What is the difference between linear momentum and kinetic energy?

Linear momentum ($vec{p} = mvec{v}$) is a vector quantity representing an object's 'mass in motion' and depends on both mass and velocity, including direction. Kinetic energy ($E_k = rac{1}{2}mv^2$) is a scalar quantity representing the energy an object possesses due to its motion and depends only on mass and the *magnitude* of velocity (speed). While momentum is conserved in all isolated interactions (like collisions), kinetic energy is only conserved in perfectly elastic collisions; it can be lost to heat, sound, or deformation in inelastic collisions.

When is the linear momentum of a system conserved?

The linear momentum of a system of particles is conserved if and only if the net external force acting on the system is zero. This means that any forces acting between the particles within the system (internal forces) do not affect the total momentum of the system, as they cancel out in action-reaction pairs. If there are external forces, but their vector sum is zero, or if they are negligible compared to internal forces during a brief interaction, then momentum is conserved.

Do internal forces affect the total linear momentum of a system?

No, internal forces do not affect the total linear momentum of a system. According to Newton's third law, internal forces always occur in equal and opposite pairs. For example, if particle A exerts a force on particle B, particle B exerts an equal and opposite force on particle A. When you sum up all these internal forces for the entire system, they cancel each other out, resulting in a net internal force of zero. Therefore, only external forces can change the total linear momentum of a system.

How does the concept of center of mass relate to linear momentum of a system?

The total linear momentum of a system of particles ($vec{P}_{sys}$) is directly related to the velocity of its center of mass ($vec{v}_{CM}$) by the equation $vec{P}_{sys} = Mvec{v}_{CM}$, where $M$ is the total mass of the system. This means that if the total linear momentum of an isolated system is conserved, its center of mass will move with a constant velocity. Even if particles within the system move chaotically, the center of mass maintains a predictable, uniform motion in the absence of external forces.

Can linear momentum be conserved in one direction but not another?

Yes, absolutely. The conservation of linear momentum is a vector principle. If the net external force acting on a system is zero along a particular direction (say, the x-axis), then the component of the total linear momentum along that specific direction will be conserved, even if there are external forces acting in other directions (e.g., y-axis). For instance, in projectile motion, horizontal momentum is conserved (neglecting air resistance) because there's no horizontal external force, but vertical momentum is not conserved due to gravity.

Linear Momentum of System

Physics

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Version 1Updated 23 Mar 2026

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The linear momentum of a particle is defined as the product of its mass and velocity, a vector quantity. For a system of particles, the total linear momentum is the vector sum of the individual linear momenta of all particles within the system. A fundamental principle in mechanics states that the total linear momentum of an isolated system (one upon which no net external force acts) remains consta…

Quick Summary

Linear momentum is a fundamental concept in physics, quantifying the 'quantity of motion' an object possesses. For a single particle, it's the product of its mass and velocity ( $vec{p} = mvec{v}$ ), making it a vector quantity.

For a system of multiple particles, the total linear momentum ( $vec{P}_{sys}$ ) is the vector sum of individual momenta. Crucially, this total momentum can also be expressed as the product of the system's total mass and the velocity of its center of mass ( $vec{P}_{sys} = Mvec{v}_{CM}$ ).

The rate of change of a system's total linear momentum is equal to the net external force acting on it ( $rac{dvec{P}_{sys}}{dt} = vec{F}_{ext}$ ). The most significant principle is the Law of Conservation of Linear Momentum: if the net external force on a system is zero, its total linear momentum remains constant.

This law is invaluable for analyzing interactions like collisions and explosions, where internal forces are dominant and external forces are negligible, allowing us to predict the motion of objects before and after such events.

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Linear Momentum: —

vec{p} = mvec{v}

(vector, unit kg·m/s)

Total System Momentum: —

vec{P}_{sys} = sum vec{p}_i = Mvec{v}_{CM}

Newton's 2nd Law for System: —

vec{F}_{ext} = \frac{dvec{P}_{sys}}{dt}

Conservation of Momentum: — If

vec{F}_{ext} = 0

, then

vec{P}_{sys} = \text{constant}

Impulse: —

vec{J} = int vec{F} dt = Deltavec{p}

Collisions:

- Elastic: Momentum conserved, Kinetic Energy conserved. - Inelastic: Momentum conserved, Kinetic Energy NOT conserved. - Perfectly Inelastic: Objects stick together, momentum conserved, max KE loss.

My Velocity Conserves Momentum:

Mass x Velocity = Momentum (

vec{p} = mvec{v}

)

Conservation of Momentum: If no External Force (

vec{F}_{ext}=0

), then total momentum is Constant (

vec{P}_{sys} = \text{constant}

Internal forces Don't Change total momentum.