1. **State the problem:** We are given the kinetic energy formula $$E = \frac{1}{2} m v^2$$ and need to make $v$ the subject. 2. **Write the formula:** $$E = \frac{1}{2} m v^2$$ 3. **Isolate $v^2$:** Multiply both sides by 2 to cancel the fraction: $$2E = \cancel{2} \times \frac{1}{\cancel{2}} m v^2 = m v^2$$ 4. **Divide both sides by $m$ to isolate $v^2$:** $$\frac{2E}{m} = \frac{m v^2}{m} = v^2$$ 5. **Take the square root of both sides to solve for $v$:** $$v = \sqrt{\frac{2E}{m}}$$ 6. **Final answer:** $$\boxed{v = \sqrt{\frac{2E}{m}}}$$ This means the velocity $v$ is the square root of twice the kinetic energy divided by the mass. This formula is useful to find velocity when kinetic energy and mass are known.