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What is the law of conservation of linear momentum?

Class 11 PhysicsNCERT chapter: Laws of Motionइसे हिंदी में पढ़ें

Exam-ready answer

The law of conservation of linear momentum states that the total momentum of an isolated system of interacting particles is conserved, that is, it remains constant when no net external force acts on the system. Internal forces between particles come in equal and opposite pairs, so they only transfer momentum within the system. Example: recoil of a gun.

Explanation

Consider two bodies A and B that interact, for example in a collision. By the third law, the force on A by B is equal and opposite to the force on B by A: F_AB = −F_BA. Both forces act for the same time Δt. By the second law, force × time equals change in momentum, so the change in momentum of A is equal and opposite to the change in momentum of B.

Writing it out: Δp_A = F_AB Δt and Δp_B = F_BA Δt, so Δp_A + Δp_B = 0. If the initial momenta are p_A and p_B and the final momenta are p′_A and p′_B, then p′_A + p′_B = p_A + p_B. The total momentum before the interaction equals the total momentum after it. This holds for every collision and explosion in an isolated system, whether the bodies stick together or bounce apart.

Recoil of a gun: before firing, the gun and bullet are at rest, so total momentum is zero. After firing, the bullet moves forward with momentum p, so the gun must move backward with momentum −p to keep the total zero. Because the gun is much heavier than the bullet, its recoil velocity is much smaller. The same idea explains how a rocket moves forward by pushing exhaust gases backward.

Points to remember

  1. Statement: the total momentum of an isolated system of interacting particles is conserved.
  2. Condition: the net external force on the system must be zero.
  3. Derived from Newton's second and third laws: Δp_A = −Δp_B.
  4. Equation: p_A + p_B = p′_A + p′_B (total momentum before = total momentum after).
  5. Examples: recoil of a gun, collisions, explosions, rocket propulsion.

Exam tip

For the proof, write F_AB = −F_BA (third law), multiply by Δt to get Δp_A = −Δp_B (second law), then add to show total momentum stays constant. Always state the condition: no net external force.

Questions students also ask

Why does the gun recoil less than the bullet moves?
The momentum of the gun and the bullet are equal in size and opposite in direction. Since momentum is mass × velocity and the gun is much heavier than the bullet, the gun gets a much smaller velocity backward.
Is momentum conserved when a moving ball hits the ground?
Not for the ball alone, because the ground exerts an external force on it. Momentum is conserved only for the whole isolated system, here the ball plus the Earth. The Earth is so massive that its change in velocity is negligible.
Is kinetic energy also conserved in every collision?
No. Total linear momentum is conserved in all collisions of an isolated system, but kinetic energy is conserved only in elastic collisions. In inelastic collisions some kinetic energy changes into heat, sound or deformation.

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