Question

A proton, initially at rest, is accelerated through a potential difference of \(500 .\) V. What is its final velocity?

Short answer

Answer: The final velocity of the proton is approximately \(3.09 \times 10^{5} m/s\).

Step-by-step solution

Identify the mass and charge of a proton

A proton has a mass of \(m_p = 1.67 \times 10^{-27} kg\) and a charge of \(q_p = 1.6 \times 10^{-19} C\).

Calculate the kinetic energy gained by the proton

Due to the potential difference, the proton will gain kinetic energy. The kinetic energy (KE) gained can be calculated using the formula: \(KE = q_p \times V\), where V is the potential difference. Plug in the values, we get: \(KE = (1.6 \times 10^{-19} C)(500V)\) \(KE = 8 \times 10^{-17} J\)

Use the kinetic energy formula to find the final velocity

The formula for kinetic energy is: \(KE = \frac{1}{2} m_p v^2\), where \(m_p\) is the mass of the proton and v is the final velocity. Let's solve for v using the obtained KE value: \(8 \times 10^{-17} J = \frac{1}{2} \times 1.67 \times 10^{-27} kg \times v^2\) Now, we can solve for v: \(v^2 = \frac{2 \times 8 \times 10^{-17}}{1.67 \times 10^{-27}}\) \(v^2 \approx 9.58 \times 10^{10}\) \(v \approx 3.09 \times 10^{5} m/s\)

Conclusion

The final velocity of the proton after being accelerated through a potential difference of 500 V is approximately \(3.09 \times 10^{5} m/s\).