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>> In this video,
we're going to take
a closer look at
the conservation of energy.
Energy is conserved.
We can't create it
and we can't destroy it.
This law is a huge help
when it comes to finding
answers to energy questions.
We're going to practice
finding some of those answers,
and as we do, keep an eye out
for how this law
is put to use.
We saw earlier that energy
can be divided into two types,
kinetic and potential.
Kinetic energy has to do with
an object's motion
and potential energy has to do
with its position.
Imagine a process
that takes place over time.
Say, the flight of a football
from when it's thrown
to when it's caught.
The law of conservation
of energy says that the total
energy of the football,
kinetic plus potential,
is constant while the ball
is in the air.
To see this,
we'll need to recall
how to calculate energy
due to motion
and energy due to position.
The kinetic energy of an object
equals one half the object's
mass times its velocity squared.
An object's potential energy,
when that stored energy
is created by the force
of gravity, equals its mass,
"m", times its height
above ground, "h,"
times "g", the acceleration
due to gravity.
Now, back to our
example problem.
The football has a mass
of 0.4 kilograms
and is thrown with
an initial velocity
of 13 meters per second
and a height of two meters
above the ground.
Assuming energy is conserved,
without any loss
to air resistance,
how fast is the football going
when it's ten meters
off the ground?
We've been saying that energy
is conserved,
but what does that really mean
when it comes to solving
equations?
It means that the number we get
when we add up an object's
kinetic energy,
plus potential energy,
is always the same.
As an equation,
we write that the initial
kinetic energy, plus the
initial potential energy
is equal to the final
kinetic energy plus
the final potential energy.
This is just like saying that
the initial energy is the same
as the final energy.
Energy is conserved.
Let's apply this
to our question.
We can substitute our equations
for kinetic and gravitational
potential energies into
the law of conservation
of energy, seen here.
We know three things.
First, our initial
kinetic energy is 1/2
times the mass,
0.4 kilograms,
times the initial velocity,
13 meters per second squared,
which equals 34 joules.
Second, our initial
gravitational potential energy
is mass, 0.4 kilograms,
times the initial height,
two meters,
times the acceleration
due to gravity,
9.8 meters per second squared,
or 7.8 joules.
And third,
our final gravitational
potential energy is the mass,
0.4 kilograms,
times the final height,
ten meters,
times the acceleration
due to gravity,
9.8 meters per second squared,
or 39 joules.
Now, let's rearrange
our equation to solve
for v final.
The football's velocity
which is ten meters
off the ground.
First, let's add the numbers
on the left side of
the equation.
Thirty-four joules
plus 7.8 joules.
That gives us 41.8 joules
on the left.
Now, subtract 39 joules
on both sides.
That gives us 2.8 joules
equals 1/2 mass times
the final velocity squared.
Multiply both sides by one
divided by 1/2 times mass.
This leaves us with 5.6 joules,
divided by the mass, equals
the final velocity squared.
Plugging in our mass
of 0.4 kilograms,
and that gives us 14 joules
per kilogram equals
the final velocity squared.
Take the square root
of both sides,
and that gives us a final
velocity of 3.7 meters
per second.
That's how fast conservation
of energy says the football
is moving when it's ten meters
above the ground.
Let's try another question.
But this time,
we'll figure out how to solve
for a potential energy term.
As a finalist in
the National Pogo Stick
High Jump Competition,
you know that if you can reach
a height of three meters
off the ground,
you'll win first prize.
After months of training,
you are able to achieve an
upward velocity of eight meters
per second as you bounce
off the ground.
Will this upward velocity
be enough to reach your target
height and win the championship?
To find out,
let's first recognize
that this is an energy
conservation question.
That means,
our starting point is to write
down that our initial energy
equals our final energy.
Since total energy equals
kinetic energy plus potential,
we then rewrite to say that
our initial kinetic energy
plus gravitational
potential energy
equals our final kinetic
plus gravitational
potential energy.
You may notice that we can even
expand the equation again,
to the basic terms that make up
kinetic and potential energy.
Mass and speed
and height and gravity.
Now, take a good look
at that equation.
Notice that the mass,
which in this case,
includes you and the pogo stick,
is in every term.
Which means we can
factor it out.
That means we can divide both
sides of the equation by
the mass and see it disappear.
This is pretty amazing.
