Momentum is also called Linear Momentum.In classical mechanics, momentum (the unit in the International System of Units is kg · m/s, and the dimension MLT ⁻ ¹) is expressed asMass of objectThe product of and speed is the physical quantity related to the mass and speed of the object, which refers to the effect of moving objects.Momentum is alsovector, its direction is the same as the speed.[1]
The momentum of the particle group is the momentum of each particle in the groupVector sum。
Momentum is aConserved quantity, which is expressed in aClosed system(No external force or the sum of external force vectors is 0)The sum of momentum remains the same。
Corporealmechanical movementIt does not occur in isolation. There is an interaction between it and the surrounding objects. This interaction is manifested in the transmission (or transfer) process of mechanical movement between the moving objects and the surrounding objects. Momentum measures the physical quantity of mechanical movement from the perspective of mechanical movement transmission. This transmission is carried out in equal quantities. How many mechanical movements of objects (i.e. momentum)When it is transferred to object 1, object 2 will lose the same amount of momentum, and the result of the transfer is that the total momentum of both will remain unchanged.
Law of conservation of momentumIs the first conservation discoveredLaws, which originated in Western Europe in the 16th and 17th centuriesphilosopherTheir philosophical thinking on the movement of the universe.
Looking at the moving objects around us, we can see that most of them, such as bouncing balls, flying bullets, moving clocks, and running machines, will stop.it seemsuniverseThe total amount of inter movement seems to be decreasing.Is the whole universe like a machine that will stop one day?But for thousands of yearsCelestial motionOur observation did not find any sign of decrease in the movement of the universe.Many philosophers who lived in the 16th and 17th centuries believed that the total amount of motion in the universe would not decrease. As long as a suitable physical quantity could be found to measure motion, the total amount of motion would be conservative.What is the proper physical quantity?
Descartes' definition
French philosopher, mathematician and physicistDescartesPut forward, quality andrateThe product of is a suitable physical quantity.But later, Dutch mathematicians and physicistsHuygens(1629~1695) in researchCollision problemsIt is found that according to Descartes' definition, the total amount of motion of two objects is not necessarily conserved before and after collision.
Newton's definition of momentum
On the basis of summarizing the work of these people, Newton made an important modification to Descartes' definition, that is, he did not use mass andrateThe product of the mass andspeedIn this way, we can find the appropriate physical quantity to measure the motion.Newton called it "quantity of motion", which is now called momentum.In 1687, Newton《Mathematical Principles of Natural Philosophy》It is pointed out in the book that the amount of motion obtained by subtracting the sum of motions in one direction from the sum of motions in the opposite direction does not change due to the interaction between objects;It also points out that the interaction between two or more objectsfocusOfmotion state , and does not change due to the interaction between these objects, always stay still or doUniform linear motion。
Scope of application
Modern scientific experiments and theoretical analysis have shown that in nature, the interactions between celestial bodies are as large asproton, neutron, etcElementary particleThe interactions between the two obey the law of conservation of momentum.Therefore, it isnatureIt is one of the most important and universal objective laws in the law of motion, and has a wider application scope than Newton's law of motion.
obtain
1. Problem posing:Momentum theoremIt reveals the reason and measurement of the change of momentum of an object, that is, if the momentum of an object changes, it will be subject to external force and continue to act for a period of time, that is, the object will be subject toimpulse。However, due to the interaction of force action, any object subject to external force will also applyReaction forceTherefore, it is an important topic of both universal and practical value to study the change law of the total momentum of the interacting object system.The following is the process of exploring the change law of the total momentum of the object system.
2. Theoretically deduce the law of conservation of momentum from the typical interaction between two bodies - collision
Question scenario: What is the relationship between the momentum changes before and after the collision of two balls?
Derivation process: four steps
Isolator analysis method: start with the reason for the change of momentum of each ball, carry out force analysis on each ball, and find the relationship between their respective impulses.
Mathematic certification: apply the momentum theorem to each ball, and then combine the third law of Newton to quantitatively deduce the relationship between the changes of momentum of two balls - equal in size and opposite in direction (that is, to offset each other).
