Momentum

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In Newtonian mechanics, linear momentum , momentum translation , or just momentum (pl. Momenta) is the product of the mass and velocity of an object. It is a vector quantity, has strength and direction in three-dimensional space. If m is the mass of the object and v is the velocity (also vector), then the momentum is

${\ displaystyle \ mathbf {p} = m \ mathbf {v},}$

In SI units, measured in kilograms of meters per second (kg? M/s). Newton's second law of motion states that the rate of body change in momentum equals the total force acting on it.

Momentum depends on the frame of reference, but within the framework of inertia it is the quantity conserved , which means that if the closed system is not influenced by external forces, the total linear momentum is unchanged. Momentum is also preserved in special relativity, (with a modified formula) and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity. This is the expression of one of the fundamental symmetries of space and time: the symmetry of translation.

Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics, allow one to choose a coordinate system that combines symmetry and boundaries. In this system the quantity preserved is general momentum , and in general it is different from the kinetic momentum defined above. The concept of general momentum is brought into quantum mechanics, where it becomes the operator of the wave function. Momentum and operator positions are related to Heisenberg's uncertainty principle.

In continuous systems such as electromagnetic fields, fluids and objects that can change shape, momentum density can be determined, and the continuum version of momentum conservation leads to equations such as the Navier-Stokes equation for fluids or Cauchy momentum equations for deformable solids or liquids.

Video Momentum

Newtonian

Momentum is a vector quantity: it has a magnitude and direction. Since momentum has a direction, it can be used to predict the direction and speed of the movement of objects produced after they collide. Below, the nature of the momentum is explained in one dimension. The vector equation is almost identical to the scalar equation (see some dimensions).

Single particle

Momentum partikel secara konvensional diwakili oleh huruf p . Ini adalah produk dari dua kuantitas, massa partikel (diwakili oleh huruf m ) dan kecepatannya ( v ):

${\ displaystyle p = mv.}  ÂÂ$

The momentum unit is the product of mass units and speed. In SI units, if the mass in kilograms and speed in meters per second then the momentum is in kilograms of meters per second (kg? M/s). In cgs units, if the mass is in grams and velocities in centimeters per second, then the momentum is in grams of centimeters per second (g? Cm/s).

Being a vector, momentum has strength and direction. For example, a 1 kg model airplane, traveling northward at 1 m/s in straight flights and levels, has a momentum of 1 kg/m/s since the north is measured by reference to the ground.

Many particles

Momentum sistem partikel adalah penjumlahan dari momentum mereka. Jika dua partikel memiliki massa masing-masing m ₁ dan m ₂ , dan kecepatan v ₁ dan v ₂ , momentum totalnya

${\ displaystyle {\ begin {aligned} p & amp; = p_ {1} p_ {2} \\ & amp; = m_ {1} v_ {1} m_ {2 } v_ {2} \,. \ end {aligned}}}  ÂÂ$

Momen lebih dari dua partikel dapat ditambahkan lebih umum dengan yang berikut:

${\ displaystyle p = \ sum _ {i} m_ {i} v_ {i}.}  ÂÂ$

Sebuah sistem partikel memiliki pusat massa, sebuah titik yang ditentukan oleh jumlah tertimbang dari posisi mereka:

${\ displaystyle r _ {\ text {cm}} = {\ frac {m_ {1} r_ {1} m_ {2} r_ {2} \ cdots} { m_ {1} m_ {2} \ cdots}} = {\ frac {\ jumlah \ limit _ {i} m_ {i} r_ {i}} {\ jumlah \ batas _ {i} m_ {i}} }.}  ÂÂ$

Jika semua partikel bergerak, pusat massa secara umum akan bergerak juga (kecuali sistem berada dalam rotasi murni di sekitarnya). Jika pusat massa bergerak dengan kecepatan v _cm , momentumnya adalah:

${\ displaystyle p = mv _ {\ text {cm}}.}  ÂÂ$

This is known as Euler's first law.

