What is Relativistic momentum: Definition and 105 Discussions

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If m is an object's mass and v is its velocity (also a vector quantity), then the object's momentum is





{\displaystyle \mathbf {p} =m\mathbf {v} .}
In SI units, momentum is measured in kilogram meters per second (kg⋅m/s).
Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is also conserved in special relativity (with a modified formula) and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity. It is an expression of one of the fundamental symmetries of space and time: translational symmetry.
Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints. In these systems the conserved quantity is generalized momentum, and in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function. The momentum and position operators are related by the Heisenberg uncertainty principle.
In continuous systems such as electromagnetic fields, fluid dynamics and deformable bodies, a momentum density can be defined, and a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids.

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  1. S

    Relativistic momentum in terms of another relativisic momentum

    I feel like this should be pretty straightforward knowing all the equations involved but my brain seems be stalling for some reason.
  2. PainterGuy

    B Converting Newtonian momentum into relativistic momentum

    Hi, I was trying to derive relativistic momentum equation using classical momentum equation but it didn't work. Could you please help me? Thank you! Where am I wrong? Or, is not possible, in any way, to derive relativistic momentum using Newtonian momentum equation? Thanks!
  3. agnimusayoti

    Differential equation on relativistic momentum (ML Boas)

    $$p=\gamma m v$$ $$F = \frac {md (\gamma v}{dt}$$ $$\int{F dt} = \int{md (\gamma v}$$ $$F t= \gamma mv$$ At this step, I don't know how to make v as explicit function of t, since gamma is a function of v too. Thankss
  4. Saptarshi Sarkar

    Relativistic momentum calculation

    The answer is required to be in terms of M,m and c only. But, I am not able to calculate the momentum of the m mass particle using the above two. Can anyone help me by telling me what I am missing?
  5. Davephaelon

    B Newtonian vs. Relativistic momentum

    Perhaps naively I assumed that for a relativistic particle the product of mass times velocity would be the same in both Newtonian and Einsteinian mechanics. The simplistic thinking was the mass increase in relativistic dynamics would balance out the non-real superluminal velocity of a particle...
  6. zehkari

    Relativistic momentum of two photons from a decay

    Homework Statement Homework Equations (1) E2 = p2c2 + m02c4 (2) E = γm0c2 (3) E = Eγ1 - Eγ2 (4) p = E / c (5) E = hf (6) λ = c / f The Attempt at a Solution a) Using eqn (1), rearranged p = (E - m0c2) / c , I obtained 2.9 MeV c-1. Not sure if I have the right answer here as I...
  7. F

    Why relativistic momentum equals the following?

    In a solution to a problem we were given, it is written that a positron momentum with energy of 2mc2 (where γ=2) is √(γ2-1)*mc = √(4-1)*mc = √3*mc How did they get that P=√(γ2-1)*mc?
  8. Curtis Cleary

    Calculating Momentum in an Observer's Frame of Reference

    Homework Statement Hi all, I'm given an electron with momentum 2.0*10-20kgm/s and was asked to convert the momentum into units of Mev/c then calculate the total energy of the electron, the lorentz factor and the speed of the electron, I did this successfully but then the question got confusing...
  9. K

    I Relativistic Momentum: Deriving an Expression

    I tried to get a relativistically correct expression of ##\int_0^t\frac{dp}{ds}dt## similar to the derivation of relativistic energy expression but I got a result which is not defined: $$\int_0^t\frac{dp}{ds}dt=\int_0^v\frac{d\left(\frac{mv}{\sqrt{1-\frac{v^2}{c^2}}}\right)}{v}$$...
  10. B

    Relativistic Momentum of photon

    Homework Statement How much work is required to accelerate a proton from rest up to a speed of 0.999c? What would be the momentum of this proton? Homework Equations p=γmv The Attempt at a Solution I got part A, which was the momentum. I found that to be 20.1 GeV. Now for part B I have to find...
  11. P

    B What is the Correct Method for Deriving p = γmv?

