What is Momentum: Definition and 1000 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





p

=
m

v

.


{\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. E

    I Bernoulli and Momentum Disconnect?

    I was playing around, and I found something unexpected. If we are analyzing a simple fluid jet: We can apply Bernoulli's (which is Conservation of Energy) and arrive at: $$ P_{1_{B}} = \frac{1}{2} \rho \left( v_2^2 - v_1^2 \right) = \frac{1}{2} \rho ( v_2 - v_1 ) ( v_2+ v_1) $$ It would...
  2. A

    Momentum in a perfectly inelastic collision

    I calculated:arctan(fy/13.0)=55=>fy=18.566 m/s Then I calculated, using the momentum equation: m1viy+m2v2iy=(m1+m2)vfy=> mv2i=2*m*18.566=>v2=37.132 m/s I thought that because the cars were stuck together, the kinetic energy from the northbound car would be lost. So, the speed would have...
  3. paulimerci

    Find the magnitude of the momentum change of the ball?

    I understand that it is a 2D momentum problem with an elastic collision; Looking at the vector diagrams below, I notice that the velocity vectors initial and final in the y direction are in the same direction, indicating that momentum does not change, whereas the velocity vectors initial and...
  4. H

    B Spin And Angular Momentum of Large Objects

    I read that quantum spin is the measure of the angular momentum of a quantum object. Suppose you have a rotating Thing 1. Quantum objects bounce off of it then collide with Thing 2. Will this transfer angular momentum from Thing 1 to 2, causing it to rotate?
  5. uxioq99

    Time Independence of the Momentum Uncertainty for a Free Particle Wave

    Mine is a simple question, so I shall keep development at a minimum. If a particle is moving in the absence of a potential (##V(x) = 0##), then ##\frac{\langle\hat p \rangle}{dt} = \langle -\frac{\partial V}{\partial x}\rangle=0## will require that the momentum expectation value remains...
  6. paulimerci

    Find momentum transfer and force on the head with and without a helmet

    Without helmet, m = 4kg, ##v_f = 10m/s## ##\Delta t = 0.0005sec## $$ \Delta p = mv_f - mv_i $$ $$ \Delta p_{without} = 40 kg m/s$$ $$ Impulse_ {without} = F_{net} \cdot \Delta t$$ $$ force_{without} = 80000 N$$ With helmet m = 4kg, ##v_f = 10m/s## ##\Delta t = 0.002sec $$ \Delta p = mv_f -...
  7. C

    Puck collision with rod using angular momentum conservation

    For this problem, Why for part (a) the solution is, Is the bit circled in red zero because since the putty is released at a very small distance above the rod it velocity is negligible? Also for part (d) the solution is I did a computation of the initial and finial kinetic energies of the...
  8. 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.
  9. L

    A Going from Cauchy Stress Tensor to GR's Energy Momentum Tensor

    Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units? Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR? If so, What...
  10. chris25

    Which system to apply conservation of momentum to?

    For this problem I was very confused whether conservation of angular momentum should be applied to the person, the swing or the person-swing system. It seems to me that there is no net torque on any of the three systems I listed above. However, it seems that the angular momentums of the three...
  11. D

    B Greater momentum on impact means greater force?

    Sorry for this beginner's question, but...if F=ma, then force is all about acceleration. But if vehicle A moving at constant velocity V hits a wall, and vehicle B moving at constant velocity greater than V hits the wall, then B hits the wall with greater momentum than A and does greater damage...
  12. H

    I Momentum and Action: Understanding Lagrangian Mechanics

    Hi, In my book I have and expression that I don't really understand. Using the definition of action ##\delta S = \frac{\partial L}{\partial \dot{q}} \delta q |_{t_1}^{t_2} + \int_{t_1}^{t_2} (\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}) \delta q dt## Where L...
  13. G

    I How can I integrate variable velocity in fluid mechanics?

