

And both of these dimensions
are independent of each other.

We can analyze the vertical

and the horizontal
velocities independently.

And finally, because the
accelerations in both the vertical

and the horizontal are a constant,

they're not changing with time.

And we can separately analyze them.

This means we can use the
equations of kinematics

to analyze their motion.

In the vertical the acceleration is G.

And as we saw earlier,

the sign purely depends upon

which direction we choose to be positive.

But in the horizontal,
the acceleration is zero.

So we can just plug that in.

And if you do that,

notice the first and the third
equation becomes trivial.

They're basically saying that
velocity at any point in time

always equals the initial
velocity, which makes sense.

If the acceleration is zero,

that's what you would expect.

But that doesn't help us,

so we can get rid of them.

And the only equation
that survives is this one.

So if I ever want to figure out

what the velocity or the
position of an object is

after some time T,

I can use this equation

to find its horizontal position

and its horizontal velocity,

which is gonna be the same
as the initial velocity.

And I can use these equations

to figure out its vertical position

and its vertical velocities.