K = p²/2m: Turn Percent Changes into Square Roots
Kinetic energy and momentum are linked by K = p²/(2m). Questions on percent changes, or on bodies of different mass falling the same distance, are solved by this one relation and by K = m g h.
Kinetic Energy and Momentum Relations is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
Percent changes
For a fixed mass, K is proportional to p², so p is proportional to the square root of K.
A 100% increase in K, so K becomes 2K, makes p increase by a factor sqrt(2) = 1.414, which is 41.4%. If the momentum becomes n times, the kinetic energy becomes n² times.
Same drop, different masses
Balls of 4 kg and 8 kg dropped together have the same speed after the same fall, but their kinetic energies are in the ratio 4 : 8 = 1 : 2. Their momenta are in the same ratio, 1 : 2, because the speeds are equal.
OlympIQ exam lens
- ✓K = p²/(2m); for fixed m, p is proportional to sqrt(K)
- ✓Small percent change: Δp/p = (1/2) ΔK/K; exact: 100% rise in K gives a 41.4% rise in p
- ✓p to n p means K to n² K
- ✓Free fall: K = m g h is proportional to m at equal drop
Do not halve or double the percentage directly for large changes. The small-change rule Δp/p = (1/2)ΔK/K works only for a few percent; for 100% use the exact square root.
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