A cliff diver dives off of a cliff 85 feet above the water.
If the diver's initial vertical velocity is negative
5 feet per second-- so he's going downward, so they tell
us right here-- 5 feet per second downward, and the
acceleration of gravity is negative 32 feet per second
squared, or 32 feet per second downward-- so that means after
every second, you're going 32 feet per second faster in the
downward direction-- how long is the diver in the air before
entering the water?
So let's just set a variable t to be what we want to solve
for, which is how long is the diver in the air?
Let me just write time in the air.
And we have the initial velocity of the diver right
here, the initial vertical velocity, and that's what we
care about.
We just care about what's going on in the up and down
direction, the vertical direction.
So our initial velocity is negative 5 feet per second.
Now, what's going to be his velocity right when he hits
the water, right when he enters the water?
What's going to be his final velocity?
He's going to start at negative 5 feet per second, so
it's going to be negative 5 feet per second, that's his
initial velocity.
And every second that goes by, he's going to be going
negative 32 feet per second faster after every second.
He's accelerating downward at that rate.
So if you multiply the acceleration times time,
that's how much his velocity is increased.
So it's going to be his initial velocity minus 32 feet
per second squared.
That's the acceleration times time-- times the time that
he's in the air, times this t.
Obviously, this will be in seconds, so we have 1 second
over seconds squared.
You'll just have a second in the denominator, so you'll
have feet per second and feet per second.
The units all work out.
But hopefully this makes sense.
He's starting off at moving downward at 5 feet per second.
The negative is telling you the direction is downward, so
he's moving downward at 5 feet per second.
So after 1 second, he'll be going 37
feet per second downward.
After 2 seconds, he'll be going-- what, 32 times 2 is
64, plus 9-- 69 feet per second downwards.
So this is his final velocity.
So what's his average velocity?
If his initial velocity is negative 5 feet per second,
his final velocity is this.
It's a linear relation.
At every second that goes by, it's linearly increasing, or
decreasing, if you want to view it as a negative number.
So the average velocity is just going to be the average
of these two values.
We're really just learning a lot of physics here, but this
will set us up with a nice quadratic equation that we can
use the quadratic formula for.
So our average velocity is going to be the average of
those two velocities.
So it's going to be-- and I'll skip the units right now--
it's going to be negative 5 minus 5, minus 32t, all of
that over 2.
I'm just averaging these two things.
That's my initial velocity.
This is my final velocity.
I'm just taking the average of the two.
So this is going to be equal to what?
This is negative 10 minus 32t over 2, which is equal to
negative 5 minus 16t.
I'm just dividing everything by 2.
So this is my average velocity.
Now, we can just apply our simple distance is equal to
rate times time.
Now, what's the distance that he's
traveling in this situation?
Well, he's diving off of a cliff 85 feet above the water,
so the distance that he's going to be
traveling is 85 feet.
He starts 85 feet above the water, but the distance he'll
travel, he's going to go 85 feet down.
So the distance he's going to travel is negative 85 feet.
We use negative for down, we use positive for up.
Everything in this problem is
happening in the down direction.
So negative 85 feet is going to be equal
to the average velocity.
But we just figured out the average velocity.
It's this thing right here.
That's our average velocity.
So it's going to be-- let me do it in another color; let me
do it in this blue-- our average velocity is
negative 5 minus 16t.
That's our average velocity, velocity average.
And then we want to multiply that times time.
This is our velocity, our rate-- I'm mixing up the
colors-- this is our rate, and then we want to multiply it by
times, so rate times time is equal to distance travelled.
Let's solve for t.
And we'll skip the units for now, just because that'll kind
of get in the way of maybe the learning.
So we have negative 85 is equal to-- when we distribute
this t, we get negative 5t minus 16t squared.
And now we can add 85 to both sides of this equation.
Let's add 85 to both sides.
And we are left with 0 is equal to--
let's rearrange this.
So we'll a negative 16t squared minus 5t, plus 85.
So this, essentially, is the equation.
If we solve for t, we'll know how long he's been in the air,
because we've used all of the other constraints.
So this is a straight up quadratic equation, so we
could use the quadratic formula here.
So it's going to be negative b.
Now what's b?
b here is negative 5, so negative b, which is negative
negative 5, plus or minus the square root of b squared.
Negative 5 squared is 25 minus 4, times a, which is negative
16, times c, which is 85.
All of that over 2 times a. a is negative 16, so 2 times
that is negative 32.
So the time is equal to positive 5, right, negative
negative 5 is positive 5, plus or minus the square root of--
we probably want to take out our calculator for this.
Let's see, let's evaluate this.
So we have a negative and a negative, so those two are
going to cancel-- let put my calculator
aside right for a second.
So we have negative out here, and then we have a negative,
so these two are going to cancel out.
So it's going to be 25 plus 4, times 16, times 85.
So let's figure out what that is.
So if we have 25 plus-- I'll put some parentheses here-- 4
times 16, times 85 is equal to-- so we got 25 plus 4,
times 16, times 85-- is equal to 5,465.
And we could take the square root of this.
So let's just take the square root of it, and we have 73.9.
So let's just stick with 73.9.
So this is going to be equal to-- let me delete this right
here-- this is going to be equal to, we already figured
out what the square root is, we just evaluated it-- let me
delete that right there-- this is going to be equal to plus
or minus 73.9, all of that over negative 32.
Now, we can add 73.9, and we can subtract 73.9, but what
happens if we subtract 73.9?
Well, actually, let's think about this carefully.
Let's do both of these.
Let's just go through them.
So the two combinations, so the two possibilities for t,
is we have 5 plus 73.9, which is 78.9, all of that over
negative 32.
That's one possibility for t.
The other one is 5 minus 73.9.
And let's see, what's 73.9 minus 5?
It is negative 68.9.
73 minus 5 is 68, swap the negatives, so it's negative
68.9 over negative 32.
In this case, the negatives cancel out.
So which of these-- can we use both of these times?
Well, this right here is going to be a negative number, and
we don't want a negative time.
It's definitely a positive amount of time
that he's in the air.
[UNINTELLIGIBLE]
negative time, are we going back in time?
Who knows?
So we can't even use that.
So the time in the air is going to be this, it's going
to be 68.9 divided by 32, so let's figure out what that is.
So it's 68.9 divided by 32 is equal to 2.15 seconds.
So his time in the air is going to be 2.15 seconds.
And we're done.