    Do you know of any place where I can look up things about the momentum (linear momentum) in fluid mechanics? It's just that when I have a variable velocity and it has to be integrated, I don't quite understand how to do it. I have looked for videos and things and I can't find that they are...
  14. haha0p1

    Inelastic collisions with constant momentum

    Kinetic energy before collision =1/2 mv² + 1/2 mv² = mv² (since energy is a scalar quantity, the direction does not matter). Kindly tell why am I not getting the required answer i.e: 1/2 mv². Am I doing the calculation wrong?
  15. uSee2

    Experimental Design: Pulley and Mass Hangers

    ^ This is my personal drawing of the diagram, I couldn't take a picture of the actual one. The setup is a pulley wrapped with a cord and mass hangers attached to each end. My first thought when approaching this problem was to first determine the rotational inertia of the pulley, then use some...
  16. uSee2

    Explosion of 2 Carts on a Platform (Momentum)

    My Explanation: This system is a closed system, so the center of mass velocity stays constant. It was initially at rest so the position of the center of mass is constant. After their collision, the 2 carts are to the right of x = 0. Center of mass originally was at x = 0, so the platform had to...
  17. gggnano

    Surely this will NOT work: violation of conservation of momentum?

    The rotating ball should push the vehicle first to the right and once it hits the airbag - to the left?? Even if this works, how are you going to automate it and repeat it?
  18. N

    Average value of components of angular momentum for a wave packet

    I have typed up the main problem in latex (see photo below) It seems all such integrals evaluates to 0, but that is apparantly unreasonable for in classical mechanics such a free particle is with nonzero angular momentum with respect to y axis.
  19. S

    Calculate orbital angular momentum

    The section Kepler’s Second Law here describes the above equation. In this problem, ##\text{r = D, m = M and v = V}## What is the way to go about finding out ##\theta## as shown in Figure 13.21?
  20. A

    Electromagnetic linear momentum for a system of two moving charges

    When you write out the equations of motion for a system of two isolated charges, you can add both of the equations and get the increase in the particles linear momentum on one side. On the other side, you get the sum of all the forces between the particles. I understand that this sum of forces...
  21. Ahmed1029

    I How is photon momentum compatible with special relativity?

    In relativity, momentum of a body is given by ##p=mv/\sqrt{1-v^2/c^2}##, but if mass is exactly zero and velocity is exactly ##c##, how is the photon momentum even defined? I don't think this problem can be resolved by simply stating the other formula relating energy to momentum, since it was...
  22. Spector989

    System of particles, impulse and conservation of angular momentum

    So i was able to solve the angular velocity part but i don't know how to find the velocity of centre of mass . For the first part i simply conserved momentum about COM because if i consider the particles as a part of the same system as rod the collision are internal forces . I am mainly...
  23. phos19

    I How do I check if the canonical angular momentum is conserved?

    Specifically given a purely magnetic hamiltonian with some associated vector potential : $$ H = \dfrac{1}{2m} (\vec{p} - q\vec{A}) $$ How can I deduce if $$ \vec{L} = \vec{r} \times \vec{p}$$ is conserved? ( $$\vec{p} = \dfrac{\partial L}{\partial x'}$$, i.e. the momentum is canonical)
  24. V

    Satellite mechanics: linear and rotational momentum

    [This is a continuation of OP's thread here: https://www.physicsforums.com/threads/satellite-mechanics-linear-and-rotational-momentum.1046963/ ] satellite mechanics: linear and rotational momentum I'm trying to better understand classical mechanics, and came up with a question: Say we have a...
  25. H

    Conservation of Momentum of Rocket Exploding after Takeoff

    -Solved for vf using equation 3 to get 20.0m/s (speed before explosion) then solved for the distance to reach the explosion using equation 4, to get 20.0m, which felt wrong having the same numbers but that may just be coincidence. -Found the distance travelled of the lighter piece using 530m -...
  26. Y

    Calculate the angular momentum of this particle in rotational motion

    i,j,k arevector I know L=P*r=m*v*r=m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k=mabωk. but why m(acosωti+bsinωtj)*(-aωsinωti+bωcosωtj)=mabw((cos^2)ωt+(sin^2)ωt)k.I need some detail. please help me.
  27. Q