    I thought this mental experiment: consider an inertial frame of reference solidal to a particle moving with velocity v respect to another inertial frame of reference, i got to find the momentum in the second frame of reference (in the first is 0 since is solidal with the particle) p= m* dx/dt...
  12. Ian Baughman

    I What does this symbol represent?

    In my nuclear engineering class we are looking at relativity right now. For one of our homework problems we have to derive an equation for a particle moving at relativistic speed showing the kinetic energy in terms of the particles momentum. In the answer they have a term mc and I have no idea...
  13. R

    Pi meson decay (relativistic momentum)

    Homework Statement A charged π meson (rest mass = 273me) decays into a neutrino (zero rest mass) and a μ meson (rest mass = 207me). Find the kinetic energies of the neutrino and the mu meson. Homework Equations E = moγc2 K = mo(γ-1)c2 v = pc2/E p = moγv The Attempt at a Solution In the rest...
  14. A

    What is the speed of the centre of momentum frame in terms of c?

    Homework Statement A particle with mass m has speed 0.802c relative to inertial frame S. The particle collides with an identical particle at rest relative to frame S. Relative to S and in terms of c, what is the speed of a frame S' in which the total momentum of these particles is zero? This...
  15. A

    A De-Broglie's Fictitious wave

    I read through DeBroglie's original paper - and also a modern explanation on the same (attached). The first contradiction that DeBroglie arrives at is simple enough - he considers the 'wave-particle' as observed from a stationary frame - and from a moving frame. The 'inner frequency' of the...
  16. Elvis 123456789

    Determining beta as a function of relativistic momentum

    Homework Statement For a fast moving particle, its momentum and energy are frequently easier to measure than its velocity. a) Show that the factor of beta (as defined by β=v/c), can also be determined by measuring the ratio of relativistic momentum (p) and total energy (E). b) Sketch...
  17. G

    I Relativistic Momentum Invariance in Perpendicular Boosts

    Hi, I've found a derivation of the formula for the relativistic momentum where they considered a car crashing into a wall in the system of the car and in an inertial system that moves parallel to the wall (and therefore perpendicularly to the movement of the car). They argue that since both...
  18. J

    Is distance traveled proportional to relativistic momentum?

    Homework Statement Hi I've been modelling a particle traveling in a particle detector that has a momentum vector Px, Py, Pz which we've conveniently been using Pperpendicular (i.e. in the xy plane) and Pz. I can calculate the distance traveled in the xy plane and I need to calculate the...
  19. Oaxaca

    B DeBroglie Wavelength with Relativistic Electron

    I am trying to find the DeBroglie wavelength of an electron moving at .8c. I have never learned special relativity but I believe the momentum is affected (mass change). I used the formula p= (mv)/(1-v^2/c^2) and got a momentum of p = 2.733 E-22 and a wavelength of lamda = 2.4149 E-12. Did I...
  20. J

    Solving for velocity in relativistic momentum

    Homework Statement the relativistic momentum of an electron is 1.6 times larger than the value computed non-relativistically. What is the speed of the electron. Homework Equations Relativistic momentum p=mv/√1-v2/c2 The Attempt at a Solution Moving most of the equation around is fairly easy...
  21. EsmeeDijk

    Lorentz transformation & relativistic momentum

    Homework Statement We now specify the velocity v to be along the positive x1-direction in S and of magnitude v. We also consider a frame \overline{S} which moves at speed u with respect to S in the positive x1-direction. question 1 : Write down the transformation law for p^\mu . question 2...
  22. N

    Relativistic momentum (Lorentz boost) low velocity limit

    Hello, If I have a momenta pμ=(E,px,py,pz) and transform it via lorentz boost in x-direction with velocity v I'll get for the new 0th component E′=γE+γvpx why is this in the limit of low velocities the same as transforming the energy by a galilei transformation with velocity v? For γvpx i get...
  23. S

