How to derive the law of conservation from Newton s third law and the impulse theorem

Let the masses of the two objects m1 m2 collide on a smooth horizontal plane, with the initial velocity v1 v2 and the post-collision velocity v1' v2'

A was taken as the research object.

f1t=m1v1'-m1v1

B was taken as the research object.

f2t=m2v2'-m2v2

by Newton's third law.

f1t=-f2t

m1v1'-m1v1=-m2v2'+m2v2

m1v1+m2v2=m1v1'=m2v2'

Law of Conservation of Momentum:

The conclusion that a system is not affected by external forces or the sum of the external forces is zero, and the total momentum of the system remains the same, is called the law of conservation of momentum.

1.The law of conservation of momentum is one of the most important and universal conservation laws in nature, which is an experimental law, which can also be deduced from Newton's third law in combination with the momentum theorem.

2.The system of things that has forces with each other is called a system, and the objects in the system can be two, three or more, and the system should be reasonably selected according to the needs and the convenience of solving the problem when solving practical problems.

Newton's third law, expressed by the formula f(a b) = f(b a) momentum theorem: impulse (i) = change in momentum (δp).

The impulse i = fδt, then fδt = δp

Set a, b, c ............ a, b, and c in the systemThe external force on each part is FA, FB, FC ......

Then according to the momentum theorem, there is (fa+f(b a)+f(c a)+....t=δPa is the same: (fb+f(ab)+f(cb)+....t=δpbfc+f(a→c)+f(b→c)+…t=δpc The left and right sides of each equation are added separately, and according to Newton's third law, the internal forces of all systems cancel each other out, then there is.

fa+fb+fc+…=δpa+δpb+δpc+…On the left is the external force on each part of the system, and on the right is the change in momentum of each part.

When the external force on the system is zero, i.e., Fa+Fb+Fc+....When =0, the change in momentum of the system δp = δpa + δpb + δpc + ....=0

i.e. the law of conservation of momentum.

When two objects collide, the time period of the impact process is as follows: t1 t2;

In this process, the two objects are each subjected to a force, which is set as: f1(t), f2(t);

After the impact, the momentum of the two objects changes:

From the impulse theorem, it can be seen that in the process of t1 t2:

The momentum increment of object 1 is: δp1 = f1(t)dt; The integration interval is: [t1,t2];

The momentum increment of object 2 is: δp2 = f2(t)dt; The integral interval is also: [t1,t2];

Since f1(t) and f2(t) are reaction forces against each other, and Newton's third law, we can see that:

f1(t) ≡f2(t)；

Bringing the above equation into the integral formula, the calculation yields:

p1 = -δp2；

That is, the increments of the momentum of the two objects are opposite to each other;

Therefore, the sum of the momentum of the two objects before and after collision does not change; That is, the law of conservation of momentum holds.

On a smooth horizontal plane, there are two objects A and B with masses of m1 and m2 respectively, moving in the same direction (or opposite direction) along the same straight line, and the velocities are v1 and v2 respectively, and the velocities become v1 after collision'and v2', derived from Newton's second law.

f1t1=m1(v1'-v1)

f2t2=m2(v2'-v2)

where f1 and f2 are the interaction forces between a and b, and Newton's third law yields f1 f2, and t1=t2

What the hell? Is it the impulse theorem, which you invented?

1. Newton's three laws: The first law explains the meaning of the balance and grinding of force: force is the cause of changing the state of motion of objects; The second law states the effect of force: force causes an object to gain acceleration; The third law reveals the nature of force: force is an interaction between objects.

2. Its scope of application is the scope of classical mechanics, and the applicable conditions are the particle point, inertial reference frame and macroscopic and low-speed transportation and intelligent movement problems. Newton's laws of motion explain the complete system of Newtonian mechanics and the basic laws of motion in classical mechanics, which are widely used in various fields. Suddenly, he replied.

Newton's Three Laws, also known as Newton's Laws of Motion, are the cornerstone of classical mechanics and were proposed by the English scientist Isaac Newton in the 17th century. These three laws describe the laws of motion of objects under the action of force and provide important guidance for us to understand the forces and motion in the world.

The first law, let seep is also known as the law of inertia. It shows that if there is no external force acting on an object or the net force is zero, then it will remain at rest or move in a straight line at a uniform speed. In other words, the object will remain in its state of motion and will not be changed by force.

This law provides insight into the concept of inertia, which states that an object retains its original state when it is not disturbed by a force that collapses.

The second law, also known as the law of motion. According to this law, when an object is subjected to an external force, it will produce an acceleration. The magnitude of this acceleration is directly proportional to the force acting on the object and inversely proportional to the mass of the object.

This law can be expressed by the formula f=ma, where f represents the force, m represents the mass of the object, and a represents acceleration. This law tells us that force is responsible for the change in the acceleration of an object, and it also shows that mass has an important influence on the state of motion of an object.

The third law, also known as the law of action-reaction. According to this law, any force results in a force acting on it, equal in magnitude and opposite in the opposite direction. In a nutshell, this law states that all forces exist in pairs, and the forces acting and reacting between them are equal in magnitude and opposite in direction.

This law is often used to explain the interaction between objects, such as the force exerted by an object on the ground versus the force exerted by the ground on the object.

Newton's three laws explain the motion of objects through concise and intuitive rules. These fixed and masked spinal laws laid the foundation for the later development of science, which not only played an important role in classical mechanics, but also influenced the research and application of other fields, such as electromagnetism and thermodynamics. Newton's contributions have given mankind a deeper understanding of the laws of motion in nature and provided a solid foundation for scientific research and technical applications.

In today's physics, Newton's three laws are still an important concept and play a vital role in the solution of many problems.

Newton's third law reveals the nature of forces in nature and is a law that describes the nature of the interaction of forces. From this law, the law of conservation of momentum, which is universally applicable in physics, can be directly derived. Or conversely, Newton's third law is the law of conservation of momentum expressed in forceful language.

In fact, Newton's third law was obtained by analyzing the law of conservation of momentum obtained by Huygens, Descartes and others in their research on collision problems. In his Principia, Newton formulated his third law as follows: "Every action has an equal reaction; Alternatively, the interaction between two objects is always equal and pointing in opposite directions.

1. Newton's first law: any object remains at rest or in a state of uniform linear motion until it is forced to change this state by the force of other objects.

2. Newton's second law: the object will produce acceleration under the action of the resultant external force, the direction of acceleration is the same as the direction of the resultant external force, and the magnitude of the acceleration is proportional to the magnitude of the combined external force and the inverse proportion of the inertial mass of the object.

3. Newton's third law: The force of action and reaction between two objects, on the same straight line, is equal in magnitude and opposite in direction.

4. Newton's laws of motion include Newton's first law of motion, Newton's second law of motion, and Newton's third law of motion, which were summarized by Isaac Newton in his book "Mathematical Principles of Natural Philosophy" in 1687. Among them, the first law states the meaning of force: force is what changes the state of motion of an object; The second law states the effect of force:

The force causes the object to gain acceleration; The third law reveals the nature of force: force is an interaction between objects.

5. The laws in Newton's laws of motion are independent of each other, and the internal logic conforms to self-consistent consistency. Its scope of application is the range of classical mechanics, and the applicable conditions are particles, inertial reference frames, and macroscopic and low-speed motion problems. Newton's laws of motion explain the complete system of Newtonian mechanics and the basic laws of motion in classical force rotation, which are widely used in various fields.