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.