Why does kinetic energy increase quadratically, not linearly, with speed? (2011)

Physics Stack Exchange users have been discussing why kinetic energy increases with the square of an object's speed, rather than increasing linearly. The answer hinges on how work and acceleration interact. When you apply force over distance, you create energy — and as you accelerate further, both the distance required and the effort needed increase together. That multiplication gives you the squared relationship. In practical terms: double your speed and you need four times the energy to reach that velocity. It's why small speed differences dramatically affect collision damage and fuel costs.

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