What is the difference between momentum and kinetic energy?

Momentum ($vec{p} = mvec{v}$) is a vector quantity representing the 'quantity of motion' and depends linearly on velocity. Kinetic energy ($E_k = rac{1}{2}mv^2$) is a scalar quantity representing the energy of motion and depends on the square of velocity. While both depend on mass and velocity, they describe different physical aspects. Momentum is conserved in all types of collisions within an isolated system, but kinetic energy is only conserved in elastic collisions.

Why is momentum conserved only in an isolated system?

Momentum is conserved only in an isolated system because an isolated system is defined as one where the net external force acting on it is zero. According to Newton's second law, the rate of change of momentum of a system is equal to the net external force acting on it ($vec{F}_{ ext{net}} = rac{dvec{P}_{ ext{total}}}{dt}$). If $vec{F}_{ ext{net}} = 0$, then $rac{dvec{P}_{ ext{total}}}{dt} = 0$, which means the total momentum $vec{P}_{ ext{total}}$ must be constant. External forces introduce an impulse that changes the system's total momentum.

Can momentum be conserved if kinetic energy is not conserved?

Yes, absolutely. This is the defining characteristic of inelastic collisions. In such collisions, the total linear momentum of the system is always conserved (assuming an isolated system), but a portion of the initial kinetic energy is converted into other forms of energy, such as heat, sound, or internal deformation energy. For example, when two cars collide and crumple, momentum is conserved, but kinetic energy is lost to the deformation and sound.

How does the conservation of momentum apply to explosions?

In an explosion, the forces causing the fragmentation are internal forces within the system (e.g., chemical reactions in a bomb). Therefore, if no external forces are acting on the object before and during the explosion, the total momentum of the system (the original object plus its fragments) remains conserved. If the object was initially at rest, its total momentum was zero. After the explosion, the vector sum of the momenta of all the fragments must still be zero.

What is the significance of the coefficient of restitution?

The coefficient of restitution ($e$) is a crucial parameter that quantifies the 'bounciness' or elasticity of a collision. It helps us classify collisions and predict the relative velocities of objects after impact. An $e=1$ signifies a perfectly elastic collision where kinetic energy is conserved. An $e=0$ indicates a perfectly inelastic collision where objects stick together and there's maximum kinetic energy loss. Values between 0 and 1 represent typical inelastic collisions, providing insight into how much kinetic energy is dissipated.

Conservation of Momentum

Physics

NEET UG

Version 1Updated 22 Mar 2026

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The principle of conservation of momentum states that if no external force acts on a system of two or more interacting objects, the total momentum of the system remains constant. In simpler terms, the total momentum before a collision or interaction is equal to the total momentum after the collision or interaction, provided the system is isolated. This fundamental law is a direct consequence of Ne…

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The conservation of momentum is a fundamental principle stating that the total momentum of an isolated system remains constant. Momentum, a vector quantity, is defined as the product of mass and velocity ( $p = mv$ ).

An isolated system is one where no net external forces act upon it. This principle is a direct consequence of Newton's third law of motion, where internal action-reaction forces cancel out, leading to no change in the system's total momentum.

It applies to all types of interactions, including collisions and explosions. In collisions, while total momentum is always conserved in an isolated system, kinetic energy may or may not be. Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions conserve momentum but lose kinetic energy.

Perfectly inelastic collisions are a special case where objects stick together, resulting in maximum kinetic energy loss. Understanding the vector nature of momentum and the conditions for an isolated system are crucial for applying this principle correctly.

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Key Concepts

Vector Nature of Momentum

Momentum is a vector quantity, meaning it has both magnitude and direction. When applying the conservation of…

Derivation from Newton's Third Law

The conservation of momentum is not an independent law but a direct consequence of Newton's third law…

Perfectly Inelastic Collisions

In a perfectly inelastic collision, the colliding objects stick together after impact and move as a single…

Momentum: —

vec{p} = mvec{v}

(vector quantity, SI unit: kg·m/s)

Conservation of Momentum: — For an isolated system (

vec{F}_{\text{net, ext}} = 0

), total momentum is constant:

sum vec{p}_{\text{initial}} = sum vec{p}_{\text{final}}

Elastic Collision: — Momentum conserved, Kinetic Energy conserved.

e=1

Inelastic Collision: — Momentum conserved, Kinetic Energy NOT conserved.

0 < e < 1

Perfectly Inelastic Collision: — Objects stick together. Momentum conserved, max KE loss.

e=0

. Formula:

m_1u_1 + m_2u_2 = (m_1+m_2)V

Impulse: —

vec{J} = Delta vec{p} = vec{F}_{\text{avg}}Delta t

. Also,

vec{J} = int vec{F} dt

Recoil/Explosion: — Initial momentum (often zero) = sum of final momenta of fragments.

MICE: Momentum Is Conserved Everywhere (in an isolated system)!