EPFO APFC 2023 General Studies · Question 52 of 120

Correct — B, (b) The momentum remains conserved, but not the kinetic energy. This is the defining property of an inelastic collision, and the two halves of the statement rest on quite different footings. Momentum is conserved in every collision in which no net external force acts on the pair while they are in contact. The reason is Newton's third law: during the impact each body pushes on the other with an equal and opposite force, so the impulse one receives is exactly the impulse the other loses, and the vector sum of the two momenta cannot change. Collisions last a very short time, and the impulsive internal forces are enormous compared with ordinary external forces such as friction or gravity, which is why the conservation holds to a very good approximation even when the surface is not frictionless. Kinetic energy has no such protection. It is conserved only when the deformation of the colliding bodies is entirely recovered — that is, only in an elastic collision. In an inelastic collision part of the kinetic energy is spent on permanent deformation, on heating the bodies, and on the sound of the impact, so it leaves the mechanical account altogether. Nothing is destroyed: total energy is conserved, as it always is, but the share of it that is kinetic falls. A worked case makes the size of the effect concrete. A body of mass m moving at speed v strikes an identical body at rest and the two stick together — a perfectly inelastic collision. Momentum conservation gives mv = 2mv′, so the common velocity is v/2. The kinetic energy before is ½mv² and after is ½(2m)(v/2)² = ¼mv², which is exactly half. Half the kinetic energy has gone into deformation and heat, while every unit of momentum is still there. The quantity that grades a collision is the coefficient of restitution, the ratio of the relative velocity of separation to the relative velocity of approach. It is 1 for a perfectly elastic collision, zero for a perfectly inelastic one in which the bodies move off together, and between the two for the real collisions of everyday life. The ask here is positive — which statement is correct — although three of the four options are phrased around a negation.

• (a)The kinetic energy remains conserved, but not the momentum. — This is the true statement with its two halves exchanged, and it describes a collision that cannot happen. Momentum conservation follows directly from Newton's third law and holds for every collision free of net external force, elastic or not; kinetic energy conservation is a special property that holds only when the deformation is fully recovered. So there is no collision in which the more fragile of the two quantities survives while the more robust one does not. If a candidate ever finds a collision problem in which momentum appears not to balance, the correct conclusion is that an external force has been left out of the system — a wall, the ground, a spring anchored outside — and not that momentum has been lost.
• (c)Both the kinetic energy and the momentum remain conserved. — This is the definition of an elastic collision, not an inelastic one, and it is the option that punishes a candidate who reads the stem too quickly. In a perfectly elastic collision the bodies separate with the same relative speed at which they approached, the coefficient of restitution is 1, and both momentum and kinetic energy are unchanged; collisions between atoms and molecules, and between subatomic particles, come very close to this ideal, and the collision of two hard steel balls approximates it. The stem, however, specifies an inelastic collision, in which some kinetic energy is converted into heat, sound and permanent deformation. Reading the first six words of the question decides between this option and the keyed one.
• (d)Neither the kinetic energy nor the momentum remains conserved. — This overstates what inelasticity costs. Inelastic means only that kinetic energy is not conserved; momentum is untouched, because the internal forces of the impact are equal and opposite and cancel in the total. Even in the extreme case — a perfectly inelastic collision in which the bodies coalesce and the loss of kinetic energy is the largest that momentum conservation permits — the total momentum after the impact is exactly what it was before. It is also worth being clear about what is never lost: total energy is conserved in every collision without exception, because the kinetic energy that disappears reappears as heat, sound and the work done in deforming the bodies. Only the kinetic share falls.

A collision is an interaction in which two bodies exert large forces on each other for a short time, and it is analysed by asking which of the conserved quantities survive it. Linear momentum, the product of mass and velocity, is a vector, and its total for an isolated pair is conserved in every collision: during the impact the two bodies exert equal and opposite forces on each other by Newton's third law, so the impulses cancel and the total momentum before equals the total after. Because the impact forces are impulsive — very large for a very short time — ordinary external forces such as friction contribute negligible impulse, so the conservation is a good approximation in practice even on a rough table. Kinetic energy, a scalar, is a different matter. It is conserved only if the bodies deform elastically and recover completely, which defines an elastic collision. When they do not, the missing kinetic energy has gone into permanent deformation, into internal energy that raises the temperature, and into the sound of the impact — so total energy is conserved, as it always is, while the kinetic share falls. Collisions are graded by the coefficient of restitution, the ratio of the relative velocity of separation to the relative velocity of approach: unity for a perfectly elastic collision, zero for a perfectly inelastic one in which the bodies move off with a common velocity, and intermediate for real ones. The perfectly inelastic case is the extreme of energy loss, and the loss is fixed by momentum conservation itself — for equal masses with one initially at rest, exactly half the kinetic energy goes.

