Kinetic energy (KE) and momentum (p) of a body are linked by KE = p^2 / (2m) and p = sqrt(2 m KE), where m is the mass. So for the same mass, KE is proportional to the square of momentum. Memory hook: "K holds P-squared" — Kinetic energy carries momentum squared, and mass sits below in the 2m denominator.
Left: for a fixed mass, kinetic energy rises as the square of momentum (a parabola), so doubling momentum gives four times the KE. Right: for a fixed kinetic energy, momentum grows as the square root of mass, so a heavier body has more momentum.
Your doubts, answered
Are kinetic energy and momentum the same thing?
No. Momentum p = mv is a vector (has direction) with unit kg m/s. Kinetic energy KE = (1/2)mv^2 is a scalar (no direction) with unit joule. They are linked by KE = p^2 / (2m), but they are different quantities. This is why momentum can be zero for a system while kinetic energy is not (two equal masses moving opposite ways).
Why is KE proportional to p squared and not to p?
Start from KE = (1/2)mv^2 and p = mv, so v = p/m. Substitute: KE = (1/2)m(p/m)^2 = p^2 / (2m). The v^2 in kinetic energy becomes p^2 after substitution. That squaring is why doubling momentum makes KE four times larger, not two times.
If momentum is doubled, what happens to kinetic energy?
For the same mass, KE = p^2 / (2m), so KE depends on p^2. If p becomes 2p, then KE becomes (2p)^2 / (2m) = 4 times the original KE. So doubling momentum makes kinetic energy 4 times bigger.
Two bodies have the same kinetic energy. Which one has greater momentum?
Use p = sqrt(2 m KE). With KE fixed, p is proportional to sqrt(m). The heavier body has larger momentum. Example: same KE, one body 4 times heavier has 2 times the momentum. This is a favourite NEET one-liner.
Two bodies have the same momentum. Which one has greater kinetic energy?
Use KE = p^2 / (2m). With p fixed, KE is inversely proportional to m. The lighter body has greater kinetic energy. So for equal momentum, the smaller mass carries more KE.
⚠️ The NEET trap ✗ Assuming the heavier body always has more kinetic energy, or plugging p directly into KE without squaring (using KE = p/2m). ✓ For equal momentum, lighter mass has MORE KE (KE = p^2/2m). For equal KE, heavier mass has MORE momentum (p = sqrt(2mKE)). Always square p in the KE formula. 🧠 Fixed KE vs fixed momentum flip the mass dependence — read the question first.
Solved Work, Energy And Power NEET PYQs
Try the real previous-year questions from this chapter — each with the answer and a full solution.
