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- [Voiceover] Two identical
spheres are released from

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

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

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Sphere A has no initial velocity
and falls straight down.

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Sphere B is given an initial
horizontal velocity of

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magnitude V sub zero, and
travels a horizontal distance D

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before it reaches the ground.

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

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even though sphere B has more distance

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to cover before landing.

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Air resistance is negligible.

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The dots below represent spheres A and B.

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

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not components, exerted on each sphere

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at time T sub F over two.

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So we can see our spheres here,

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when I guess this little
this thing releases,

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sphere A goes straight down.

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

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

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

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but it has some horizontal
velocity that makes it move out

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and hit the ground D to the right.

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

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

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

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

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So this is while both
of them are in flight.

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So while both of them are in flight,

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the only force acting on each of them,

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is just going to be the force of gravity.

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And since the spheres are identical,

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the force of that gravity
is going to be identical.

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They have the same mass, so let me draw.

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So that right over there
is the force of gravity

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

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sphere B.

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And so we could write, force of gravity,

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force of,

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force of gravity.

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

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

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

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So this is equal to,

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is equal to M times the
gravitational field.

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

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the only force acting on them,

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we're assuming air
resistance is negligible,

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

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

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

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On the axes below,
sketch and label a graph

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of the horizontal components
of the velocity of sphere A

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and of sphere B as a function of time.

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Alright, I'll do sphere A first.

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This is pretty straight forward.

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

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Sphere A has no horizontal
velocity the entire time

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we're talking about it, it only,

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

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It's going to be accelerated downwards.

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So sphere A has no horizontal velocity,

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so I will draw a line like this.

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So sphere A has no horizontal
velocity the entire time.

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Now sphere, sphere B,

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sphere B is going to be

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a little bit more interesting,
slightly more interesting.

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

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

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has a magnitude of V sub zero.

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And since air resistance is negligible,

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

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until it hits the ground.

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So, so sphere B, this is,

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

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

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

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Well sphere B is going
to be at that velocity,

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actually let me just make
it a little bit clearer.

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

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

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until the final time, until T F.

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So this is T equals zero to T F.

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The entire time while the ball's in the,

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while that sphere is in the air,

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it's going to have the
horizontal component

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of its velocity is just
going to be constant.

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It's not going to be
slowed down by anything

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because we're assuming air
resistance is negligible.

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And then right when it hits the ground,

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it essentially, if you
think about the force

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that is stopping it is
essentially friction,

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

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a magnitude of velocity,

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of horizontal magnitude velocity of zero.

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

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Now you could label this if you want,

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this is, let me actually let me label it,

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

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that is sphere A right over there.

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In sphere B if you want, you could show,

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it would overwrite sphere A, so your B

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would be zero after that.

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

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

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Finally, in a clear, coherent, paragl

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(laughs)

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clear, coherent,
paragraph-length response,

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explain why the spheres reach
the ground at the same time

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even though they travel
different distances.

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Include references in your
answers to parts A and B.

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Alright, so let me think about it.

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I'll try to write a clear, coherent,

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paragraph-length response.

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So I'll say, the entire time the,

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or let me say from,

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from T equals zero...

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to T equals T sub F,

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the only force

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acting on the spheres

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is the downward force of gravity.

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Is the downward force,

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force of (mumbles) of gravity.

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At T equals zero,

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at T equals zero, they both,

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they both have zero vertical velocity or

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the magnitude of the velocity
in the vertical direction

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is zero for both of em.

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Let me write it that way.

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The, the magnitude

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of both of their velocities,

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both of their velocities,

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velocities,

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in the vertical

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direction

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(writes sentence)

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is zero.

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After T equals zero,

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they are accelerated,

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they are accelerated at the same rate.

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Accelerated

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(writing)

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at the same,

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they're accelerated at the same rate.

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so their vertical component of velocity,

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their vertical

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(writing)

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component, components of velocity,

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velocity are always the same.

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Of velocity are always

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(writing)

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the same.

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And they have the same
vertical distance to cover,

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and they have the same,

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(writing)

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the same vertical distance to cover.

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Vertical distance to cover.

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So they hit the ground at the same time.

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Let me make sure that makes sense.

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

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so their vertical
components of velocity are

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always the same.

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

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Since they have the same, since,

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actually let me,

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since they have the same

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vertical distance to cover,

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vertical distance to cover,

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(writing)

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they will hit the ground at the same time.

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They will hit the ground

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(writes sentence)

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at the same time.

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Same time.

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They do have different
horizontal velocities,

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

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their velocities or the distance
in the vertical direction.

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They have different horizontal,

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(writes sentence)

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horizontal velocities,

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but that

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(writes sentence)

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does not effect

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the time in which they,

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they cover the same vertical distance,

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effect the time in which

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(writes sentence)

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they cover the same

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vertical distance.

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And you could write
something to that effect,

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

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the components of spheres Bs velocities,

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it would actually have a larger velocity

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if you were to add the components.

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If you're not thinking you needed

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the horizontal or the vertical direction,

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and so it does indeed cover
more distance and space

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over the same amount of time.

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

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

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at any given point in time
in the vertical direction.

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It actually has the same velocity

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It's being accelerated in the same way

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