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. I_Try_Math

    Velocities of four masses | Conservation of Momentum

    ##\vec{\rho_{D1,i}}+\vec{\rho_{D2,i}}+\vec{\rho_{D3,i}}+\vec{\rho_{R,i}} = \vec{\rho_{D1,f}} +\vec{\rho_{D2,f}} +\vec{\rho_{D3,f}} +\vec{\rho_{R,f}}## ##\vec{0} = \vec{\rho_{D1,f}} +\vec{\rho_{D2,f}} +\vec{\rho_{D3,f}} +\vec{\rho_{R,f}}## ##\vec{0} = m_D(15\hat{i} + 5\hat{j}) + m_D(-12\hat{i} +...
  2. I_Try_Math

    Average momentum of an avalanche

    ##m_s## = mass of snow ##V_s## = volume of snow ##\vec{v}## = velocity of snow D = density of snow ##\rho_{avg} = \frac{\rho}{V_s}## ##=\frac{m_s \vec{v}}{V_s}## ##=\frac{V_s D \vec{v}}{V_s}## ##=D \vec{v}## ##=350 \frac{1,000}{5.5}## ##=63,636.4## The textbook's answer is ##1.3 \times 10^9##...
  3. I_Try_Math

    B More convenient mathematical notation for a simple use case

    So in my textbook there's a basic problem where you solve for the final velocities of two hockey pucks, which happen to have different colors which are red and blue, using conservation of momentum. The notation that the textbook uses to express the final velocities of the pucks is ##v_{1,f}##...
  4. W

    Conservation of Momentum FRQ

    (bottom graph relates only to c) (a) i. The students can calculate the area under the graph to find the impulse exerted on the block. This is because the area under a force vs. time graph is the change in momentum or the impulse. ii. Knowing that the graph is linear and begins at around 3 N...
  5. S

    I Einstein - de Haas Effect

    This effect is (apparently) always explained in terms of a "book-keeping" need to conserve angular momentum. I totally get that (as the kids say these days), but it doesn't provide a chain of cause and effect that leads to the observed rotation of the iron rod. Is there a classical thought...
  6. S

    Collison between two particles

    I am stuck with this problem. Intuition tells me the answer is no, but I am struggling to prove it. If we consider two particles travelling in the same direction, the 2nd particle will gain velocity (impulse is in same direction to velocity), and the first particle will lose velocity (if it...
  7. S

    Fast and Furious Scene Analysis

    The collision seems to be an inelastic collision meaning momentum is conserved however, energy is not due to likely thermal energy from the collision. Using conservation of momentum, we can maybe somehow find the initial momentum of each vehicle and set it equal to the final momentums? However...
  8. H

    British physics olympiad problem: A ball bearing bouncing off a steel cylinder

    I am struggling to find correct approximation for the problem. Is momentum conserved at the immediate impact of... (1) Can I ignore gravitational force and potential for the spring which is connected to ground and vertically upholding a mass . ( using equilibrium)
  9. G

    B Ignoring the motion of the Earth for energy vs. momentum conservation

    Hi. If I drop an inelastic body, its potential energy first gets converted to kinetic, then to deformation energy. We use conservation of energy without taking into account the kinetic energy gain of the earth during the fall. However, at first sight conservation of momentum seems to be...
  10. S

    I Maxwell's equations and the momentum of charge

    There appears to be a conservation of charge momentum (qv) analogous to that for mass (mv) although in the case of charge it is more potential in nature. A change in the flow of charge (or current) produces changing magnetic and electrics fields according to Maxwell's equations. These in...
  11. M

    I What part of momentum moves the boat in the inclined plane paradox experiment?

    Hello and thank you for welcoming me to your forum! To get started, I would like to give you a little help: In the pattern experiment, when firing the cannon, if part of the momentum reaches the bird, and if another part of the momentum manages to stun the fish, what part of the momentum would...
  12. S

    B Does shortness from length contraction have a real physical effect?

    Do the contractions affect physics in any frame? Examples: If length contraction reduces mass in the direction of motion, and therefore reduces the total momentum. (from observer's perspective) If in the reference frame of a station, the moving train weighs less than it did when parked...
  13. pineapplebanana

    Elastic Collision troubles...

