- [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,
(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.