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# What is conservation of energy?

Learn what conservation of energy means, and how it can make solving problems easier.

# What is the principle of conservation of energy?

In physics, the term conservation refers to something which doesn't change. This means that the variable in an equation which represents a conserved quantity is constant over time. It has the same value both before and after an event.
There are many conserved quantities in physics. They are often remarkably useful for making predictions in what would otherwise be very complicated situations. In mechanics, there are three fundamental quantities which are conserved. These are energy, momentum and angular momentum.
If you have looked at examples in other articles—for example, the kinetic energy of charging elephants—then it may surprise you that energy is a conserved quantity. After all, energy often changes in collisions. It turns out that there are a couple of key qualifying statements we need to add:
• Energy, as we'll be discussing it in this article, refers to the total energy of a system. As objects move around over time, the energy associated with them—e.g., kinetic, gravitational potential, heat—might change forms, but if energy is conserved, then the total will remain the same.
• Conservation of energy applies only to isolated systems. A ball rolling across a rough floor will not obey the law of conservation of energy because it is not isolated from the floor. The floor is, in fact, doing work on the ball through friction. However, if we consider the ball and floor together, then conservation of energy will apply. We would normally call this combination the ball-floor system.
In mechanics problems, we are likely to encounter systems containing kinetic energy (E, start subscript, K, end subscript), gravitational potential energy (U, start subscript, g, end subscript), elastic—spring—potential energy (U, start subscript, s, end subscript), and heat (thermal energy) (E, start subscript, H, end subscript). Solving such problems often begins by establishing conservation of energy in a system between some initial time—subscript i—and at some later time—subscript f.
E, start subscript, K, i, end subscript, plus, U, start subscript, g, i, end subscript, plus, U, start subscript, s, i, end subscript, equals, E, start subscript, K, f, end subscript, plus, U, start subscript, g, f, end subscript, plus, U, start subscript, s, f, end subscript, plus, E, start subscript, H, f, end subscript
Which could be expanded out as:
start fraction, 1, divided by, 2, end fraction, m, v, start subscript, i, end subscript, squared, plus, m, g, h, start subscript, i, end subscript, plus, start fraction, 1, divided by, 2, end fraction, k, x, start subscript, i, end subscript, squared, equals, start fraction, 1, divided by, 2, end fraction, m, v, start subscript, f, end subscript, squared, plus, m, g, h, start subscript, f, end subscript, plus, start fraction, 1, divided by, 2, end fraction, k, x, start subscript, f, end subscript, squared, plus, E, start subscript, H, f, end subscript

# What do we mean by system here?

In physics, system is the suffix we give to a collection of objects that we choose to model with our equations. If we are to describe the motion of an object using conservation of energy, then the system should include the object of interest and all other objects that it interacts with.
In practice, we always have to choose to ignore some interactions. When defining a system, we are drawing a line around things we care about and things we don't. The things we don't include are usually collectively termed the environment. Ignoring some of the environment will inevitably make our calculations less accurate. There is no indignity in doing this however. In fact, being a good physicist is often as much about understanding the effects you need to describe as it is about knowing which effects can be safely ignored.
Consider the problem of a person making a bungee jump from a bridge. At a minimum, the system should include the jumper, bungee, and the Earth. A more accurate calculation might include the air, which does work on the jumper via drag, or air resistance. We could go further and include the bridge and its foundation, but since we know that the bridge is much heavier than the jumper, we can safely ignore this. We wouldn't expect the force of a decelerating bungee jumper to have any significant effect on the bridge, especially if the bridge is designed to bear the load of heavy vehicles.

