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Let's figure out how long this
object is going to be in the

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air given that its vertical
velocity, or the magnitude of

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the vertical velocity
is s sine of theta.

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So its speed in the vertical
direction is s sine of theta.

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So how long is it going
to be in the air?

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Well if I told you that
something is going upwards at

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10 meters per second and gravity
is decelerating it at

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10 meters per second squared.

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So every second it's going to
slow it down by 10 meters per

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second, how long will it take
for that object to get to 0,

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to stop moving?

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Let me write that down.

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Let's say that some object is
moving upwards at 10 meters

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per second.

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And let's say that the gravity
is slowing it down.

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Slowing it down at 10
meters per second.

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So every second that goes by,
it'll slow this thing down by

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10 meters per second.

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Well, it'll take it exactly 1
second to make it go from 10

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meters per second to 0
meters per second.

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And then it's going to be at
some height in the air, and

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then the thing's going to
start accelerating.

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Gravity is going to start
accelerating it downward.

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And then it'll take another
second for it to go from 0--

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from having no velocity,
to having 10

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meters per second again.

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So in this case, it'll take the
time in the air-- we could

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say time sub air, I guess, is
going to be equal to this 10

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meters per second,
your velocity.

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10 meters per second divided
by the acceleration.

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Divided by this 10 meters
per second.

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10 meters per second times 2.

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This is how long it'll take for
the object to go from 10

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meters per second to 0 at
some point in the air.

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And then it's going to take the
exact same amount of time

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for it to fall back
to the ground.

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So times 2.

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If the object was moving upwards
at 20 meters per

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second and gravity is still
slowing it down at 10 meters

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per second per second, then it's
going to take 2 seconds.

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If this was 20, then
this would be 20.

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And it'll take 2 seconds to slow
it down to 0 and then 2

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more seconds until it hits
the ground again.

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For it to speed back up as
it approaches the ground.

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So no matter what your upward
velocity, the time in the air

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is going to be your speed, your
vertical speed, divided

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by the acceleration
of gravity.

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And this is the amount of time
it's going to take you to go

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from this point to that point.

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To have some vertical velocity
and then slow down to 0.

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And it's going to take the exact
same amount of time for

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you to speed back up by gravity
and get to your

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original speed.

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We're assuming no
air resistance.

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So it's kind of a
pure problem.

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So this is the time up, the time
down is going to be the

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same thing.

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So we can multiply that by 2.

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Now, we already know what the
vertical component for our

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problem is.

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It is s sine of theta.

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So we could just substitute
that back in there.

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And we know how long we're
going to be in the air.

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The time in the air is going to
be our speed-- or I should

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maybe put the 2 out front.

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2 times s sine theta.

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Let me make it clear.

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This 2 right here is
this 2 right there.

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All of that over the
acceleration of gravity.

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So if you told me that I'm
shooting this object off at--

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I don't know-- 100 meters
per second.

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So if this is 100 meters per
second and if theta were-- I

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don't know-- let's say theta
were 30 degrees, then sine of

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theta would be 1/2.

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So it'd be 100 meters per second
times 1/2 divided by

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the acceleration of gravity
times 2 would tell you exactly

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how long you would
be in the air.

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How long it takes to go up all
the way, become stationery,

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and then fall back down
to the ground.

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