    So i started off breaking up the problem into two sequences, right before the collision and after the collision has happened. I need to find the first ball's speed immediately before the collision which is no problem. PEi = KEf > mghi = (1/2)mvf (vf being the velocity right before the collision...
  14. A

    Collision problem, two marbles, one starts moving and one starts at rest, find the speeds of both after the collision

    Solved equation 1 for v1f and then substituted into equation 2 and solved for v2f. Got 2.22 as the answer, but it said the answer is incorrect.
  15. resurgance2001

    Conservation of momentum for a robot on a space platform

    The momentum of the robot is 95.0 x 1.4 m/s towards the platform. This must be equal and opposite to the momentum imparted to the beam. Dividing 133 kg m/s by 330.0 Kg gives a velocity of 0.403 m/s for the beam. So the relative velocity of the robot relative to the platform is 1.40 - 0.403 =...
  16. cianfa72

    I About the use of nominal definitions in physics

    Hi, I was reading the interesting lecture of Feynman about Characteristics of Force -- https://www.feynmanlectures.caltech.edu/I_12.html He basically says that nominal definitions like mathematical definitions of "abstract" objects have actually no physical meaning. For instance take the...
  17. Kyuubi

    Expected value of momentum P in terms of k

    Now if I'm given a ##\phi(k)##, and I'm asked to find ##\langle p \rangle##, ##\langle p^2 \rangle##, etc. Am I justified to say that ##\langle p \rangle = \hbar \langle k \rangle## and that ##\langle p^2 \rangle = \hbar^2 \langle k^2 \rangle## ?
  18. R

    Rocket propelled by a beam of photons

    Before the engine is switched off: $$P_{Initial rocket} = (m_{i}c, 0)$$ $$P_{photons} = (E/c, -p_{photons})$$ where E is the energy of the photons and ##p_{photons}## is the 3 momentum of the photons. Rocket after engine switched off: $$P_{Final rocket} = (m_{f}c, m_{f}v)$$ By conservation of...
  19. Spooky123

    Using Momentum Principle to Find Ratio of Speeds?

    Given that the ions are initially at rest my initial velocity is 0. Therefore my Vavg is equal to vf/2 Using the formula Vavg = Change in positon/time, I can solve vf to be equal to 2r/t. Using the momentum principle, I get an equation of 2r/t = FnetT/12m -> Given that the mass of the ion is...
  20. W

    Attempting to teach this: Momentum and Impulse

    For question 1. I am stuck. I know that the equation involves time and possibly rate, should solve for distance. But not sure how to set it up with information given. 2. Ft= m 🔺️ v F(3)= (100kg)(30m/s) 3 s= 3000 kg m/s Same applied to question 3. 3. F(2)= (100kg)(-30m/s) F(2) = -3000 kg m/s...
  21. H

    Change in momentum: Child jumping from a swing on a playground

    800 - (32 x 9.8) = 32v/0.18 where v = velocity this gives me v = 2.736 m/s The answer given, however, is 800 = 32v/0.18, i.e. v = 4.5 m/s The difference, of course, is the weight of the child. I don't understand why this is not allowed for in the net force acting on the child. Can someone put me...
  22. L

    Why is momentum equal to mass times velocity?

    I tried searching on the internet for hours to find an answer, but I didn't find any.
  23. G

    I Question about the velocity of the center of mass reference frame

    I'm looking into center of mass and I saw the derivation of: ## V = \frac{\sum\limits_{i = 1}^{n} m_iv_i}{\sum\limits_{i = 1}^{n} m_i} ## I understand how it's derived, so no need to explain this further. It's a velocity of the frame in which total momentum of our objects is zero. Forget...
  24. C

    Spring momentum conservation problem

    For this problem, The reason why I am not sure whether it is a valid assumption whether momentum is conserved because during the collision if we consider the two masses to be the system, then there will be a uniform gravitational field acting on both masses, and a spring force that is acting...
  25. Lotto

    B What will be the ball's velocity after a perfectly elastic collision?

    From the bus driver's point of view, who is at rest, the ball's initial velocity is ##u+v##. After the collision, its velocity has to have the same value, but an opposite direction, so ##-(u+v)##. So that means that relative to me standing on the ground at rest, the ball's new velocity is...
  26. S

    Why is momentum not conserved?

    Here is question + drawing.
  27. revix

    I Why momentum is conserved when a gun fires? (conceptual question)

    I understand that conservation of motion comes from the action and reaction pairs of newton's third law. When it is triggered, two forces appear that cancel when analyzed as a system. My question is how is it that momentum is conserved if before the shot there was no force in the system and...
  28. D

    I Can We Accurately Measure Momentum in the Lab?

    In the lab, how accurately can we measure momentum? What is the max value of the uncertainty in position as the uncertainty in momentum approaches zero? Or vice versa. What experiments do these types of measurements?
  29. L

    I Angular Momentum problem v2 (mass moving inward or outward)

    Hello, simplified the Angular momentum problem that comes up when i try to solve a mass moving inward or outwards and it does not conserver the angular momentum properly. I have tried this is many software by now, or by someone else and we all have found that there is no angular momentum...
  30. F

    Why is the thrust equation same under gravitational force?