    Integration of structure function F2 to calculate quark momentum

    I study particle physics with “Particles and Nuclei” / Povh et al. and “Modern particle physics” / Mark Thomson and I am currently at “Deep-Inelastic scattering”. After introducing several scattering equations, such as Rosenbluth, that all include terms for electric AND magnetic scattering, i.e...
  28. Spector989

    Conservation of momentum and mechanical energy on an inclined plane

    So i am tried to conserve momentum and use conservation of mechanical energy but won't there be psuedo force acting on the block if i am solving from non inertial frame ?. If i ignore the pseudo force and simply use C.O.M.E and include the K.E of the wedge and solve normally i do get the...
  29. B

    Why does an electron have minimum kinetic energy when its momentum is 2h/λ?

    Solution given: The minimum kinetic energy electrons will arise from a change in photon energy on scattering that is a minimum and this will arise from the smallest wavelength change of the photon. The Compton scattering formula is ∆λ = (h/mc)(1 − cos φ) which is minimised when 1 = cos φ. This...
  30. V

    I Satellite mechanics: linear and rotational momentum

    satellite mechanics: linear and rotational momentum I'm trying to better understand classical mechanics, and came up with a question: Say we have a squared satellite weighting 100kg, 1 meter on each side. it has a thruster on it's side, shown in picture thruster quickly ejects 100g of propellant...
  31. O

    Using Momentum, KE and PE to solve this skier velocity problem

    See a picture of the question above. My thoughts are: dp(y)/dy is negative such that when going up the slope, the momentum in the y direction is equal to 0 just as the skier reaches the top of the circular section. Given that there is no friction on the slopes, the energy of the skier...
  32. A

    Orbital angular momentum Hamiltonian

    I think that the quantum numbers are l=1 and ml=0, so I write the spherical harmonic Y=Squareroot(3/4pi)*cos(theta). I would like to know how to compute the wave function at t=0, then I know it evolves with the time-evolution operator U(t), to answer the first request.
  33. G

    Tainter Damper Figure: Analyzing Forces

    Figure: Attempt at a solution: $$b=12\, \textrm{m},\quad H=8\, \textrm{m}$$ a) $$F_H=p_{CG}A=3767040\, \textrm{N}=\boxed{3767,04\, \textrm{kN}}$$ $$A=8\cdot 12=96\, \textrm{m}^2$$ $$p_{CG}=\rho_g h_{cg}=39240\, \textrm{Pa}$$ b) $$F_V=mg=\rho_g V$$ We calculate ##\theta \rightarrow 8=10\cdot...
  34. VVS2000

    I Other ways of finding expectation value of momentum

    Apart from the usual integral method, are there any other ways to find expectation value of momentum? I know one way is by using ehrenfest theorem, relating it time derivative of expectation value of position operator. Even using the uncertainty principle, we might get it if we know the...
  35. VVS2000

    Expectation value in momentum space

    so from Fourier transform we know that Ψ(r)=1/2πℏ∫φ(p)exp(ipr/ℏ)dp I proved that <p>= ∫φ(p)*pφ(p)dp from <p>=∫Ψ(r)*pΨ(r)dr so will the same hold any operator??
  36. person123

    I Momentum of a Water Jet Impacting Plate

    Suppose you have a jet of fluid (say water) traveling vertically upward at a constant velocity. It impacts a stationary horizontal plate and so moves radially outward in all directions. Assume that there's no energy loss during the impact, so the speed of the fluid remains constant. Is momentum...
  37. Christian Thom

    I Double-slit experiment with momentum entangled pair of photons

    In Kaur, M., Singh, M. Quantum double-double-slit experiment with momentum entangled photons. Sci Rep 10, 11427 (2020). https://doi.org/10.1038/s41598-020-68181-1 and in C. K. Hong and T. G. Noh, "Two-photon double-slit interference experiment," J. Opt. Soc. Am. B 15, 1192-1197 (1998) it is...
  38. SMOKEYWC

    Conservation of momentum (wrecking ball hits a stationary object)

    I have a wrecking ball with a mass of .5kg traveling at 3.03 m/s that hits a stationary block .9 meters high, weighing .06kg. I calculated the ball's exit velocity after it hits the block to be -3.00 m/s . I calculated the final velocity of th block to be 4.2 m/s Vf = Sqrt 2(g)(h) = sqrt...
  39. Math Amateur

    I The Inertia Tensor .... Determining Components of Angular Momentum ....

