CDS (II) 2020 General Knowledge · Question 86 of 120

Correct — B, increase by four times. Write both quantities for a body of mass m moving at speed v. Momentum is p = mv and kinetic energy is E = mv²/2, so eliminating v gives E = p²/2m. Kinetic energy therefore goes as the square of momentum for a fixed mass. Double p and E rises by a factor of 2² = 4. The same result falls out arithmetically: if momentum doubles at constant mass, the speed must double, and since energy depends on v², doubling v multiplies the energy by four. Take a 10 kg body at 10 m/s, where p = 100 kg m/s and E = 500 J; at 20 m/s the momentum is 200 kg m/s and the energy is 2,000 J — momentum twice as large, energy four times as large.

• (a)remain same — Kinetic energy cannot stay the same while momentum changes at fixed mass, because both are determined by the one variable that is changing, the speed.
• (c)increase by two times — This is the answer you get by treating energy as linear in momentum. It would be right only if E were proportional to p, but E is proportional to p squared.
• (d)increase by eight times — Eight times would follow from a cube relation. Kinetic energy goes as the square of the speed, not the cube.

Momentum and kinetic energy are the two quantities a moving body carries, and they behave differently. Momentum p = mv is a vector, is conserved in every collision, and depends linearly on speed. Kinetic energy E = mv²/2 is a scalar, is conserved only in elastic collisions, and depends on the square of speed. The bridge between them, E = p²/2m, is worth memorising in its own right, because it turns any question relating one to the other into a single substitution.

The general rule is more useful than the particular answer here. If momentum changes by a factor k at constant mass, kinetic energy changes by k squared; if kinetic energy changes by a factor k, momentum changes by the square root of k. That handles every version of this question the examiners set. The reason the square appears is also worth holding: doubling the speed doubles the momentum but doubles both the force needed to stop the body and the distance over which it acts, and work is force times distance.

• Momentum p = mv; kinetic energy E = mv²/2; the two are related by E = p²/2m.
• At constant mass, multiplying momentum by k multiplies kinetic energy by k squared.
• Momentum is a vector and is conserved in all collisions; kinetic energy is a scalar and is conserved only in elastic collisions.
• A 10 kg body at 10 m/s has momentum 100 kg m/s and kinetic energy 500 J.
• The SI unit of momentum is kg m/s, equivalently the newton second; kinetic energy is measured in joules.

• Treating kinetic energy as proportional to momentum rather than to its square.
• Forgetting that the relation E = p²/2m holds only when the mass stays the same.
• Confusing the units — momentum in kg m/s or newton seconds, energy in joules.

As a proportionality item in either direction, or as a numerical asking for momentum and kinetic energy of a given body.

• practice — not a real PYQ

  If the kinetic energy of a body of fixed mass is made nine times its original value, its momentum becomes

  • (a)three times
  • (b)nine times
  • (c)eighty-one times
  • (d)unchanged

  Answer(a) three times — since E is proportional to p squared, p is proportional to the square root of E, and the square root of nine is three.

• practice — not a real PYQ

  Two bodies of different masses have the same momentum. The one with the greater kinetic energy is

  • (a)the heavier body
  • (b)the lighter body
  • (c)both have equal kinetic energy
  • (d)cannot be determined

  Answer(b) the lighter body — from E = p²/2m, at equal momentum the kinetic energy is inversely proportional to the mass.