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Work Done = $Fd$

Power = $Fv$

If the (net) Work Done = Change in Kinetic Energy, and the object starts from rest:
Work Done = $\frac{1}{2} mv^2$

Power = $\frac{1}{2} m av$

Power = $\frac{1}{2} F v$

This isn't correct though. How do I remove factor of $\frac{1}{2}$?

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It looks like you accidentally forgot the "power rule" in your third step.

Suppose that there is only kinetic energy and the object starts from rest.

Then $$ W = \frac{1}{2} mv^2 $$

Then, since power is $P= \frac{dW}{dt}$ we have that

$$ P = \frac{dW}{dt} = \frac{1}{2} m \frac{d}{dt} v^2 = mv \dot{v} = (ma)v = F\cdot v $$

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