Relationship with style

Jika gaya total yang diterapkan pada partikel adalah konstanta F , dan diterapkan untuk interval waktu ? t , momentum partikel berubah dengan jumlah

${\ displaystyle \ Delta p = F \ Delta t \,.}  ÂÂ$

Dalam bentuk diferensial, ini adalah hukum kedua Newton; tingkat perubahan momentum partikel sama dengan gaya seketika F yang bekerja padanya,

${\ displaystyle F = {\ frac {dp} {dt}}.}  ÂÂ$

Jika gaya total yang dialami oleh partikel berubah sebagai fungsi waktu, F (t) , perubahan momentum (atau impuls J ) antara waktu t ₁ dan t ₂ adalah

${\ displaystyle \ Delta p = J = \ int _ {t_ {1}} ^ {t_ {2}} F (t) \, dt \ ,.}  ÂÂ$

The impulse is measured in units of derivatives of the second newton (1 N? S = 1Ã, kg? M/s) or dyne seconds (1 dyne? S = 1 g? M/s)

Di bawah asumsi massa konstan m , itu setara dengan menulis

${\ displaystyle F = {\ frac {d (mv)} {dt}} = m {\ frac {dv} {dt}} = ma,}  ÂÂ$

then the total force is equal to the particle mass, its acceleration times.

Example : The 1 kg mass model plane accelerates from rest to speed of 6 m/sec as north within 2 seconds. The total force required to produce this acceleration is 3 newtons to the north. The momentum change is 6 kg/m/s. The rate of momentum change is 3Ã, (kg? M/s)/s = 3Ã, N.

Preservation

In a closed system (which does not exchange any problem with its environment and is not acted upon by external forces), its total momentum is constant. This fact, known as the momentum conservation law, is implied by Newton's laws of motion. Suppose, for example, that two particles interact. Because of the third law, the forces between them are the same and opposite. If the particles are numbered 1 and 2, the second law states that F ₁ = dp ₁ / dt and F > ₂ = d ₂ / dt . Therefore,

${\ displaystyle {\ frac {dp_ {1}} {dt}} = - {\ frac {dp_ {2}} {dt}},} Â Â$

dengan tanda negatif yang menunjukkan bahwa kekuatan menentang. Secara setara,

${\ displaystyle {\ frac {d} {dt}} \ kiri (p_ {1} p_ {2} \ right) = 0.}  ÂÂ$

Jika kecepatan partikel adalah u ₁ dan u ₂ sebelum interaksi, dan setelah itu mereka v ₁ dan v ₂ , lalu

${\ displaystyle m_ {1} u_ {1} m_ {2} u_ {2} = m_ {1} v_ {1} m_ {2} v_ {2}.}  ÂÂ$

This law applies no matter how complicated the force is between the particles. Similarly, if there are multiple particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and segregation caused by explosive forces. It can also be generalized to situations where Newton's law does not apply, for example in the theory of relativity and in electrodynamics.

Dependence on the terms of reference

Momentum is a measurable quantity, and the measurement depends on the observer's motion. For example: if an apple sits in a downed glass elevator, an outside observer looks at the elevator, sees the apple moving, so for the observer the apple has a nonzero momentum. For someone in the elevator, the apple does not move, so, it has zero momentum. Both observers each have a frame of reference, where they observe the movement, and, if the elevator goes down steadily, they will see behavior consistent with the same physical law.

Misalkan sebuah partikel memiliki posisi x dalam kerangka acuan stasioner. Dari sudut pandang kerangka acuan lain, bergerak dengan kecepatan seragam u , posisi (diwakili oleh koordinat terpusat) berubah seiring waktu sebagai

${\ displaystyle x '= x-ut \ ,.}  ÂÂ$

Ini disebut transformasi Galilea. Jika partikel bergerak dengan kecepatan dx / dt = v dalam bingkai referensi pertama, di detik, itu bergerak dengan kecepatan

${\ displaystyle v '= {\ frac {dx'} {dt}} = v-u \ ,.}  ÂÂ$

Karena u tidak berubah, percepatannya sama:

${\ displaystyle a '= {\ frac {dv'} {dt}} = a \ ,.}  ÂÂ$

Thus, momentum is preserved in both terms of reference. In addition, as long as the style has the same shape, in both frames, Newton's second law has not changed. A force like Newton's gravity, which relies only on the scalar distance between objects, satisfies this criterion. This independent frame of reference is called Newton's relativity or Galilean invariance.

