All formulas are according to the law of conservation of energy. Kinetic and potential energy
An absolutely inelastic impact can also be demonstrated using plasticine (clay) balls moving towards each other. If the masses of the balls m 1 and m 2, their speed before impact, then, using the law of conservation of momentum, we can write:
If the balls were moving towards each other, then together they will continue to move in the direction in which the ball with greater momentum was moving. In a particular case, if the masses and velocities of the balls are equal, then
Let us find out how the kinetic energy of the balls changes during a central absolutely inelastic impact. Since during the collision of balls between them forces act that depend not on the deformations themselves, but on their velocities, we are dealing with forces similar to friction forces, therefore the law of conservation of mechanical energy should not be observed. Due to deformation, there is a “loss” of kinetic energy, converted into thermal or other forms of energy ( energy dissipation). This “loss” can be determined by the difference in kinetic energies before and after the impact:
.
From here we get:
 |
(5.6.3) |
If the struck body was initially motionless (υ 2 = 0), then
When m 2 >> m 1 (the mass of a stationary body is very large), then almost all the kinetic energy upon impact is converted into other forms of energy. Therefore, for example, to obtain significant deformation, the anvil must be more massive than the hammer.
When then, almost all the energy is spent on the greatest possible movement, and not on residual deformation (for example, a hammer - a nail).
An absolutely inelastic impact is an example of how “loss” of mechanical energy occurs under the influence of dissipative forces.
The total mechanical energy of a closed system of bodies remains unchanged
The law of conservation of energy can be represented as
If friction forces act between bodies, then the law of conservation of energy is modified. The change in total mechanical energy is equal to the work done by friction forces
Consider the free fall of a body from a certain height h1. The body is not moving yet (let's say we are holding it), the speed is zero, the kinetic energy is zero. The potential energy is maximum because the body is now higher off the ground than in state 2 or 3.
In state 2, the body has kinetic energy (since it has already developed speed), but the potential energy has decreased, since h2 is less than h1. Part of the potential energy turned into kinetic energy.
State 3 is the state just before stopping. The body seemed to have just touched the ground, while the speed was maximum. The body has maximum kinetic energy. Potential energy is zero (the body is on the ground).
The total mechanical energies are equal if we neglect the force of air resistance. For example, the maximum potential energy in state 1 is equal to the maximum kinetic energy in state 3.
Where does the kinetic energy then disappear? Disappears without a trace? Experience shows that mechanical movement never disappears without a trace and it never arises by itself. During the braking of the body, heating of the surfaces occurred. As a result of the action of friction forces, kinetic energy did not disappear, but turned into internal energy of thermal motion of molecules.
During any physical interactions, energy does not appear or disappear, but only transforms from one form to another.
The main thing to remember
1) The essence of the law of conservation of energy
The general form of the law of conservation and transformation of energy has the form
Studying thermal processes, we will consider the formula 
When studying thermal processes, the change in mechanical energy is not considered, that is,
The law of conservation of energy is one of the most important laws, according to which the physical quantity - energy is conserved in an isolated system. All known processes in nature, without exception, obey this law. In an isolated system, energy can only be converted from one form to another, but its quantity remains constant.
In order to understand what the law is and where it comes from, let’s take a body of mass m, which we drop to the Earth. At point 1, our body is at height h and is at rest (velocity is 0). At point 2 the body has a certain speed v and is at a distance h-h1. At point 3 the body has maximum speed and it almost lies on our Earth, that is, h = 0
Law of Conservation of Energy
At point 1 the body has only potential energy, since the speed of the body is 0, so the total mechanical energy is equal.
After we released the body, it began to fall. When falling, the potential energy of a body decreases, as the height of the body above the Earth decreases, and its kinetic energy increases, as the speed of the body increases. In section 1-2 equal to h1, the potential energy will be equal to
And the kinetic energy will be equal at that moment
Body speed at point 2):
The closer a body becomes to the Earth, the less its potential energy, but at the same moment the speed of the body increases, and because of this, kinetic energy. That is, at point 2 the law of conservation of energy works: potential energy decreases, kinetic energy increases.