The science block of this paper tests whether a candidate can apply a conservation law rather than manipulate an equation, and this item is the archetype: no numbers, four statements built by pairing two quantities with two outcomes, and the answer decided by knowing which conservation is universal and which is conditional. The option set is exhaustive by construction — both conserved, neither conserved, and each one conserved without the other — so there is no way to eliminate by plausibility and the physics has to be known. That construction is worth recognising in itself, because the Commission uses it elsewhere: when four options exhaust the logical possibilities, the examiner is testing a definition and not a calculation. Note the printed form as well. Three of the four options carry a negation inside them, printed without emphasis, and the ask itself is positive — it asks which statement is correct. A candidate trained to hunt for the word not in a stem gets no signal here, because the negations belong to the options rather than to the question.

• Total linear momentum is conserved in every collision free of net external force, elastic or inelastic, because the internal impact forces are equal and opposite by Newton's third law.
• Kinetic energy is conserved only in an elastic collision; in an inelastic collision part of it becomes heat, sound and permanent deformation.
• Total energy is always conserved — an inelastic collision loses kinetic energy, not energy.
• The coefficient of restitution is the relative velocity of separation divided by the relative velocity of approach: 1 for perfectly elastic, 0 for perfectly inelastic, and between the two for real collisions.
• In a perfectly inelastic collision the bodies move off with a common velocity and the loss of kinetic energy is the maximum that momentum conservation allows.
• For equal masses with one initially at rest, a perfectly inelastic collision leaves a common velocity of half the original and loses exactly half the kinetic energy.
• Momentum is a vector and must be conserved component by component; kinetic energy is a scalar, which is why a two-dimensional collision needs the vector equation and one energy equation at most.

• Reading the stem as elastic and choosing the option in which both quantities are conserved; the first six words of the question decide the answer
• Believing that an inelastic collision destroys energy; it converts kinetic energy into other forms, and total energy is untouched
• Assuming momentum can be lost in a violent collision; if the books do not balance, an external body has been left out of the system
• Adding kinetic energies as vectors or momenta as scalars — momentum has direction and must be resolved, kinetic energy has none
• Treating perfectly inelastic as meaning all the kinetic energy is lost; the bodies still move off together, and the loss is only as large as momentum conservation permits

Mechanics reaches this paper as a statement-selection item on a conservation law, or as a short numerical application of the equations of motion. Expect the four options to exhaust the possible pairings of two quantities with two outcomes, which means guessing is worthless, and expect the discriminating word — elastic or inelastic — to sit early in the stem. The preparation that pays is a one-line statement of each conservation law together with the condition under which it holds.

No directly related past PYQ was found.

• practice — not a real PYQ

  A bullet of mass 10 g moving at 400 m/s embeds itself in a wooden block of mass 390 g resting on a frictionless surface. The common velocity of the block and the bullet immediately after the impact is

  • (a)40 m/s
  • (b)10 m/s
  • (c)4 m/s
  • (d)400 m/s

  Answer(b) 10 m/s — the bullet embeds itself, so the collision is perfectly inelastic and the two move off together. Momentum is conserved: 0·01 × 400 = 4 kg m/s before, and the combined mass is 0·4 kg, so the common velocity is 4/0·4 = 10 m/s. Kinetic energy is not conserved here, most of it having gone into heat and the work of penetrating the wood.

• practice — not a real PYQ

  The coefficient of restitution for a perfectly inelastic collision is

  • (a)greater than 1
  • (b)equal to 1
  • (c)between 0 and 1
  • (d)equal to 0

  Answer(d) equal to 0 — in a perfectly inelastic collision the bodies do not separate at all but move off with a common velocity, so the relative velocity of separation is zero and the ratio that defines the coefficient of restitution vanishes. A value of 1 marks a perfectly elastic collision, and real collisions fall between the two.