### Video Transcript

In this video, we’re talking about
energy conversion and conservation. These two terms go together because
when energy is converted from one type to another, then for the objects under
consideration, energy is also conserved.

Let’s learn now what these terms
energy conversion and energy conservation mean. Let’s say we have a situation where
we have a ball at the top of a ramp a height ℎ above ground level. Our ball is at rest; it isn’t
moving. But even so, it has some amount of
energy. It has gravitational potential
energy we can call it GPE by virtue of the fact that it’s this height above ground
level. Now, if we were to give the ball a
push so that it starts to roll down this ramp, then by the time the ball got to the
bottom of the ramp. It would have lost that
gravitational potential energy. That’s because it’s now at ground
level.

So then, where did the ball’s
gravitational potential energy when it was at the top of the ramp go? It’s a fair question because energy
conservation tells us that energy in general is not lost. It’s just transferred from one type
to another. So, if the ball’s initial
gravitational potential energy is transferred to something else, what was it? Looking at these four snapshots of
the ball as it rolls down the ramp, we can see that the ball’s velocity is indicated
by these orange arrows.

Starting out, the ball had no
velocity. But we can see that, at the bottom
of the ramp, the ball’s velocity has reached its maximum value. Since this ball has mass and is in
motion. That means it has energy due to
motion, kinetic energy. And this answers the question of
what the ball’s gravitational potential energy was converted into. As the ball lost elevation and
gained speed, this energy was transferred to kinetic energy.

This is an example of energy
conversion, energy going from one type to another. And along with this, if we assume
that the ball didn’t lose any energy to friction as it rolled down the ramp. Then we could also make this
statement that the ball’s gravitational potential energy, starting out when it was
at the top of the ramp, is equal to its kinetic energy when it’s at the ramp’s
bottom. This is an example of energy
conservation. The energy the ball initially had
was converted to another type, but the total amount of energy was conserved.

What we mean by that is if the ball
started out with 100 joules of gravitational potential energy, then by the time it
reached the bottom of the ramp, it would still have 100 joules of energy, the same
amount overall. But it’s just that the energy type
would have changed from gravitational potential to kinetic. We can say then that, in this
example, energy is both converted and it’s also conserved because the total amount
doesn’t change.

Combining these ideas of energy
conversion and energy conservation is often helpful and letting us solve
energy-related exercises. For example, let’s say that here we
knew the speed of the ball when it reached the bottom of the ramp. The speed is 𝑣. And that we wanted to solve for the
height ℎ that the ball descended. We could solve for this height,
using this equality here. To write out the equation, we would
use the fact that gravitational potential energy is equal to an object’s mass times
the acceleration due to gravity of the field it’s in multiplied by its height above
some minimum height value.

So, in the context of our ball
rolling down the ramp, we would see that the ball’s mass, we can call it 𝑚, times
the acceleration due to gravity times the height of the ramp ℎ is equal to the
ball’s kinetic energy when it’s at the bottom of the ramp. And that kinetic energy we can
recall is equal to one-half the ball’s mass times its speed squared. So then, 𝑚 times 𝑔 times ℎ when
the ball is at the top of the ramp is equal to one-half 𝑚𝑣 squared when the ball
is at the bottom.

And notice this, because one factor
of the ball’s mass appears on both sides of this equation, that means we can divide
both sides by that value, the ball’s mass. And it will completely cancel
out. In other words, once we know the
ball’s speed at the bottom of the ramp, we don’t need to know its mass to figure out
how far it fell. When we work through energy
conversion and energy conservation exercises, it’s often the case that an object’s
mass does cancel out. It’s something to keep an eye out
for because it simplifies our work.

So anyway, we want to solve for the
height ℎ that this ball fell from. And we can see that to do that in
general, we could divide both sides of this equation by the gravitational
acceleration 𝑔. Canceling that term out on the
left, and we arrive at this expression: the height the ball fell from is equal to
its final speed 𝑣 squared divided by two times 𝑔. This is an example of how we can
use this principle of energy conversion and energy conservation practically to solve
for particular variables in a scenario.