At point 3 (on the Earth’s surface) the potential energy is zero (since h = 0), and the kinetic energy is maximum
(where v3 is the speed of the body at the moment of falling to the Earth). Because
Then the kinetic energy at point 3 will be equal to Wk=mgh. Consequently, at point 3 the total energy of the body is W3=mgh and is equal to the potential energy at height h. The final formula for the law of conservation of mechanical energy will be:
The formula expresses the law of conservation of energy in a closed system in which only conservative forces act: the total mechanical energy of a closed system of bodies interacting with each other only by conservative forces does not change with any movements of these bodies. Only mutual transformations of the potential energy of bodies into their kinetic energy and vice versa occur.
In the Formula we used:
W - Total body energy
Body potential energy
Body kinetic energy
m - Body mass
g - Gravity acceleration
h - The height at which the body is located
\upsilon - Body speed
This video lesson is intended for self-acquaintance with the topic “The Law of Conservation of Mechanical Energy.” First, let's define total energy and a closed system. Then we will formulate the Law of Conservation of Mechanical Energy and consider in which areas of physics it can be applied. We will also define work and learn how to define it by looking at the formulas associated with it.
Topic: Mechanical vibrations and waves. Sound
Lesson 32. Law of conservation of mechanical energy
Eryutkin Evgeniy Sergeevich
The topic of the lesson is one of the fundamental laws of nature -.
We previously talked about potential and kinetic energy, and also that a body can have both potential and kinetic energy together. Before talking about the law of conservation of mechanical energy, let us remember what total energy is. Full of energy is the sum of the potential and kinetic energies of a body. Let's remember what is called a closed system. This is a system in which there is a strictly defined number of bodies interacting with each other, but no other bodies from the outside act on this system.
When we have decided on the concept of total energy and a closed system, we can talk about the law of conservation of mechanical energy. So, the total mechanical energy in a closed system of bodies interacting with each other through gravitational or elastic forces remains unchanged during any movement of these bodies.
It is convenient to consider the conservation of energy using the example of a free fall of a body from a certain height. If a body is at rest at a certain height relative to the Earth, then this body has potential energy. As soon as the body begins to move, the height of the body decreases, and the potential energy decreases. At the same time, speed begins to increase, and kinetic energy appears. When the body approaches the Earth, the height of the body is 0, the potential energy is also 0, and the maximum will be the kinetic energy of the body. This is where the transformation of potential energy into kinetic energy is visible. The same can be said about the movement of the body in reverse, from bottom to top, when the body is thrown vertically upward.
Of course, it should be noted that we considered this example taking into account the absence of friction forces, which in reality act in any system. Let's turn to the formulas and see how the law of conservation of mechanical energy is written: .
Imagine that a body in a certain frame of reference has kinetic energy and potential energy. If the system is closed, then with any change a redistribution has occurred, a transformation of one type of energy into another, but the total energy remains the same in value. Imagine a situation where a car is moving along a horizontal road. The driver turns off the engine and continues driving with the engine turned off. What happens in this case? In this case, the car has kinetic energy. But you know very well that over time the car will stop. Where did the energy go in this case? After all, the potential energy of the body in this case also did not change; it was some kind of constant value relative to the Earth. How did the energy change happen? In this case, the energy was used to overcome friction forces. If friction occurs in a system, it also affects the energy of that system. Let's see how the change in energy is recorded in this case.
The energy changes, and this change in energy is determined by the work against the friction force. We can determine the work using the formula, which is known from grade 7: A = F.* S.
So, when we talk about energy and work, we must understand that each time we must take into account the fact that part of the energy is spent on overcoming friction forces. Work is being done to overcome friction forces.
To conclude the lesson, I would like to say that work and energy are essentially related quantities through acting forces.
Additional task 1 “On the fall of a body from a certain height”
Problem 1
The body is at a height of 5 m from the surface of the earth and begins to fall freely. Determine the speed of the body at the moment of contact with the ground.
Given: Solution:
H = 5 m 1. EP = m* g*.H
V0 = 0 ; m * g * H =
_______ V2 = 2gH
VK - ? Answer:
Let's consider the law of conservation of energy.
Rice. 1. Body movement (task 1)
At the top point the body has only potential energy: EP = m * g * H. When the body approaches the ground, the height of the body above the ground will be equal to 0, which means that the potential energy of the body has disappeared, it has turned into kinetic energy.
According to the law of conservation of energy, we can write: m * g * H =. Body weight is reduced. Transforming the above equation, we get: V2 = 2gH.
The final answer will be: 
. If we substitute the entire value, we get: .
Additional task 2
A body falls freely from a height H. Determine at what height the kinetic energy is equal to a third of the potential.
