1. The problem states that an electron is accelerated from rest by an electric potential difference $V$ and reaches a speed of $9.4 \times 10^6$ m/s. 2. The kinetic energy gained by the electron is given by the formula: $$E_k = \frac{1}{2} m v^2$$ where $m$ is the mass of the electron and $v$ is its velocity. 3. The work done on the electron by the electric field is equal to the change in kinetic energy, which is also equal to the charge $Q$ times the potential difference $V$: $$E_k = QV$$ 4. Rearranging to find $V$: $$V = \frac{E_k}{Q}$$ 5. Calculate the kinetic energy: $$E_k = \frac{1}{2} \times 9.11 \times 10^{-31} \times (9.4 \times 10^6)^2$$ $$= \frac{1}{2} \times 9.11 \times 10^{-31} \times 8.836 \times 10^{13}$$ $$= 4.0555 \times 10^{-17} \text{ J}$$ 6. Calculate the potential difference $V$: $$V = \frac{4.0555 \times 10^{-17}}{1.6 \times 10^{-19}}$$ $$= 253.47 \text{ V}$$ 7. Your calculation of kinetic energy was off by a factor of 2 because you did not include the $\frac{1}{2}$ factor correctly or made a calculation error in squaring the velocity. 8. The correct potential difference $V$ is approximately $253$ volts, not $503$ volts. Therefore, the mistake was in the calculation of kinetic energy, which led to an incorrect value of $V$.