System analysis method: on the basis of the above, take the whole (system) composed of two balls as the research object, and get the change rule of the total momentum of the system - the change of the total momentum is zero (the total momentum conservation).TotalMomentum conservationExpression for.(Giveinternal force, external forceconcept）
System Stress Analysis Diagram
Conclusion: From the further inquiry of conservation conditions, improve the content of the law of conservation of momentum, and obtain the law of conservation of momentum completely.Give systemforce analysisFigure to draw specific conclusions.
Regardless of the form of collision, the vector sum of mv of two objects before and after the collision remains unchanged.The total momentum of the interacting objects is conserved as long as the system is not subject to external force, or the resultant external force is zero.
3. Experimental verification of the law of conservation of momentum: use the interaction of two sliding blocks on the air cushion guide rail to verify it.
One is divided into two verification: two sliders with equal mass interact through the metal elastic ring (the system was originally stationary, and the rubber band that tied the two sliders was burned). The experiment shows that the total momentum vector of the two sliders after action is also zero.In specific operation, use twoPhotoelectric gate(Connected to the digital timer s1 gear) Measure the time of the two sliders (that is, the time when the two sliders are equipped with a light shield of the same width passing through the photoelectric door) after the action is equal (use the "conversion" gear in the digital timer to call up the time recorded each time).
Verification of combining two into one: two objects of equal mass, one moving and the other stationary, collide and combine into one, and measure the time before and after the collision respectively.(Only one slider is equipped with a light shield).
Key points of knowledge
1. Since velocity is a vector, momentum is a vector, and its direction is the same as that of velocity, namelyp=mv.
twoimpulseIt is also a vector. The direction of impulse is the same as that of force,
F can be a constant force or a variable force.
3. Impulse theorem isTime accumulation effect of descriptive powerOf,I=mv₂-mv₁.
4. The momentum theorem can be directly derived from Newton's law of motion, so the momentum theorem is consistent with Newton's law of motion. Many problems that can be solved by Newton's law of motion can be solved by momentum theorem.In some problems, the momentum theorem is better thanNewton's law of motionIt is much easier to solve problems.
5. For a system composed of multiple interacting particles, if the system is not subject to external force or the vector sum of the external force is always zero during the force process, then the total momentum of the system is conserved.It can be expressed as:m₁v₁+m₂v₂=m₁v₁'+m₂v₂'．
content
If a system is free from external forces or the vector sum of external forces is zero, then the total momentum of the system remains unchanged. This conclusion is called the law of conservation of momentum.The law of conservation of momentum is one of the most important and universal conservation laws in nature, which is applicable tomacroscopicObjects, also applicable toMicroparticle；It is applicable to both low speed moving objects and high speed moving objects. It is an experimental law, and can also be applied to Newton's third law andMomentum theoremDerived.
Conditions for establishment
The conservation of momentum is conditional, that is, the resultant external force is zero.There are three specific types: the system is not subject to external forces at all（idealConditions);The sum of external forces acting on the system is zero, or the sum of external forces in a certain direction is zero (non ideal condition);The external force on the system is much smaller than the internal force, and the action time is extremely short (approximate condition).
Definitions and mathematical expressions
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The product of the mass of an object and its center of mass velocity is also called linear momentum.For a particle with mass m, if its velocity is v, its momentum p=mv.The momentum of a particle is a vector whose direction is the same as that of the velocity vector.For an object, its momentum P is the momentum of each mass element
The sum of
。Substitute the position vector of the center of mass C (where M is the mass of the object) into the above equation to get:
，
That is, the momentum of the object is equal to the product of the mass of the object and its centroid velocity, and its direction is the same as the direction of the centroid velocity vector.
Common units of momentum areG · cm/s、Kg · m/sEtc.
Newton's second law
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Newton's second law points out that the rate of change of body momentum is equal to the resultant force of all external forces acting on the body, that is
。
When the object is free from external force, its momentum remains unchanged (seeMomentum conservation）。The change of body momentum is particularly obvious in the collision phenomenon.
