A hydrogen ion (H^+) and a singly ionized hel | vedclass

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(D) The kinetic energy gained by a charged particle accelerated through a potential difference $V$ is given by $K = qV = \frac{1}{2}mv^2$.From this,th

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$2:1$

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(D) The kinetic energy gained by a charged particle accelerated through a potential difference $V$ is given by $K = qV = \frac{1}{2}mv^2$.From this,the speed $v$ is given by $v = \sqrt{\frac{2qV}{m}}$.For a hydrogen ion $(H^+)$,charge $q_H = e$ and mass $m_H \approx m$.For a singly ionized helium atom $(He^+)$,charge $q_{He} = e$ and mass $m_{He} \approx 4m$.The ratio of speeds is $\frac{v_H}{v_{He}} = \frac{\sqrt{2q_H V / m_H}}{\sqrt{2q_{He} V / m_{He}}} = \sqrt{\frac{q_H}{q_{He}} \cdot \frac{m_{He}}{m_H}}$.Substituting the values: $\frac{v_H}{v_{He}} = \sqrt{\frac{e}{e} \cdot \frac{4m}{m}} = \sqrt{4} = 2$.Thus,the ratio is $2:1$.

$A$ hydrogen ion $(H^+)$ and a singly ionized helium atom $(He^+)$ are accelerated from rest through the same potential difference $(V)$. The ratio of the final speeds of the hydrogen ion and the helium ion is close to:

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