    Kaon Decay -- Special relativity

    Homework Statement A K0 particle is unstable and has a mass of 8.87x10-28 kg. It can decay into π+ and π- particles, each of mass 2.49x10-28 kg. Suppose that a K0 is moving in the +x direction and decays by this process, with the π+ moving off at 0.9c and the π- moving off at 0.8c. a) What was...
  24. Cell_52797

    Calculating Relativistic Momentum for Physics Students

    Homework Statement If a mass of 1 kg has a velocity of 0.7c, what is the relativistic momentum? Homework Equations I don't know what equation to use...Please help..
  25. M

    Solve Relativistic Momentum PROBLEM - Extra Factor of u Explained

    PROBLEM SOLVED - the worked example I was referring too was wrong :/ ---------------------------- Hello, I've been stuck on a question in one of my SR problem sets for some time now, and managed to find a worked solution to a similar problem online. I've attached an image of the problem (the...
  26. F

    Relativistic momentum conservation

    Homework Statement Good morning/ afternoon I have a doubt about relativistic collisions, any help will be welcome. Consider the process of annihilation electron-positron to two photons. Calculate in the centre of mass system the wavelength of photons produced using momentum of e+e-...
  27. M

    Relativistic momentum and kinetic energy

    Homework Statement What is the kinetic energy of an electron with a momentum of 40 GeV/c? The Attempt at a Solution Kinetic energy involves velocity of the particle so my first thought was to write momentum in terms of velocity. p = \frac{mv}{(1-(v/c)^2)^{1/2}} p^2 = \frac{(mv)^2}{(1 -...
  28. A

    Derivation of relativistic momentum

    Hi all, Is it possible to derive the equation p = ymv, and hence based on this, kinetic energy formula, without referring to 4-vectors or 2-dimensional collisions, that is derive it in one dimension? I tried this website/pdf but the mathematics is beyond my understanding. So could some one...
  29. V

    Please explain what is wrong with my relativistic momentum problem

    Hello! First, please, zoom in the attached file and take a look on it. Image 1 demonstrates the situation: two identical balls collide, horizontal component of velocity of both remains unchanged and vertical component becomes opposite as a result of the collision. Table in image 3 shows all...
  30. T

    Conservation of relativistic momentum for identical particles

    If a particle of mass M is at rest in a lab when it decays into 3 identical particles of mass m with: particle 1: having a velocity of 4c/5 in the -i direction vector particle 2: having a velocity of 3c/5 in the -j direction vector particle 3: having an unknown velocity in a direction defined...
  31. O

    Find relativistic momentum of electron given kinetic energy.

    Homework Statement At what energy does an electron become “relativistic”? Consider electrons with kinetic energies of 50 eV, 50 keV, and 50 MeV. For each case, calculate the momentum of the electron first using the non-relativistic formula for kinetic energy, and then using the correct...
  32. J

    Relativistic Momentum of a photon

    [b]1. An elementary particle of mass M completely absorbs a photon, after which its mass is 1.01M. (a) What was the energy of the incoming photon? (b) Why is that energy greater than 0.01Mc2? Homework Equations p (photon) = E/c p (particle) = γmv p(i) = p(f) Ek = γmc^2 -mc^2...
  33. PsychonautQQ

    Comparing relativistic momentum to classical

    EDIT: Okay I don't expect an answer for this because of my crappy attempt at LaTex, i'll work on making it look prettier sorry Homework Statement If the kinetic energy of a particle is equal to twice its rest energy, what percentage error is made by using p = mu for the magnitude of its...
  34. L

    Solving a Relativistic Momentum Problem with Differential Equations

    I've been trying to work on differential equations using several books and one of the first exercise questions I encountered already has me stuck. The momentum p of an electron at speed v near the speed c of light increases according to the formula p=mv/√(1-v^2/c^2), where m is a constant...
  35. S

    Interpret Relativistic Momentum: Facts & Popular Opinion

    I understood the derivation of relativistic momentum, but I am uncertain of how to exactly interpret it. One could interpret the arrangement of terms to be relativistic mass times velocity, and this appears to be in agreement with data from particle accelerators (or so I have been led to...
  36. R