# What is mechanical energy?

Mechanical energy, E, start subscript, M, end subscript, is the sum of the potential energy and kinetic energy in a system.
start box, E, start subscript, M, end subscript, equals, E, start subscript, P, end subscript, plus, E, start subscript, K, end subscript, end box
Only conservative forces like gravity and the spring force that have potential energy associated with them. Nonconservative forces like friction and drag do not. We can always get back the energy that we put into a system via a conservative force. Energy transferred by nonconservative forces however is difficult to recover. It often ends up as heat or some other form which is typically outside the system—in other words, lost to the environment.
What this means in practice is that the special case of conservation of mechanical energy is often more useful for making calculations than conservation of energy in general. Conservation of mechanical energy only applies when all forces are conservative. Luckily, there are many situations where nonconservative forces are negligible, or at least a good approximation can still be made when neglecting them.

# How can conservation of energy describe how objects move?

When energy is conserved, we can set up equations which equate the sum of the different forms of energy in a system. We then may be able to solve the equations for velocity, distance, or some other parameter on which the energy depends. If we don't know enough of the variables to find a unique solution, then it may still be useful to plot related variables to see where solutions lie.
Consider a golfer on the moon—gravitational acceleration 1.625 m/ssquared—striking a golf ball. By the way, Astronaut Alan Shepard actually did this. The ball leaves the club at an angle of 45degrees to the lunar surface traveling at 20 m/s both horizontally and vertically—total velocity 28.28 m/s. How high would the golf ball go?
We begin by writing down the mechanical energy:
E, start subscript, M, end subscript, equals, start fraction, 1, divided by, 2, end fraction, m, v, squared, plus, m, g, h
Applying the principle of conservation of mechanical energy, we can solve for the height h—note that the mass cancels out.
start fraction, 1, divided by, 2, end fraction, m, v, start subscript, i, end subscript, squared, equals, m, g, h, start subscript, f, end subscript, plus, start fraction, 1, divided by, 2, end fraction, m, v, start subscript, f, end subscript, squared
\begin{aligned} h &= \frac{\frac{1}{2}v_i^2-\frac{1}{2}v_f^2}{g} \\ &=\frac{\frac{1}{2}(28.28~\mathrm{m/s})^2-\frac{1}{2}(20~\mathrm{m/s})^2}{1.625~\mathrm{m/s^2}} \\ &= 123~\mathrm{m}\end{aligned}
As you can see, applying the principle of conservation of energy allows us to quickly solve problems like this which would be more difficult if done only with the kinematic equations.
Exercise 1: Suppose the ball had an unexpected collision with a nearby american flag hoisted to a height of 2 m. How fast would it be traveling at the time of collision?
Exercise 2: The image below shows a plot of the kinetic, gravitational potential and mechanical energy over time during the flight of a small model rocket. Points of interest such as maximum height, apogee, and the time of motor stop, burnout, are noted on the plot. The rocket is subject to several conservative and nonconservative forces over the course of the flight. Is there a time during the flight when the rocket is subject to only conservative forces? Why?

# Why can perpetual motion machines never work?

The perpetual motion machine is a concept for a machine which continues its motion forever, without any reduction in speed. An endless variety of weird and wonderful machines have been described over the years. They include pumps said to run themselves via their own head of falling water, wheels which are said to push themselves around by means of unbalanced masses, and many variations of self-repelling magnets.
Though often interesting curiosities, such a machine has never been shown to be perpetual, nor could it ever be. In fact, even if such a machine were to exist, it wouldn't be very useful. It would have no ability to do work. Note that this differs from the concept of the over-unity machine, which is said to output more than 100% of the energy put into it, in clear violation of the principle of conservation of energy.
From the most basic principles of mechanics, there is nothing that strictly makes the perpetual motion machine impossible. If a system could be fully isolated from the environment and subject to only conservative forces, then energy would be conserved and it would run forever. The problem is that in reality, there is no way to completely isolate a system and energy is never completely conserved within the machine.
It is possible today to make extremely low friction flywheels which rotate in a vacuum for storing energy. Yet, they still lose energy and eventually spin down when unloaded, some over a period of years . The earth itself, rotating on its axis in space is perhaps an extreme example of such a machine. Yet, because of interactions with the moon, tidal friction, and other celestial bodies, it too is gradually slowing. In fact, every couple of years, scientists have to add a leap second to our record of time to account for variation in the length of day.