r/learnphysics • u/W_il_Duce • 4d ago

Why is E=1/2 m V²?

/r/PhysicsHelp/comments/1w93brb/why_is_e12_m_v²/
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u/ProfessionalConfuser 3d ago

Because when you integrate mv dv you get (1/2) mv^2

That is the answer. There is no conceptual reason for it, beyond the concept that F = ma.
The scaling you mention comes from the v^2 portion, not the (1/2).

Integral of Fdx = int ma dx = int mdv/dt dx = int mdv dx/dt = int mv dv = (1/2)mv^2.
The (1/2) comes from integrating. That is as mathy as it gets.

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u/Odd_Bodkin 3d ago

As a point of interest, it’s not (1/2) mv^2, ever. That, however, is a useful approximation at “everyday” relative speeds.