1. The problem states the kinetic energy formula: $$K = \frac{1}{2}mv^2$$ where $m$ is mass and $v$ is speed. 2. We check each manipulation: - Option A: Multiply both sides by 2: $$2K = 2 \times \frac{1}{2}mv^2 = mv^2$$ This is correct. - Option B: Solve for $m$ by dividing both sides by $\frac{1}{2}v^2$: $$m = \frac{K}{\frac{1}{2}v^2} = \frac{K}{1} \times \frac{2}{v^2} = \frac{2K}{v^2}$$ This is correct. - Option C: Solve for $v^2$ by dividing both sides by $\frac{1}{2}m$: $$v^2 = \frac{K}{\frac{1}{2}m} = K \times \frac{2}{m} = \frac{2K}{m}$$ This is correct. - Option D: Solve for $v$ by taking the square root of $v^2$: $$v = \sqrt{v^2} = \sqrt{\frac{2K}{m}}$$ The option states $v = \sqrt{2K/m}$ without parentheses, which is ambiguous but mathematically equivalent to $v = \sqrt{\frac{2K}{m}}$. 3. However, if option D is written as $v = \sqrt{2K}/m$, it would be incorrect. Since the problem states $v = \sqrt{2K/m}$, it is correct. 4. Therefore, all options A, B, C, and D correctly manipulate the equation. 5. Since the question asks which incorrectly manipulates the equation, none of the options are incorrect. Final answer: None of the options incorrectly manipulate the equation.