2015 AP Physics 1 free response 4

- [Voiceover] Two identical
spheres are released from

a device at time equals
zero, from the same height H,

as shown above, or T
equals zero I should say.

Sphere A has no initial velocity
and falls straight down.

Sphere B is given an initial
horizontal velocity of

magnitude V sub zero, and
travels a horizontal distance D

before it reaches the ground.

The spheres reach the ground
at the same time T sub F,

even though sphere B has more distance

to cover before landing.

Air resistance is negligible.

The dots below represent spheres A and B.

Draw a free-body diagram
showing and labeling the forces,

not components, exerted on each sphere

at time T sub F over two.

So we can see our spheres here,

when I guess this little
this thing releases,

sphere A goes straight down.

Sphere B, it it will
go, well it's vertical,

and the vertical direction,
it'll go down just the same way.

It'll be accelerated in just
the same way as sphere A,

but it has some horizontal
velocity that makes it move out

and hit the ground D to the right.

And when it hits the
ground, that's T sub F.

When they're up here, that's
right when they're released,

it's T equals zero, and then
this is at T equals T sub F.

And they say a free-body
diagram at T sub F over two.

So this is while both
of them are in flight.

So while both of them are in flight,

the only force acting on each of them,

is just going to be the force of gravity.

And since the spheres are identical,

the force of that gravity
is going to be identical.

They have the same mass, so let me draw.

So that right over there
is the force of gravity

on sphere A, and that is
the force of gravity on

sphere B.

And so we could write, force of gravity,

force of,

force of gravity.

And if we want, we could,
we could say the magnitude

is F sub G, if we want. F sub G.

Or we could label it as M
times the gravitational field.

So this is equal to,

is equal to M times the
gravitational field.

And that's it, while they're mid flight,

the only force acting on them,

we're assuming air
resistance is negligible,

is the force of gravity's
going to be the same because

they have the same mass,
they're identical spheres.

Alright, let's tackle
the next part of this.

On the axes below,
sketch and label a graph

of the horizontal components
of the velocity of sphere A

and of sphere B as a function of time.

Alright, I'll do sphere A first.

This is pretty straight forward.

Sphere A, if you will
remember, let's go up here.

Sphere A has no horizontal
velocity the entire time

we're talking about it, it only,

it's only going to be accelerated
in the vertical direction.

It's going to be accelerated downwards.

So sphere A has no horizontal velocity,

so I will draw a line like this.

So sphere A has no horizontal
velocity the entire time.

Now sphere, sphere B,

sphere B is going to be

a little bit more interesting,
slightly more interesting.

It's velocity, they tell us,
that it's initial velocity is

V sub zero, it's initial
horizontal velocity I should say,

has a magnitude of V sub zero.

And since air resistance is negligible,

it's gonna continue going
to the right at V sub zero

until it hits the ground.

So, so sphere B, this is,

and I'm just gonna pick
one of these as V sub zero.

Let's say that this right
over here is V sub zero.

That's the magnitude of
it's horizontal velocity.

Well sphere B is going
to be at that velocity,

actually let me just make
it a little bit clearer.

It's gonna be at that
velocity until, until V F.

So if we say this right
over here, or not V F,

until the final time, until T F.

So this is T equals zero to T F.

The entire time while the ball's in the,

while that sphere is in the air,

it's going to have the
horizontal component

of its velocity is just
going to be constant.

It's not going to be
slowed down by anything

because we're assuming air
resistance is negligible.

And then right when it hits the ground,

it essentially, if you
think about the force

that is stopping it is
essentially friction,

but then it very quickly
goes down to a velocity of a

a magnitude of velocity,

of horizontal magnitude velocity of zero.

Alright, alright now let's
tackle the last part of this.

Now you could label this if you want,

this is, let me actually let me label it,

this is B, sphere B, and this is sphere,

that is sphere A right over there.

In sphere B if you want, you could show,

it would overwrite sphere A, so your B

would be zero after that.

It's not continuing to
move on to the right,

or at least they don't tell
us anything about, about that.

Finally, in a clear, coherent, paragl

(laughs)

clear, coherent,
paragraph-length response,

explain why the spheres reach
the ground at the same time

even though they travel
different distances.

Include references in your
answers to parts A and B.

Alright, so let me think about it.

I'll try to write a clear, coherent,

paragraph-length response.

So I'll say, the entire time the,

or let me say from,

from T equals zero...

to T equals T sub F,

the only force

acting on the spheres

is the downward force of gravity.

Is the downward force,

force of (mumbles) of gravity.

At T equals zero,

at T equals zero, they both,

they both have zero vertical velocity or

the magnitude of the velocity
in the vertical direction

is zero for both of em.

Let me write it that way.

The, the magnitude

of both of their velocities,

both of their velocities,

velocities,

in the vertical

direction

(writes sentence)

is zero.

After T equals zero,

they are accelerated,

they are accelerated at the same rate.

Accelerated

(writing)

at the same,

they're accelerated at the same rate.

so their vertical component of velocity,

their vertical

(writing)

component, components of velocity,

velocity are always the same.

Of velocity are always

(writing)

the same.

And they have the same
vertical distance to cover,

and they have the same,

(writing)

the same vertical distance to cover.

Vertical distance to cover.

So they hit the ground at the same time.

Let me make sure that makes sense.

After T equals zero, they are
accelerated at the same rate,

so their vertical
components of velocity are

always the same.

Let me, actually let me,
let me write this this way.

Since they have the same, since,

actually let me,

since they have the same

vertical distance to cover,

vertical distance to cover,

(writing)

they will hit the ground at the same time.

They will hit the ground

(writes sentence)

at the same time.

Same time.

They do have different
horizontal velocities,

but that does not affect their,
that does effect the time

their velocities or the distance
in the vertical direction.

They have different horizontal,

(writes sentence)

horizontal velocities,

but that

(writes sentence)

does not effect

the time in which they,

they cover the same vertical distance,

effect the time in which

(writes sentence)

they cover the same

vertical distance.

And you could write
something to that effect,

and you could also write
that yes, if you were to add

the components of spheres Bs velocities,

it would actually have a larger velocity

if you were to add the components.

If you're not thinking you needed

the horizontal or the vertical direction,

and so it does indeed cover
more distance and space

over the same amount of time.

But if you think about it just
in the vertical direction,

it's covering the same
distance, in the same time,

at any given point in time
in the vertical direction.

It actually has the same velocity

It's being accelerated in the same way

that starts off at, of
magnitude of velocity of zero.