## Step 1: Identify Key Information and Relativistic Formula - **Given:** - Kinetic energy (\(K\)) = 5.00 MeV for both particles - \(\beta\) ray = electron, mass \(m_\beta = m_e = .511\,\text{MeV}/c^2\) - \(\alpha\) particle mass \(m_\alpha = 4 \times 938.3\,\text{MeV}/c^2 = 3727.2\,\text{MeV}/c^2\) (since it's 4 protons/neutrons) - **Relativistic Kinetic Energy Equation:** \[ K = (\gamma - 1)mc^2 \] where \(\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}\). Rearranged: \[ \gamma = 1 + \frac{K}{mc^2} \] To find \(v\): \[ v = c \sqrt{1 - \frac{1}{\gamma^2}} \] ### **For the \(\beta\) particle (electron):** \[ \gamma_\beta = 1 + \frac{5.00}{.511} = 10.78 \] \[ v_\beta = c \sqrt{1 - \frac{1}{(10.78)^2}} = c \sqrt{1 - .00860} = c \times .9957 \] **Explanation:** We calculated the relativistic \(\gamma\) factor for the electron using its rest mass and kinetic energy, then solved for velocity as a fraction of \(c\). --- ## Step 2: Repeat for Alpha Particle and Find Ratio ### **For the \(\alpha\) particle:** \[ \gamma_\alpha = 1 + \frac{5.00}{3727.2} = 1.00134 \] \[ v_\alpha = c \sqrt{1 - \frac{1}{(1.00134)^2}} \] Calculate: \[ (1.00134)^2 \approx 1.00268 \] \[ \frac{1}{1.00268} \approx .99733 \] \[ v_\alpha = c \sqrt{1 - .99733} = c \sqrt{.00267} = c \times .0517 \] ### **Velocity Ratio:** \[ \frac{v_\alpha}{v_\beta} = \frac{.0517}{.9957} \approx .052 \] --- ## **Final Answer** \[ \boxed{\frac{v_\alpha}{v_\beta} \approx .052} \] **Conclusion:** The velocity of a 5.00-MeV \(\alpha\) particle is only about 5.2% that of a 5.00-MeV \(\beta\) particle (electron), confirming that even with the same kinetic energy, \(\beta\) rays travel much faster than \(\alpha\) particles due to their much smaller mass.