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Do I then have to use: $$\frac{1}{2}m(V_f^2-V_i^2)$$?

I am assuming I should use this formula for change in kinetic energy if the object under examination doesn't start from rest.

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    $\begingroup$ Yes, that's right. Just subtract the initial kinetic energy from the final kinetic energy. $\endgroup$ – Samuel Weir Nov 29 '15 at 21:10
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The kinetic energy is defined as:

$E_{k}=\frac{1}{2}m v^{2}$

Then the change of kinetic energy between, lets say, state 1 and 2 is:

$\Delta E_{k}=\frac{1}{2}m v_{2}^{2} - \frac{1}{2}m v_{1}^{2}$

$\Delta E_{k}=\frac{1}{2}m (v_{2}^{2} - v_{1}^{2})$

as you correctly write.

If you want to consider that the object starts from rest you just set $v_{1}=0$ and if not, you just set the nonzero value to $v_{1}$.

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