It means that, in this case,
the mass of the system
doesn't change the outcome.
Using this streamlined statement
of energy conservation,
what's our next step?
Think about bouncing on
a pogo stick.
When you're headed down,
right as you are touching
the ground, just about
to leave the ground,
your height above the ground
is zero.
At that point,
all of your energy is kinetic.
And when you're at a maximum
height, at that point,
your velocity is zero,
and all of your energy
is potential.
So, we know that h initial,
our starting height, is zero.
And our velocity at our
highest point is also zero.
This eliminates initial
potential energy
and final kinetic energy
from our energy
conservation equation.
Which now looks like this.
One half our initial velocity
squared equals our final height
times the acceleration
due to gravity.
In our situation,
we want to know the final
height, if it is above
three meters.
So, let's rearrange our terms
to get an equation
that says h final
equals something.
To do that,
we just need to divide both
sides of the equation by g.
And there we have it.
Our final height equals
our initial velocity squared,
divided by two times g,
the acceleration due to gravity.
Plugging in eight meters
per second for our initial
velocity and 9.8 meters
per second squared for g,
we find our final height
is 3.3 meters.
With this height, you will
be able to win the competition.
Let's work one more example.
This time, including
spring energy into the scenario.
A large, powerful spring
is set up vertically with
one end fixed to the ground.
The spring's constant
is 2,000 newtons per meter.
And at an equilibrium,
it is three meters long.
You set an eight kilogram
bowling ball on the spring
and compress it 1.5 meters.
Then the spring is released,
and the bowling ball launches
straight up in the air.
What is the maximum height
the ball reaches?
Go ahead, press pause,
give it a try.
And when you think you have
your answer, come back
and check your work.
Did you get 30 meters?
If so, great work.
If not, let's solve it together.
The law of conservation of
energy applies here.
All the spring energy
is turned into kinetic energy
of the bowling ball,
which turns into gravitational
potential energy as the ball
slows to a stop as the ball
reaches its maximum height
in the air.
To figure out how high
the ball goes,
we'll focus on two points
in time.
When the spring is compressed
to half its length, two,
and when the ball is at
its maximum height
in their air, three.
The law of conservation
of energy says the total energy
when the spring is compressed
is equal to the total energy
when the ball is at
its highest point.
As an equation,
we can write it as
the initial potential energy,
plus the initial kinetic energy
equals the final potential
energy, plus the final
kinetic energy.
In this example,
we have two kinds of
potential energy,
gravitational and spring
potential energy.
The kinetic energy plus
gravitational potential energy
plus spring potential energy
initially equals
the final kinetic energy
plus gravitational
potential energy,
plus spring potential energy.
So, we can rewrite
the energy equation
to include those
separate terms.
One half the bowling ball's
mass, m,
times the initial velocity,
v sub i squared,
plus the mass, m,
times the ball's height
above ground, h sub i,
times the acceleration
due to gravity, g,
plus 1/2 the spring constant, k,
times the displacement of
the spring's initial compression
from the equilibrium position,
x sub i squared,
is equal to 1/2 m
times the final velocity,
v sub f squared,
plus m times g
times the ball's final height,
h sub f,
plus 1/2 k
times the spring's
final displacement
from equilibrium position,
x sub f squared.
Notice that the ball's initial
velocity, v sub i,
and final velocity, v sub f,
are both zero.
That means we can drop those
terms out of the equation.
We also see that,
when the ball is up in the air,
x sub f is zero.
That means we can remove
this term from the equation.
We want to solve for h sub f,
the maximum height
of the bowling ball.
So, let's rearrange
this equation to get h sub f
on one side by itself.
Divide both sides of
the equation by m times g.
We now have an equation that
will let us solve for h sub f,
the maximum height
of the bowling ball.
To solve for h sub f,
we add our initial height,
which is 1.5 meters
to 1/2 times the constant k,
which is 2000 newtons per meter
times the initial displacement
of the spring from
the equilibrium position,
1.5 meters squared,
divided by our mass
of eight kilograms,
times the acceleration
due to gravity,
9.8 meters per second squared.
Plugging all these numbers in,
we find the maximum height
above ground of the bowling ball
is 30 meters.
That's a closer look
at conservation of energy.
For more "Physics in Motion",
go to our homepage,
where our entire series
is available to you.
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