Now, we’ve considered a couple
types of energy, gravitational, potential, and kinetic. But we know that there are
more. For example, when a spring is
compressed, like this, energy is transferred to the spring. And it’s called elastic potential
energy. Abbreviated EPE, it’s equal
mathematically to one-half a spring’s constant, 𝑘, times its displacement from its
equilibrium length, 𝑥, squared. There are yet more types of energy
we could consider. But if we focus on these three, we
can see that this idea of energy conversion applies between all of them.

For example, elastic potential
energy could be converted to kinetic energy. And then that energy type could be
converted to gravitational potential energy. And that could then be converted to
elastic potential energy. And the transitions could go the
opposite way as well. Each one of these transitions is an
energy conversion. Now, when we talk about energy
being converted from one type to another, in general, that can happen in one of two
ways. To see this. Let’s consider a specific
conversion. Let’s think about elastic potential
energy being converted to gravitational potential.

Let’s say that instead of a ball
rolling down a hill, we have a mass on the end of a spring that’s compressed but is
then extended to its equilibrium length. We’ll say that, at the outset, this
spring was compressed a distance 𝑥 from its natural or its equilibrium length. But then, after it was released and
came to rest, it was extended back to that length. Let’s say that, at the outset, when
the spring was compressed, our mass on the end of the spring had no gravitational
potential energy. That means what we’re saying is
that this altitude here is a height of zero.

So then, at first, the only energy
involved is the energy possessed by the spring. It’s the spring’s elastic potential
energy, one-half its spring constant 𝑘, whatever that is, times 𝑥 squared. So, we’re saying that this is all
the energy possessed by our spring and our mass at the start. But then our spring is released,
the mass moves up and the spring extends back to its natural length. Now, because it’s extended as it
normally would be, our spring no longer has potential energy, but instead our mass
does, gravitational potential energy. And as we’ve seen, that energy is
equal to the mass of our block times 𝑔 times its height ℎ. And in this case, we’ve called that
height 𝑥. So, we can replace ℎ with 𝑥 in
this expression.

Now, we mentioned earlier that
there are two ways that energy conversion can occur. One way is for energy of one type
to go completely and totally into energy of another type. In this example, this would happen
if 100 percent of the elastic potential energy in the spring was converted to
gravitational potential energy. If that happened, we would have a
total energy conversion from elastic potential to gravitational potential. But there’s another kind of energy
conversion, and that’s where energy is dissipated. That means that some energy is
lost, meaning it’s not converted into the type of energy that we’re mainly
considering.

In our mass-on-a-spring example,
we’re mainly considering elastic potential energy being converted into gravitational
potential energy. But what if, as this energy was
being converted, some of it was lost due to friction in the coils of the extending
spring? In that case, not all of the energy
we would start out with over here would go into the gravitational potential energy
of our mass over here. Instead, we would have a second
term on the right-hand side of our equation. There’s an energy dissipated
term.

In this type of conversion, some of
the energy goes into gravitational potential, and some of it is dissipated. Typically, that energy would be
lost as heat. It’s important to see that even if
all of the initial energy isn’t converted into, in this case, gravitational
potential energy, that is, even if some of it is dissipated still in the process
overall. If we account the total energy on
the left and right sides of this equation, they’ll still be equal to one
another. In other words, energy is still
conserved.

So, when we talk about energy being
converted from one type to another, the energy could either be converted totally
from one type into another specific type. Or it could be converted and spread
across several types, in which case some of the energy would be considered
dissipated or lost. Even if this happens, though,
energy in the process we’re considering is typically conserved. Meaning that the total energy we
end up with considering all those things it’s converted to is equal to the total
energy we started with.

Knowing all this, let’s turn now to
an example exercise.

A car is initially at rest before
it starts to roll along a downward-sloping road with its engine turned off. While rolling, the car’s velocity
increased by 1.4 meters per second. What vertically downward distance
does the car travel? Gravity is the only force that acts
on the car.

All right, so, we have this car
that is initially at rest on this downward-sloping road. And then, with its engines turned
off, it starts to roll downhill. Say the break was released and the
car simply starts to roll under the influence of gravity. After it’s been rolling for some
time, we’re told that the car’s velocity had increased by 1.4 meters per second. Now since the car was initially at
rest, that means that, at the start, its velocity was zero meters per second. So, if its velocity has increased
by 1.4 meters per second, then we can say it simply is 1.4 meters per second.

