# What is conservation of energy?

**In physics, the term conservation corresponds to the absence of change. This means that the variable representing a conserved quantity is constant over time. It will always have the same value no matter what.**

**Many quantities are conserved in physics, and this is extremely useful for making predictions in situations that would otherwise have been very complicated to solve. In mechanics, three fundamental quantities are conserved: energy(Opens in a new window) , momentum and angular momentum .**

**In other articles you may have seen examples where the energy changes during collisions, especially in the one on kinetic energy with elephants charging at full speed. It is therefore difficult to imagine that the energy is conserved. It is that it is necessary to bring some important precisions:**

**Energy , as we will discuss it here, refers to the total energy of a system. As objects move through time, the energy associated with them, e.g. kinetic , gravitational potential , thermal , can change shape, but if the energy conserves, then the sum of these energies will remain the same.**

**Conservation of energy only applies to isolated systems . A ball rolling over a rough floor will not obey the law of conservation of energy because it is not isolated from the ground. The floor acts on the ball by friction. However, if we consider the ball and the floor together, then the conservation of energy will apply. We would then call this set the ball/floor system .**

**In mechanics problems, systems often include kinetic energy (E_cE **

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** E, start subscript, c, end subscript), gravitational potential energy (E_pE **

**p**

** **

** E, start subscript, p, end subscript), elastic (spring) potential energy (E_p{_e}E **

**p**

** **

** **

**e**

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** E, start subscript, p, end subscript, start subscript, e, end subscript), and thermal energy (heat) (QQQ). To solve these problems, one often begins by establishing the conservation of energy in a system between a certain initial instant — denoted by the index i — and a final instant — denoted by the index f.**

**E_\mathrm{c_i} + E_\mathrm{p_i} + E_\mathrm{pe_i} = E_\mathrm{c_f} + E_\mathrm{p_f} + E_\mathrm{pe_f} + Q_\mathrm{f}E **

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** E, start subscript, c, start subscript, i, end subscript, end subscript, plus, E, start subscript, p, start subscript, i, end subscript, end subscript, plus, E, start subscript, p, e, start subscript, i, end subscript, end subscript, equals, E, start subscript, c, start subscript, f, end subscript, end subscript, plus, E, start subscript, p, start subscript, f, end subscript, end subscript, plus, E, start subscript, p, e, start subscript, f, end subscript, end subscript, plus, Q, start subscript, f, end subscript**

**Or in its full form:**

**\frac{1}{2}mv_i^2 + mgh_i + \frac{1}{2}kx_i^2 = \frac{1}{2}mv_f^2 + mgh_f + \frac{1}{2}kx_f^ 2 + Q_\mathrm{f} **

**2**

**1**

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** m v **

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** +m g h **

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** + **

**2**

**1**

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** k x **

**I**

**2**

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** = **

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**1**

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** m v **

**f**

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** +m g h **

**f**

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** + **

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**1**

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** k x **

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** +Q **

**f**

** **

** 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, Q, start subscript, f, end subscript**

**What do we mean by system ?**

**In physics, a system is a set of objects whose interactions we study and to which we apply the formulas at our disposal. To describe the motion of an object according to the principle of conservation of energy, the system must include the object in question and all the other objects with which it interacts.**

**In practice, we will generally choose to ignore certain interactions. To define a system, we separate the objects that interest us from the others. These form what is called the environment . By ignoring the interactions of the environment, we will necessarily obtain less precise calculations, but this is not catastrophic. In reality, in physics, to understand certain effects, it is necessary to be able to ignore others. The key is to carefully select those that are not taken into account.**

**Let’s take as an example a person doing a bungee jump from the top of a bridge. The system must include at least the person, the rubber band and the Earth. For a more precise calculation, one should also consider the air, which exerts a resistance or drag force on the person . We could go further and include the bridge and its foundations. However, knowing that the bridge is much heavier than the person, we can safely ignore it. The force exerted on the bridge by a decelerating bungee jumper is unlikely to be significant, particularly if the bridge is designed to withstand the weight of heavy vehicles.**

**There is always a low level of interaction between even distant objects. It is therefore necessary to define the limit of the system intelligently.**

**There is always a low level of interaction between even distant objects. It is therefore necessary to define the limit of the system intelligently.**

**There is always a low level of interaction between even distant objects. It is therefore necessary to define the limit of the system intelligently.**

**What is mechanical energy?**

**mechanical energy,E_mE **

**m**

** **

** E, start subscript, m, end subscript, is the sum of the potential energy and the kinetic energy of a system.**

**\boxed{E_\mathrm{m} = E_\mathrm{p} + E_\mathrm{c}} **

**E **

**m**

** **

** =E **

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** +E **

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** start box, E, start subscript, m, end subscript, equals, E, start subscript, p, end subscript, plus, E, start subscript, c, end subscript, end box**

