Question

Hydrogen ion and singly ionized helium atom are accelerated, from rest, through the same potential difference. The ratio of final speeds of hydrogen and helium ions is close to :

• A. 2 : 1
• B. 1 : 2
• C. 5 : 7
• D. 10 : 7

Hint:

Both ions carry charge +e, so \(qV = \frac12 mv^2\) gives \(v \propto 1/\sqrt{m}\). The helium ion is about 4 times as heavy as the hydrogen ion.

Answer 1

The Correct Option is A

Energy gained = work done by the field: \(qV = \frac12 mv^2\), so \(v = \sqrt{\frac{2qV}{m}}\).
H⁺ and He⁺ both have charge e and the same V, so \(\frac{v_H}{v_{He}} = \sqrt{\frac{m_{He}}{m_H}}\).
\(m_{He} \approx 4\,m_H\), so \(\frac{v_H}{v_{He}} = \sqrt{4} = 2\).
Ratio = 2 : 1.
Answer: (1) 2 : 1