Given: Solution:
N EP = m. g. H; ; 
M.g.h = m.g.h + m.g.h
h - ? Answer: h = H.
Rice. 2. To task 2
When a body is at a height H, it has potential energy, and only potential energy. This energy is determined by the formula: EP = m * g * H. This will be the total energy of the body.
When a body begins to move downward, the potential energy decreases, but at the same time the kinetic energy increases. At the height that needs to be determined, the body will already have a certain speed V. For the point corresponding to the height h, the kinetic energy has the form: . The potential energy at this height will be denoted as follows: .
According to the law of conservation of energy, our total energy is conserved. This energy EP = m * g * H remains a constant value. For point h we can write the following relation: (according to Z.S.E.).
Remembering that the kinetic energy according to the conditions of the problem is , we can write the following: m.g.Н = m.g.h + m.g.h.
Please note that the mass is reduced, the acceleration of gravity is reduced, after simple transformations we find that the height at which this relationship holds is h = H.
Answer: h= 0.75H
Additional task 3
Two bodies - a block of mass m1 and a plasticine ball of mass m2 - are moving towards each other with the same speeds. After the collision, the plasticine ball sticks to the block, the two bodies continue to move together. Determine how much energy is converted into the internal energy of these bodies, taking into account the fact that the mass of the block is 3 times the mass of the plasticine ball.
Given: Solution:
m1 = 3. m2 m1.V1- m2.V2= (m1+m2).U; 3.m2V- m2.V= 4 m2.U2.V=4.U; .
This means that the speed of the block and plasticine ball together will be 2 times less than the speed before the collision.
The next step is this.
.
In this case, the total energy is the sum of the kinetic energies of two bodies. The bodies that have not yet touched do not hit. What happened then, after the collision? Look at the following entry: 
.
On the left side we leave the total energy, and on the right side we must write kinetic energy bodies after interaction and take into account that part of the mechanical energy turned into heat Q.
Thus we have: 
. As a result, we get the answer .
Please note: as a result of this interaction, most of the energy is converted into heat, i.e. turns into internal energy.
List of additional literature:
Are you so familiar with the laws of conservation? // Quantum. - 1987. - No. 5. - P. 32-33.
Gorodetsky E.E. Law of conservation of energy // Quantum. - 1988. - No. 5. - P. 45-47.
Soloveychik I.A. Physics. Mechanics. A manual for applicants and high school students. – St. Petersburg: IGREC Agency, 1995. – P. 119-145.
Physics: Mechanics. 10th grade: Textbook. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakisheva. – M.: Bustard, 2002. – P. 309-347.
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One of the most important laws, according to which the physical quantity - energy is conserved in an isolated system. All known processes in nature, without exception, obey this law. In an isolated system, energy can only be converted from one form to another, but its quantity remains constant.
In order to understand what the law is and where it comes from, let’s take a body of mass m, which we drop to the Earth. At point 1, our body is at height h and is at rest (velocity is 0). At point 2 the body has a certain speed v and is at a distance h-h1. At point 3 the body has maximum speed and it almost lies on our Earth, that is, h = 0
At point 1 the body has only potential energy, since the speed of the body is 0, so the total mechanical energy is equal.
After we released the body, it began to fall. When falling, the potential energy of a body decreases, as the height of the body above the Earth decreases, and its kinetic energy increases, as the speed of the body increases. In section 1-2 equal to h1, the potential energy will be equal to
And the kinetic energy will be equal at that moment ( - the speed of the body at point 2):
The closer a body becomes to the Earth, the less its potential energy, but at the same moment the speed of the body increases, and because of this, kinetic energy. That is, at point 2 the law of conservation of energy works: potential energy decreases, kinetic energy increases.
At point 3 (on the surface of the Earth), the potential energy is zero (since h = 0), and the kinetic energy is maximum (where v3 is the speed of the body at the moment of falling to the Earth). Since , the kinetic energy at point 3 will be equal to Wk=mgh. Consequently, at point 3 the total energy of the body is W3=mgh and is equal to the potential energy at height h. The final formula for the law of conservation of mechanical energy will be:
The formula expresses the law of conservation of energy in a closed system in which only conservative forces act: the total mechanical energy of a closed system of bodies interacting with each other only by conservative forces does not change with any movements of these bodies. Only mutual transformations of the potential energy of bodies into their kinetic energy and vice versa occur.
In Formula we used.