    I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in following his logic regarding proceeding to derive the components of Angular Momentum and from there the components of the Inertia Tensor ... On page 36 we read the following: In the above text...
  40. O

    Understanding the Steps to Relativistic Momentum

    How does this end up +? Can't work out steps.
  41. sachin

    Choosing what consists of a "system" in Newton's laws of motion

    The question is solved in a single step by taking the blocks as a system and using conservation of linear momentum in the horizontal direction as there is no net force acting in the horizontal direction. Conserving the momentum we get, m x v + M x 0 = (m+M)v', so,,v' = mv/(m +M).where v' is the...
  42. D

    Getting wrong answer in an (angular) impulse momentum problem

    I have tried this same approach three times and I got the same answer. I can't figure out what's wrong. Btw answer is 12mu/(3+cos2α) And yes, sorry for my shitty handwriting. If you can't understand the reasoning behind any step then please let me know.
  43. gggnano

    I Is it possible to "violate" momentum at the expense of more energy?

    This is in fact a shamelessly simple question to a point the reason it puzzles me is because it's too simple: So basically you have a closed empty/hollow cylinder filled with either gas or even an ordinary solid ball...and then on the left side of the cylinder you put a force on the "fuel"...
  44. E

    A Probability flux integrated over all space is mean momentum?

    In Sakurai Modern Quantum Mechanics, I came across a statement which says probabiliy flux integrated over all space is just the mean momentum (eq 2.192 below). I was wondering if anybody can help me explain how this is obtained. I can see that ##i\hbar\nabla## is taken as the ##\mathbf{p}##...
  45. G

    Tong QFT sheet 2, question 6: Normal ordering of the angular momentum operator

    My attempt/questions: I use ##T^{0i} = \dot{\phi}\partial^i \phi##, ##\dot{\phi} = \pi##, and antisymmetry of ##Q_i## to get: ##Q_i = 2\epsilon_{ijk}\int d^3x [x^j \partial^k \phi(\vec{x})] \pi(\vec{x})##. I then plug in the expansions for ##\phi(\vec{x})## and ##\pi(\vec{x})## and multiply...
  46. Fatboyx

    I Pulling cable through underground ductwork.

    Hello all, I am an Engineering dropout turned Cable Splicer. In my job we do a lot of Heavy Duty underground cable pulling. Usually plastic jacketed cable through some type of ductwork (typically plastic as well). We use a winch truck and a heavy rope to pull this cable through the ducts...
  47. B

    Velocity of a relativistic particle in a uniform magnetic field

    d(ɣmv)/dt = qvB (dɣ/dt)mv + ɣm(dv/dt) = qvB Substituting gamma in and using the chain rule, it ends up simplifying to the following: ɣ^3*m(dv/dt) = qvB Now, I am confused on how to solve for v.
  48. M

    B Why is KE not conserved when momentum is?

    Its clear in elastic collision that both KE and momentum is conserved. Bodies exchange their velocities. It is seen clearly in this video. There is no decrease in speed. Total KE is constant. But in an inelastic collision momentum is conserved again but not the KE. There is loss in KE (I guess...
  49. mr_sparxx

    I Kepler's second law derivation from angular momentum conservation

    Many texts state that in an elliptic orbit you can find angular momentum magnitude as $$ L = r m v = m r^2 \frac {d \theta} {dt} $$ I wonder if $$ v = r \frac {d \theta} {dt} $$ is valid at every point. I understand this approximation in a circumference or radius r but what about an arc...
  50. C

    I Angular momentum and turning a bicycle

    Hello everyone! I've been watching the following Walter Lewin lecture, the part that illustrates my question is part 17:19 of the video Most things have made sense during this lecture, but one persistent question I have is the following: why does the bicycle tilt toward the inside of the...
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