Two objects, x and y, have equal mass and are moving with speeds u and 3u respectively. Their kinetic energies kx and ky are related as

Correct — C, 9kx = ky. Kinetic energy is one-half of mass times the square of speed, so for two objects of equal mass the energies stand in the ratio of the squares of their speeds. Object x moves at u, so kx = ½mu². Object y moves at 3u, so ky = ½m(3u)² = 9 × ½mu². That is ky = 9kx, which the option writes as 9kx = ky. Tripling the speed multiplies the kinetic energy nine times, not three, because the speed enters squared.

• (a)kx = ky — Equal energies would need equal speeds, since the masses are already equal. The stem gives u against 3u.
• (b)2kx = ky — A doubling of energy would follow from a speed ratio of the square root of two, not three. Nothing in the stem produces a factor of two.
• (d)3kx = ky — The classic error — treating kinetic energy as if it rose in proportion to speed. That is how momentum behaves; energy carries the square, so the factor is nine.

Kinetic energy is ½mv² and momentum is mv. One depends on the square of speed, the other on the first power of it, and almost every question in this family is built on the gap between the two. Triple the speed of a body and its momentum triples while its kinetic energy becomes nine times as great. The two quantities are linked directly by the relation that kinetic energy equals momentum squared divided by twice the mass.

The masses are stated to be equal precisely so that the mass cancels and the item reduces to the speed dependence alone. Write the ratio rather than the two energies and the work is a single line: ky ÷ kx = (3u ÷ u)² = 9. The reason the square matters physically is that doubling a vehicle's speed does not double its stopping distance but roughly quadruples it, since the brakes must dissipate four times the energy over the same retarding force. One caution on reading the options: they are written as equations rather than as ratios, so 9kx = ky and 'ky is nine times kx' say the same thing. A candidate who computes the ratio correctly and then reads the equation the wrong way round can still mark option (d) by accident.

• Kinetic energy is ½mv², so at constant mass it varies as the square of the speed.
• Tripling the speed multiplies the kinetic energy by nine.
• Momentum is mv and is linear in speed, so tripling the speed only triples the momentum.
• Kinetic energy and momentum are linked by kinetic energy equal to p² divided by 2m.
• The same square law explains why stopping distance grows far faster than speed.

• Scaling energy linearly with speed, which is the behaviour of momentum.
• Reading the equation backwards after computing the ratio correctly.
• Forgetting to check whether the masses are equal before cancelling them.

Asked as a ratio item with the masses fixed equal, so the square dependence of kinetic energy on speed is the only thing being tested.

• practice — not a real PYQ

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

  • (a)half
  • (b)one-fourth
  • (c)double
  • (d)unchanged

  Answer(b) one-fourth — kinetic energy varies as the square of the speed.

• practice — not a real PYQ

  Two bodies of equal mass have momenta in the ratio 1 : 3. The ratio of their kinetic energies is

  • (a)1 : 3
  • (b)1 : 6
  • (c)1 : 9
  • (d)3 : 1

  Answer(c) 1 : 9 — with equal masses, kinetic energy equal to p² ÷ 2m makes the energy ratio the square of the momentum ratio.