CDS (II) 2022 General Knowledge · Question 24 of 120

Correct — A, 100 N.s and 500 J. Two substitutions settle it. Linear momentum is mass times velocity, so p equals 10 kg times 10 m/s, which is 100 kg m/s; since one kilogram metre per second equals one newton second, that is 100 N.s. Kinetic energy is half the mass times the square of the speed, so it is half of 10 times 100, which is 500 J. A useful cross-check is the relation between the two: kinetic energy equals p squared divided by 2m, and 100 squared divided by 20 is 500, which agrees. The trap in this question lies entirely in the factor of one half — a candidate who computes mass times speed squared without halving it gets 1000 J and walks into option B.

• (b)100 N.s and 1000 J — The momentum is right and the energy is doubled. This is what you get by writing kinetic energy as mv squared and forgetting the factor of one half.
• (c)200 N.s and 500 J — The energy is right and the momentum is doubled. Momentum is mv, not 2mv; the 2 belongs in the denominator of the energy formula, not in the momentum.
• (d)200 N.s and 1000 J — Both quantities doubled — the result of misplacing the factor of two twice over. Neither number can be reached from m equal to 10 kg and v equal to 10 m/s by any correct route.

Momentum and kinetic energy are two different measures of the same motion and they behave differently. Momentum, p equal to mv, is a vector: it has direction, and it is conserved in every collision. Kinetic energy, half mv squared, is a scalar: it has no direction, it depends on the square of the speed, and it is conserved only when a collision is elastic. Because of the square, doubling the speed doubles the momentum but quadruples the kinetic energy — the reason stopping distance rises so steeply with speed.

Arithmetic questions of this kind are marked as much on unit sense as on the formula. The newton second is worth understanding rather than memorising: force equals rate of change of momentum, so a newton second is exactly a kilogram metre per second, and the two labels are interchangeable. That equivalence is also why impulse, force times time, comes out in the same units as momentum. Once the two numbers are computed, notice that this question can be answered without computing both — momentum alone splits the four options into two pairs, and kinetic energy alone splits them the other way, so either calculation done correctly narrows the field to two and the second confirms.

• Linear momentum p equals mv and is a vector quantity, measured in kg m/s or equivalently N.s.
• Kinetic energy equals one half mv squared and is a scalar quantity, measured in joules.
• Kinetic energy can also be written as p squared divided by 2m.
• Doubling the speed doubles the momentum but multiplies the kinetic energy by four.
• Impulse is force multiplied by time and equals the change in momentum, which is why it shares the unit N.s.

• Dropping the factor of one half in the kinetic energy formula.
• Doubling the mass or the velocity somewhere in the momentum step, which produces the 200 N.s options.
• Treating N.s and kg m/s as different units; they are the same.

As a two-quantity numerical like this one, as a proportional-change question when speed or momentum is doubled, or as a units-matching item.

• practice — not a real PYQ

  A body of mass 2 kg moves with a speed of 5 m/s. Its kinetic energy is

  • (a)10 J
  • (b)25 J
  • (c)50 J
  • (d)100 J

  Answer(b) 25 J — half of 2 times 25 is 25 J; forgetting the half would give 50 J.

• practice — not a real PYQ

  If the speed of a moving body is doubled, its kinetic energy becomes

  • (a)half
  • (b)twice
  • (c)four times
  • (d)unchanged

  Answer(c) four times — kinetic energy varies as the square of the speed, while momentum only doubles.