**Only conservative forces , such as gravity and restoring force, are associated with potential energy. This is not the case for non-conservative forces, such as friction and drag. It is always possible to recover the energy supplied to a system by means of a conservative force, while the energy transferred by non-conservative forces is difficult to recover. It often ends up as heat or some other form usually outside the system. In other words, it gets lost in the environment.**

**Thus, in practice, the special case of conservation of mechanical energy is more often used than the general principle of conservation of energy to solve problems. The general principle applies only when all the forces are conservative. Fortunately, there are many situations where the nonconservative forces are negligible, or where one obtains at least a good approximation by ignoring them.**

**How to describe the movement of objects thanks to the conservation of energy?**

**When energy is conserved, one can pose equations stating the equality of the sum of the different forms of energy in a system. One can then isolate and determine the speed, the distance or any other parameter on which the energy depends. If some variables are missing to solve the equation, we can try to express these variables in terms of others to arrive at the solution.**

**Imagine a golfer on the Moon (where the acceleration due to gravity is 1.625 m/s^2 **

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** squared) hitting a golf ball. This is what astronaut Alan Shepard did during the Apollo 14 mission. The ball leaves the club at an angle of 45^\circ **

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** degreesrelative to the lunar surface and advances at 20 m/s horizontally and vertically, for a total speed of 28.28 m/s. How high will the ball go?**

**Let’s start by writing the mechanical energy:**

**E_\mathrm{m} = \frac{1}{2} mv^2 + mghE **

**m**

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** = **

**2**

**1**

** **

** m v **

**2**

** +m g hE, start subscript, m, end subscript, equals, start fraction, 1, divided by, 2, end fraction, m, v, squared, plus, m, g, h**

**By applying the principle of conservation of energy, we can determine the heighthhh(the mass simplifies).**

**\frac{1}{2} m v_i^2 = mgh_f+\frac{1}{2} m v_f^2 **

**2**

**1**

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** m v **

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**2**

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** =m g h **

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** m v **

**f**

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** 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} **

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**= **

**1 , 6 2 5 m / s **

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** ( 2 8 , 2 8 m / s ) **

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** − **

**2**

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** ( 2 0 m / s ) **

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**=1 2 3 m**

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**How do we know that the final speed was 20 m/s?**

**We therefore see that by applying the principle of conservation of energy, we can quickly solve this type of problem, which would have been much more difficult if we had only used the equations of motion .**

**Exercise 1: Suppose the ball collides with an American flag hoisted nearby at a height of 2 m. What is its speed at the time of the collision? See the answer.**

**Exercise 2: In the figure below, the kinetic energy, the gravitational potential energy and the mechanical energy during the flight of a model rocket are represented as a function of time. Interesting points are noted on the graph: the maximum height (apogee) and the moment when the engine stops (end of combustion). The rocket is subjected to several conservative and non-conservative forces throughout the flight. Is there a time during flight when the rocket is subject to only conservative forces? Why ?**

**Transfert d’énergie pendant le vol d’un modèle réduit de fusée [1].**

**Energy transfer during the flight of a model rocket [1].**

**Energy transfer during the flight of a model rocket [1].**

**See the answer.**

**Why can perpetual motion machines ever exist?**

**Perpetual motion machines are machines which, once set in motion, continue to move indefinitely, without loss of speed. Over the centuries, many models, each stranger and more fabulous than the next, have been imagined. We’ve seen pumps that are supposed to work on their own with their own downward head of water, wheels that are supposed to spin indefinitely thanks to unbalanced masses, and many variations of magnets that repel each other.**

**As interesting as they are, these machines remain myths. We have never seen a perpetual motion machine and we will never see one. Truth be told, even if it existed, it wouldn’t be very useful. Such a machine could not provide work. Attention, it should not be confused with the concept of superunitary machine, which would produce more energy than it consumes, which is clearly contrary to the principle of conservation of energy.**

**If we stick to the fundamental principles of mechanics, in theory, nothing prevents us from realizing a perpetual motion machine. If a system could be perfectly isolated from the environment and subject only to conservative forces, then the energy would remain constant and the machine would run indefinitely. The problem is that it is impossible to completely isolate a system. Therefore, the energy cannot actually be stored in the machine.**

**Today we know how to make flywheels. These devices are rotated in a vacuum chamber which greatly limits friction. It is therefore possible to store energy there, but there is still a loss of energy. They eventually decelerate and stop rotating when unloaded, after several years [2]. The Earth itself is an extreme example of such a machine, as it rotates around its axis in space. However, due to interactions with the Moon and other celestial bodies, as well as tidal friction, Earth’s rotational speed is gradually decreasing. In fact, every two years or so, scientists add a leap secondto our timesheet to account for the variation in the length of a day**

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