Changes to the terms of reference, can, often, simplify the calculation of motion. For example, in a two-particle collision, a reference frame can be selected, in which, one particle begins at rest. Another commonly used reference frame, is the center of the mass frame - which moves with the center of mass. In this frame, the total momentum is zero.

Apps to collision

By itself, the law of conservation of momentum is insufficient to determine the movement of particles after a collision. Other properties of motion, kinetic energy, must be known. This is not necessarily preserved. If conserved, the collision is called elastic collision ; if not, it is an inelastic collision .

Elastic collision

Elastic collision is one in which no kinetic energy is absorbed in a collision. Very elastic "collisions" can occur when they are not touching each other, such as in atomic or nuclear scattering where electrical repulsion sets them apart. A catapult maneuver from satellites around the planet can also be seen as a perfectly elastic collision. The collision between two pool balls is a good example of a nearly completely elastic collision, because of its high stiffness, but when the body comes in contact there is always dissipation.

Changes to the terms of reference can simplify crash analysis. For example, suppose there are two bodies with the same mass m , one stationary and another approaching the other at speed v (as shown). The center of the mob is moving at the speed of v / 2 and the two bodies move towards it at the speed of v / 2 . Because of symmetry, after the collision they must move away from the center of mass with the same speed. Adding the mass center velocity to both, we find that the moving body now stops and the other moves away at the speed of v . The bodies have been exchanging speed. Regardless of the speed of the body, the transition to the center of the mass frame leads us to the same conclusion. Therefore, the final velocity is given by

${\ displaystyle {\ begin {aligned} v_ {1} & amp; = u_ {2} \\ v_ {2} & amp; = u_ {1} \ ,. \ end {aligned}}}$

If one body has a mass that is much larger than the other, its speed will be less affected by the collision while the other body will undergo major changes.

Inelastic collision

In an inelastic collision, some of the kinetic energy of the colliding body is converted into another form of energy (such as heat or sound). Examples include traffic collisions, where the effects of kinetic energy lost can be seen in damage to the vehicle; electrons lose some of their energy to atoms (as in the Franck-Hertz experiment); and particle accelerators in which the kinetic energy is converted into mass in the form of new particles.

In perfect inelastic collision (like insects that hit the windshield), both bodies have the same motion afterward. If one body does not move to start, the equation for momentum conservation is

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begitu

${\ displaystyle v = {\ frac {m_ {1}} {m_ {1} m_ {2}}} u_ {1} \ ,.}  ÂÂ$

In terms of reference moving at the speed of v ), the objects are hit by a collision and 100% of kinetic energy is converted into other energy forms.

One of the sizes of collision inelasticity is the restoration coefficient of C _R , which is defined as the relative speed ratio of the separation to the relative velocity of the approach. In applying this size to the ball that bounces off the solid surface, it can be easily measured using the following formula:

${\ displaystyle C _ {\ text {R}} = {\ sqrt {\ frac {\ text {bounce height}} {\ text {drop height} }}} \,.} Â Â$

Momentum and energy equations also apply to the movement of objects that begin together and then move apart. For example, an explosion is the result of a chain reaction that converts potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. The rocket also utilizes momentum preservation: propellants are pushed out, gaining momentum, and the same and opposite momentum is given to the rocket.

Multiple dimensions

Gerak nyata memiliki arah dan kecepatan dan harus diwakili oleh vektor. Dalam sistem koordinat dengan x , y , z sumbu, kecepatan memiliki komponen v _x dalam x -direction, v _y dalam y -direction, v _z dalam z -direction. Vektor diwakili oleh simbol tebal:

${\ displaystyle \ mathbf {v} = \ kiri (v_ {x}, v_ {y}, v_ {z} \ kanan).}  ÂÂ$

Demikian pula, momentumnya adalah kuantitas vektor dan diwakili oleh simbol tebal: