Q.State the relation between the kinetic energy and momentum of a body.
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Kinetic Energy and Speed: The Intuition
Imagine pushing a heavy box across the floor. The harder you push, the faster it moves. But here's the surprising part: doubling the speed does not require double the work — it requires four times the work. That's the core of the kinetic energy–speed relation.
Why? Because kinetic energy isn't about how fast you're moving — it's about how much effort it took to get you moving that fast. And effort (work) depends on both force and distance. When you push something to a higher speed, you have to apply force over a longer distance, and that extra distance multiplies the work required.
Kinetic energy is the energy an object possesses because of its motion. A stationary object has zero kinetic energy.
The Precise Statement
The kinetic energy K of an object of mass m moving with speed v is:
K=21mv2
This is the kinetic energy–speed relation. The key point: kinetic energy is proportional to the square of the speed, not the speed itself.
K=21mv2
Why the Square? A Simple Derivation
Start from Newton's second law: F=ma. If a constant force F acts on an object initially at rest over a displacement s, the work done is W=Fs.
From kinematics, for constant acceleration starting from rest: v2=2as. So s=2av2.
Substitute into work:
W=Fs=(ma)(2av2)=21mv2
That work becomes the object's kinetic energy. The square comes from the kinematic relation v2=2as — a direct consequence of how distance and speed are linked under constant acceleration.
A common mistake: thinking kinetic energy is 21mv or mv2. The factor 21 is essential — it comes from the integration of force over distance.
What This Means in Practice
| Speed change | Kinetic energy change |
|---|---|
| Double speed (2v) | K becomes 4× original |
| Triple speed (3v) | K becomes 9× original |
| Halve speed (v/2) | K becomes 1/4 of original |
This explains why:
- A car crash at 100 km/h is four times as destructive as one at 50 km/h (four times the energy to dissipate).
- Braking distance quadruples when speed doubles (because brakes must do four times the work).
- A bullet at twice the speed penetrates much deeper than twice as far.
Kinetic energy depends only on mass and speed — not on direction. Two objects with the same mass and speed have the same kinetic energy, even if moving in opposite directions.
Units
In SI units:
- Mass m in kilograms (kg) …
Kinetic energy and momentum are both defined in terms of a body's mass and velocity, so one can be expressed directly in terms of the other.
…
Kinetic energy and momentum are linked by KE = p^2/2m, derived directly from their definitions.
Momentum: p = mv → v = p/m
Kinetic energy: KE = (1/2) m v^2
Substituting v = p/m:
KE = (1/2) m (p/m)^2 = (1/2) (p^2/m) = p^2 / (2m)
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Showing the 12 most recent of 28 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.If the momentum of a body is increased by 100%, then the percentage increase in the kinetic energy is(a) 150%(b) 200%(c) 300%(d) None of these
›Reveal solutionSolution
KE = p^2 / (2m), so KE scales with the SQUARE of momentum; doubling p means KE becomes 4 times as large, i.e. a 300% increase over the original.
Kinetic energy in terms of momentum: KE = p^2 / (2m)
Let the initial momentum be p, so initial KE = p^2/(2m).
Momentum increased by 100% means the new momentum is p' = 2p.
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- CBSE 2026Set ANNUAL1 markMCQQ.When the momentum of body is increased by three times, its K.E. becomes:(a) Twice(b) Half(c) Four times(d) Nine times
›Reveal solutionSolution
KE=p2/2m, so KE∝p2 — tripling p makes KE nine times as large.
Momentum is p=mv and kinetic energy is KE=21mv2. Writing KE in terms of p: KE=21mv2=2m(mv)2=2mp2.
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- CBSE 2026Set ANNUAL1 markMCQQ.If the linear momentum of the object is increased by 0.1% then the kinetic energy is increased by:(a) 0.4%(b) 0.1%(c) 0.01%(d) 0.2%
›Reveal solutionSolution
Since KE = p^2/2m depends on the square of momentum, a small percentage change in p produces twice that percentage change in KE: 2 x 0.1% = 0.2%.
Kinetic energy can be written in terms of linear momentum p = mv as
KE = (1/2)mv^2 = p^2/(2m)
Taking logarithms to relate fractional changes (mass m is constant):
ln(KE) = 2 ln p - ln(2m)
Differentiating:
delta(KE)/KE = 2(deltap/p)
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- CBSE 2026Set ANNUAL1 markMCQQ.Kinetic energy of a body of mass 2 kg is 36 joule. Its momentum in kg m/s will be(a) 6(b) 12(c) 24(d) 36
›Reveal solutionSolution
p = sqrt(2 m KE) = sqrt(144) = 12 kg m/s. Answer (B).
Kinetic energy in terms of momentum: KE = p^2/(2m), so p = sqrt(2 m KE).
Given m = 2 kg and KE = 36 J:
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- CBSE 2026Set ANNUAL1 markMCQQ.For constant momentum (p), the correct graph between kinetic energy (E) and mass (m) will be(a) decreasing hyperbola-like E-m curve(b) straight line increasing with mass(c) upward-increasing E-m curve(d) none of these
›Reveal solutionSolution
With p constant, E = p^2/(2m) is proportional to 1/m -> a hyperbola-like decreasing curve. Answer (A).
Kinetic energy can be written as E = p^2/(2m).
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- CBSE 2025Set ANNUAL1 markMCQQ.Kinetic energies of two moving bodies of masses 1 gm and 4 gm are equal. The ratio of their linear momentum is (A) 4 : 1 (B) sqrt(2) : 1 (C) 1 : 2 (D) 1 : 16
›Reveal solutionSolution
With equal kinetic energies, the ratio of momenta of the two bodies is 1:2.
Kinetic energy in terms of momentum: Ek=2mp2⇒p=2mEk.
Since both bodies have the same kinetic energy Ek:
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- CBSE 2025Set ANNUAL1 markMCQQ.Two bodies of masses 2 kg and 32 kg have equal kinetic energy. What is the ratio of their momentum?(a) 1 : 1(b) 1 : 2(c) 1 : 4(d) 2 : 1
›Reveal solutionSolution
Since KE = p^2/2m, momentum for a given kinetic energy is p = sqrt(2mKE); with equal KE, the momentum ratio reduces to sqrt(m1/m2).
Kinetic energy: KE = p^2 / (2m), so p = sqrt(2m * KE).
Both bodies have the same KE (call it E), so: …
- CBSE 2025Set ANNUAL1 markMCQQ.If the linear momentum of a body increases by 20%, what will be the percentage increase in K.E.?(a) 40%(b) 36%(c) 44%(d) 20%
›Reveal solutionSolution
Since KE=p2/(2m), kinetic energy grows with the square of momentum — so a 20% rise in momentum produces a 44% rise in kinetic energy, not 20% or 40%.
Kinetic energy expressed in terms of momentum p and mass m:
KE=p2/(2m)
If momentum increases by 20%, the new momentum is p′=1.2p.
New kinetic energy:
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- CBSE 2024Set ANNUAL1 markMCQQ.The kinetic energy of a body of mass 1 kg is 1 joule. Its velocity will be (A) 0.45 ms^-1 (B) 1 ms^-1 (C) 1.4 ms^-1 (D) 4.4 ms^-1
›Reveal solutionSolution
v=2KE/m≈1.4m/s.
Kinetic energy: KE=21mv2. Rearranging, v=m2KE.
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- CBSE 2024Set ANNUAL1 markMCQQ.The linear momentum of a body of mass m is p. Then its kinetic energy will be (A) mp (B) mp^2 (C) p^2/m (D) p^2/2m
›Reveal solutionSolution
In terms of momentum, KE=p2/2m.
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- CBSE 2024Set ANNUAL1 markMCQQ.The momentum of any body is increased by 20%. Then its kinetic energy will increase by (A) 20% (B) 66% (C) 44% (D) 88%
›Reveal solutionSolution
A 20% rise in momentum raises kinetic energy by 44%.
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- CBSE 2024Set ANNUAL1 markMCQQ.What is the kinetic energy of a 2 kg object moving at a velocity of 3 m/s ?(a) 6 J(b) 9 J(c) 18 J(d) 27 J
›Reveal solutionSolution
Kinetic energy of a moving object is KE = ½mv²; here it works out to 9 J.
The kinetic energy of an object of mass m moving with speed v is:
KE=21mv2
Substituting m = 2 kg, v = 3 m/s:
…
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