Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

Published by Pearson
ISBN 10: 0321740904
ISBN 13: 978-0-32174-090-8

Chapter 28 - The Electric Potential - Exercises and Problems - Page 836: 54

Answer

$4.1\times 10^7\;\rm m/s$

Work Step by Step

Using the conservation of energy principle and assuming that there are no external forces exerted on this system [the Alpha particle+ the thorium nucleus]. $$U_i+K_i=U_f+K_f$$ where $_i\rightarrow$ when the particle is still on the thorium nucleus surface, and $_f\rightarrow$ when it is far away from it. The thorium nucleus remains constant in both cases, but the alpha particle was initially at rest, and finally when $r\rightarrow\infty$, its speed is $v_\alpha$. $$\dfrac{kq_{\alpha}q_{\rm thorium}}{r_{\rm thorium}}+0=0+\frac{1}{2}m_{\alpha}v_{\alpha}^2$$ $$v_\alpha=\sqrt{\dfrac{2kq_{\alpha}q_{\rm thorium}}{m_{\alpha}r_{\rm thorium}}}$$ where $q_{\alpha}=2e$, and $q_{\rm thorium}=90e$ $$v_\alpha=\sqrt{\dfrac{360ke^2}{m_{\alpha}r_{\rm thorium}}}$$ Plug the known; $$v_\alpha=\sqrt{\dfrac{360(9\times 10^9)(1.6\times 10^{-19})^2}{(4\times 1.67\times 10^{-27})(7.5\times 10^{-15})}}$$ $$v_\alpha=\color{red}{\bf 4.1\times 10^7}\;\rm m/s$$
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