The equation for final velocity, expressed as {v_{f}}^2 = {v_{i}}^2 + 2a\Delta d, is derived using kinematic equations. The derivation begins with the equations v = v_{0} + at and r = r_{0} + v_{0}t + \frac{1}{2}at^2. By isolating time (t) from the first equation and substituting it into the second, the algebraic manipulation leads to the final equation. This method effectively connects initial velocity, acceleration, and displacement in a straightforward manner.
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