    The homework statement isn't exactly as is mentioned above. The actual problem statement is as follows: This is problem 3.8 from John R. Taylor's Classical Mechanics; however, my question is not related to the main problem itself but one particular aspect of it. Now, in the same textbook (John...
  31. M

    Find power needed to fly this airplane using momentum considerations

    I just don't understand should I take u relative to the plane or relative to the ground. I tried to solve it like this: $$p_{final}=m_{0}v-m(u-v)-M(u-v)$$ $$p_{initial}=m_{0}v$$ $$\Delta p=-m(u-v)-M(u-v)$$ ##m_0## is mass of the plane. $$F=\Delta p$$ $$F=-m(u-v)-M(u-v)=(m+M)(v-u)$$...
  32. Mohmmad Maaitah

    Deriving force from momentum using d(mv)/dt

    How did the d(mv)/dt become the other two? Can someone explain how do we derive for new formulas in physics?
  33. C

    General form of Newton II -- Not understanding this step in the derivation

    For this, Does someone please know how do we derive equation 9.9 from 9.8? Do we take the limits as t approach's zero for both sides? Why not take limit as momentum goes to zero? Many thanks!
  34. gurbir_s

    I Angular momentum associated with a current carrying circular wire

    How should I calculate the angular momentum carried by a current carrying circular wire? Is it correct to consider the angular momentum of the electrons moving with drift velocity? Like ##L = n m_e v_{drift} r## where ##r## is radius of the loop, and ##n## is total number of electrons moving in...
  35. Kyuubi

    Solving Orbital Speed with Energy & Angular Momentum Conservation

    I've already solved the orbital speed by equating the kinetic and potential energy in the circle orbit case. $$\frac{1}{2}mv^2 = \frac{1}{2}ka^2.$$And so $$v^2 = \frac{k}{m}a^2$$Now when the impulse is added, the particle will obviously change course. If we set our reference point in time just...
  36. Daniel Guh

    AP Physics C Mechanics: Linear Momentum for Colliding Billiard Balls

    I'm guessing this question can be solved using the law of conservation of momentum Vi = 5 m/s (5 m/s) M = (4.33 m/s) cos30 M + V sinθ M I don't know what to do after this... I'm also not sure if I use the sin and cos correctly.
  37. M

    Final Angular Momentum of a Space Station

    Li = Lrf +Ltf Iωo = Iωf + mvRsinθ I = MR^2 (MR^2)ωo = (MR^2)ωf + mvRsinθ ωf = (MR^2ωo -mvRsinθ)/MR^2 = 3.99
  38. C

    B Is it possible to measure both position and momentum simultaneously?

    A simultaneous measurement of both a particle's position and momentum may be successfully accomplished if more than one photon were utilized for the measurement. A non-demolishing measurement is possible if the emitters were aligned such that each would offset the other’s recoil of the target...
  39. E

    I Ballentine Equation 5.13 on conservation of momentum

    In Chapter 5.3, Ballentine uses geometrical arguments to obtain the initial magnitude of a hydrogen atom's bound electron momentum. How does equation (5.13) obtain? I tried to naively compute $$p_e^2 \equiv \textbf{p}_e\cdot \textbf{p}_e = p_a^2+p_b^2+p_o^2 + 2\textbf{p}_a\cdot \textbf{p}_b -...
  40. D

    I Momentum eigenfunctions in an infinite well

    Hi For an infinite well , solving the Schrodinger equation gives wavefunctions of the form sin(nπx/L). These are not eigenfunctions of the momentum operator which means there are no eigenvalues of the momentum operator. Does this mean momentum cannot be measured ? Inside the infinite well the...
  41. J

    2-D Momentum Problem -- Elastic collision of two spheres

    Hi, Here is the problem What is required to answer this question is two assumptions. Firstly, the component of the momentum normal to the centre line is the same before and after. Therefore, secondly, A must recoil entirely in the horizontal plane. This is the only way to answer this question...
  42. snoopies622

    B How to show that particle spin includes angular momentum?

    I understand how a massive, electrically charged spinning ball would have both angular momentum and a magnetic dipole, and i can see how the Stern–Gerlach experiment shows that the magnetic dipole of an electron is quantized. What kind of experiment demonstrates a connection between electron...
  43. Z

    A Momentum operator -- Why do we use the plane wave solution?

    Why in order to derive the QM momentum operator we use the plane wave solution. Why later on in field theory and particle physics, the plane wave ansatz is so physically important?
  44. P

    Angular momentum <Lx^2> and <Ly^2>

    Hi, I have a question. Let us say we have the wave function as with eigen value and base eigen value of: ##!\psi >\:=\:\frac{1}{6}\left(4!1,0,0>\:+\:3!2,1,1>\:-1!2,1,0\:+\:\sqrt{10}!2,1,-1>\right)## I need to find <Ly^2> the solution of the problem according to answers, is demanding that...
  45. K

    I Angular momentum and rotations

    Cohen tannoudji. Vol 1.pg 702"Now, let us consider an infinitesimal rotation ##\mathscr{R}_{\mathbf{e}_z}(\mathrm{~d} \alpha)## about the ##O z## axis. Since the group law is conserved for infinitesimal rotations, the operator ##R_{\mathbf{e}_z}(\mathrm{~d} \alpha)## is necessarily of the form...
  46. 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...
  47. 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...
  48. 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...
  49. 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?
  50. 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...
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