In order to pick up the speed, the
car has rolled this distance down along the road. But it’s not that distance exactly
that we want to solve for. Instead, we want to calculate the
vertically downward distance that the car travels. So, if the car starts out at this
elevation here and then it ends up at this elevation here, by the time it’s reached
1.4 meters per second of speed, we want to know what this distance here is, the
vertically downward distance the car has traveled. We can call that distance 𝑑.

Now, because the car was elevated,
we could say this distance 𝑑 at the outset, that means that it began with some
amount of gravitational potential energy. That energy is equal to the car’s
mass times the acceleration due to gravity times its height above some minimum
value. And in terms of our variables, we
can say that that energy is 𝑚, the car’s mass, times 𝑔, the acceleration due to
gravity, times 𝑑. Because the car starts out at rest,
that means the only energy it has at the outset is gravitational potential energy,
𝑚 times 𝑔 times 𝑑. But then as the car rolls along,
this energy is converted from gravitational potential energy to kinetic energy. And we can see that because the car
acquires this velocity, 1.4 meters per second.

Now, the kinetic energy of an
object, we can recall, is equal to one-half its mass times its speed squared. In the case of our car, we can
write that simply as one-half 𝑚𝑣 squared. But now, look at this. Over here, we’ve called 𝑣 equal to
zero, and here we’ve called 𝑣 equal to 1.4 meters per second. So, to avoid any confusion, let’s
call this final velocity 𝑣 sub f. And then, we’ll go over and add
that subscript to our 𝑣 over here. That way we know which car velocity
we’re talking about.

So, the car’s initial energy is 𝑚
times 𝑔 times 𝑑. And its final energy after it’s
rolled down the hill some vertical distance 𝑑 is equal to one-half 𝑚 times 𝑣 sub
f squared. And this is where energy
conservation comes in. Because energy is conserved in this
process and none of the initial energy is dissipated into other energy types besides
kinetic. We can say that 𝑚 times 𝑔 times
𝑑 is equal to one-half 𝑚𝑣 sub f squared. That is, there’s been a complete
and total energy conversion from gravitational potential to kinetic energy and
energy is conserved, meaning the total amount we start with is equal to the total
amount we end up with.

All this is good news. And now, we move on to solving for
𝑑, the vertically downward distance that our car travels. As we do, notice that there’s a
single factor of the car’s mass 𝑚 on both sides of the equation. Therefore, if we divide both sides
by that mass, then that term will cancel out completely. And we’ll see that the result that
we will find has nothing to do with the mass of the car. Anyway, let’s continue on. We want to isolate 𝑑 on one side
of the equation. So, to do that, let’s divide both
sides by the gravitational acceleration 𝑔. This cancels that term on the
left.

And we now have an expression that
tells us that the vertical downward distance the car moves is equal to its final
velocity squared divided by two times 𝑔. 𝑣 sub f is equal to 1.4 meters per
second and the acceleration due to gravity is equal to 9.8 meters per second
squared. And when we substitute these values
into our equation for 𝑑, we’re now ready to calculate this distance.

When we do, we find a result of 0.1
meters. That’s the vertically downward
distance the car travels so that its velocity increases by 1.4 meters per
second.

And note that it doesn’t have to
roll down a constant incline to do this. But could instead follow a curved
or bumpy path so long as the vertical height difference between the starting and
ending points is the same.

Let’s summarize now what we’ve
learned about energy conversion and conservation. Starting out, we saw that energy
can be transferred from one type to another. That’s called energy
conversion. We also saw that the total energy
at the start of a process can equal the total at the end. That’s energy conservation. Lastly, we learned that the
formulas used to represent energies such as gravitational potential energy or
kinetic energy can be used in equations.

When a pure energy conversion takes
place, where no energy is dissipated in the process, we can write the energy of type
one is equal to the energy of the converted type, type two. On the other hand, if energy is
dissipated, that introduces another term on the right-hand side. We then show that our initial
energy type is converted into a second type and some amount of dissipated
energy. We learned that in both of these
cases, with or without energy being dissipated, energy can be conserved.