How to prove equation of kinetic energy (KE = 1/2mv^2)?

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Answer 1 · 2018-01-20T23:52:44

See below.

The work done is accelerating an object is given by:

$W=FDeltax$

Where $F$ is the force and $Deltax$ the displacement.

If the object started from rest and all of the work was converted to kinetic energy then this will be equal to the kinetic energy of the object:

$K = FDeltax$

Using Newton's 2nd law:

$K = maDeltax=m(aDeltax)$

Now using the equation of motion:

$2aDeltax=v^2-v_0^2->aDeltax=v^2/2-v_0^2/2$

Substitute this into the equation for kinetic energy to get:

$K = m(v^2/2-v_0^2/2)$

If the object started from rest then the initial velocity will be:

$v_0=0$ so $K$ simplifies to:

$K = (mv^2)/2$