    Relativistic Momentum: 880kg Spacecraft at 1/2 C

    Hi everyone so I have a couple of questions I would like to get some help on. The first one is as follows: A 880-kg spacecraft is traveling forward at 1/2 the speed of light. What is the relativistic momentum? I know the equations I am just having trouble plugging the numbers in I think. Thanks
  37. F

    Conservation of relativistic momentum and energy

    Homework Statement The K0 meson is an uncharged member of the particle "zoo" that decays into two charged pions according to K0 ---> π+ + π-. The pions have opposite charges as indicated, and the same mass, mπ=140MeV/c^2. Suppose that a K0 at rest decays into two pions in a bubble chamber in...
  38. D

    Help with relativistic momentum question

    Homework Statement a Pion traveling at 0.93c decays into a muon which travels in the same direction and a neutrino which travels in the opposite direction. Use conservation of energy and momentum to find the energy of the muon as determined in the rest frame of the original pion. You should...
  39. D

    Help with relativistic momentum question

    Homework Statement a D meson is at rest and decays into a Kaon and a Pion. The Kaon moves with speed 0.867c and has a mass of 0.494 GeV/C^2. The pion has a mass of 0.140 GeV/C^2. use conservation of momentum to calculate the speed of the Pion. Homework Equations Relativistic Momentum P =...
  40. 7

    Relativistic Momentum: Deriving Why p=mγv

    I have been trying to derive why relativistic momentum is defined as ##p=\gamma mv##. I set up a collision between 2 same balls (##m_1 = m_2 = m##). Before the collision these two balls travel one towards another in ##x## direction with velocities ##{v_1}_x = (-{v_2}_x) = v##. After the...
  41. P

    Relativistic Momentum Not Conserved after a Collision?

    Hello, I've just registered to ask this question. This is something that's been bothering me for some time. I have a non-elastic collision of 2 masses in 1-dimension observed by two different frames of reference, and the total momentum of the system is conserved for one of them but not for the...
  42. N

    Understanding Relativistic Momentum Transformations - Homework Help Needed

    Homework Statement Show that 1/\sqrt{1-(u'/c)^2}=\gamma(1-(vu/c^2)/\sqrt{1-(u/c)^2}Homework Equations The Attempt at a Solution Since I'm terrible with Latex I took a picture of what I have so far. I'm not sure where to go next or if I'm even on the right track. My professor gave a hint...
  43. P

    Relativistic Momentum and Energy

    Hello guys, The question is: What is the momentum, in conventional SI units, of a proton of momentum 685 MeVc-1? So, I tried two methods which yielded slightly different answers, both proximate to the actual answer (3.66 x 10-19)What I first tried to do was to rearrange: ρ=γm0v To find the...
  44. J

    A question about linear accelerator and relativistic momentum

    A 3.2km linear accelerator(linac) accelerates electrons constantly down the linac,each electron will have 50GeV of energy at the exit point. what is the speed of the electron after going 1m down the accelerator? After electrons exit the linac,magnets are used to curve the electron beams. the...
  45. P

    How do I calculate the new speed given a doubled momentum?

    Hi, I was given a question in which I had to work out the speed of an object given its momentum and its mass. Now I had to answer what would be the new speed if the momentum doubled. So, from what I understand, the rest mass can't change, it is like a constant for a given body. So, in...
  46. K

    Relativistic momentum and gamma factor - differential equation

    Homework Statement I am supposed to show that the force on a relativistic particle when a) it's perpendicular to particle's velocity is F=γm_0\frac{dv}{dt} b) it's parallel to particle's velocity is F_x=m_0γ^3\frac{dv_x}{dt} I may make use of the fact that...
  47. K

    Relativistic Momentum problem

    Homework Statement Mass of particle = 5GeV/C^2 Momentum of particle = 5GeV/c Find total energy of particle and speed of particle. Homework Equations E=λmc^2 p=λmu λ=(√(1-(u/c)^2))^-1 The Attempt at a Solution When I try so solve for speed, I realize λ is a function of u, so...