Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
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Kinetic energy is quadratic in speed due to Galilean invariance and conservation laws.
A Physics Stack Exchange thread asks why kinetic energy (½mv²) increases quadratically, not linearly, with speed. Top answer uses Galilean invariance: consider two clay balls of mass m moving at speed v colliding head-on, stopping and producing heat 2mE(v). In a moving frame, one ball is stationary, the other moves at 2v; conservation of energy forces E(2v)=4E(v), implying quadratic dependence. Another answer uses work: constant braking force F takes twice the time for a double-speed object, with mean velocity also doubled, giving four times the braking distance and thus four times the work. A third notes that dropping a ball from 2 m yields less than double speed because the second meter is traversed faster, limiting velocity gain.
What commenters are saying
Commenters appreciated an anecdote: two identical cars braking equally from 70 and 100 units; the 70-unit car stops, but the 100-unit car sheds the same kinetic energy (4900 units), retaining 5100 units, hitting the obstacle at ~71 units. A correction noted that vehicles with downforce can brake harder at higher speeds, altering the result. A second thread discussed frustration with physics pedagogy-it feels like a bag of tricks versus the axiomatic clarity of math and CS-with some noting that physics models are constrained by reality, not human choice, suggesting historical context might help.