If the linear momentum of a moving object gets doubled due to application of a force, then its kinetic energy will

Q: What is the answer to CDS (II) 2020 GK Q86?

B. 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
(b)increase by four times
(c)increase by two times
(d)increase by eight times

Answer

Why

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.

Why the others are wrong

(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.

Concept

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.

Key facts

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.

Study next

Work-energy theorem, and how net work relates to the change in kinetic energyConservation of momentum and its use in collisions and recoil problemsElastic and inelastic collisions, and what survives in eachImpulse as the change in momentum, and force as the rate of change of momentumPotential and kinetic energy exchange in projectile and pendulum motion

Common traps

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.

Related PYQs

CDS_GK_2022_II_Q242022Open & attempt →

If an object of mass 10 kg is moving with a uniform speed of 10 m/s, then the linear momentum and the kinetic energy of the object, respectively, are

(a) 100 N.s and 500 J
(b) 100 N.s and 1000 J
(c) 200 N.s and 500 J
(d) 200 N.s and 1000 J

Answer(a) 100 N.s and 500 J

The same pair of formulas as a straight calculation. Working out that 10 kg at 10 m/s gives 100 units of momentum but only 500 J of energy is the concrete version of the relation this question asks about in the abstract.

Practice

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.