Which one of the following conservation laws is a consequence of the Newton's third law of motion?

Correct — B, conservation of momentum. The third law says that when two bodies interact, the force each exerts on the other is equal in size and opposite in direction. The second law says force is the rate of change of momentum. Put them together: the two forces act for exactly the same length of time, so they deliver equal and opposite impulses, and the momentum one body gains is precisely the momentum the other loses. Adding the two changes gives zero. Extend the argument to a whole system and every internal force cancels against its partner, leaving only external forces able to alter the total — so a system with no net external force keeps its momentum unchanged. Gun recoil, rocket thrust and the backward shove a swimmer gives the water are the same sentence written three ways.

• (a)Conservation of energy — The most tempting option, because energy conservation is the more familiar law, but it comes from elsewhere and one experiment separates them cleanly. Two lumps of clay that collide and stick together obey the third law exactly and conserve their total momentum, yet lose a large part of their kinetic energy to heat and sound.
• (c)Conservation of charge — An independent experimental law belonging to electromagnetism. Newtonian mechanics contains no statement about electric charge at all, so nothing in the third law could imply it.
• (d)Conservation of mass — In classical physics this is a separate assumption rather than a deduction, and relativity later replaced it with the conservation of mass and energy together. The third law is silent about how much matter there is.

Newton wrote the second law in terms of what he called quantity of motion, which is momentum, so force being the rate of change of momentum is the original statement rather than a later rewriting. The impulse delivered by a force is the force multiplied by the time for which it acts, and it equals the change in momentum it produces. Because the third-law pair share the same contact time, their impulses are equal and opposite whatever the two masses happen to be — which is why a rifle recoils slowly and a bullet leaves fast, the same momentum divided between very different masses.

A point that trips candidates is that the third-law forces never cancel each other in a single body's equation of motion, because they act on two different bodies. They cancel only when the two bodies are taken together as one system, and that is exactly the step which yields the conservation law. The elastic-versus-inelastic test is the quickest way to be sure of the answer under time pressure: momentum survives every collision, kinetic energy survives only the elastic ones, so momentum is the quantity tied to the force law itself.

• Newton's third law: forces occur in equal and opposite pairs acting on two different bodies.
• Force is the rate of change of momentum, so equal and opposite forces acting for equal times produce equal and opposite momentum changes.
• Internal forces within a system cancel in pairs, so only an external force can change a system's total momentum.
• Momentum is conserved in every collision; kinetic energy is conserved only in a perfectly elastic one.
• Recoil of a gun, thrust of a rocket and the motion of a swimmer are all direct applications of the same result.

• Answering energy because it is the best-known conservation law, when an inelastic collision disproves the link.
• Believing the two third-law forces cancel out and leave no motion; they act on different bodies.
• Assuming a heavier body must gain more momentum in an interaction, when both gain equal and opposite amounts.

Asked as a derivation-recall item: a named law is given and the candidate must identify which conservation principle follows from it rather than merely coexists with it.

• practice — not a real PYQ

  In a perfectly inelastic collision between two bodies, which one of the following is conserved?

  • (a)Kinetic energy only
  • (b)Linear momentum only
  • (c)Both kinetic energy and linear momentum
  • (d)Neither of them

  Answer(b) Linear momentum only — the bodies stick together and dissipate part of their kinetic energy as heat and sound, while the total momentum is untouched.

• practice — not a real PYQ

  A rifle of mass 4 kg fires a bullet of mass 20 g at 400 m/s. The recoil speed of the rifle is

  • (a)1 m/s
  • (b)2 m/s
  • (c)4 m/s
  • (d)8 m/s

  Answer(b) 2 m/s — total momentum starts at zero, so 4 × v equals 0.02 × 400 = 8, giving v = 2 metres per second backwards.