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 :

Solution

Physics sir · 4 steps

Both ions carry the same charge e, so they gain equal kinetic energy and speed goes as one over root mass.

1. Energy from the field. An ion of charge q that falls through a potential difference V from rest gains kinetic energy qV.

   qV = 1/2mv² ⇒ v = √(2qV/m)

2. Same charge. H⁺ has charge e. Singly ionized helium, He⁺, has lost one electron, so its charge is also e. With the same q and V, the speed depends only on mass: v ∝ 1/√m.
3. Compare masses. A helium nucleus has 2 protons and 2 neutrons, so m_(He) ≈ 4m_H. Then v_H/v_(He) = √(m_(He)/m_H) = √4 = 2.
4. So. The ratio of speeds of the hydrogen ion and the helium ion is 2 : 1, which is option (a).

The idea to keep. Accelerated from rest through a potential V, every particle has v = √(2qV/m). Read the charge from how many electrons are removed, not from the atomic number.

Answer from the NTA final answer key. Solution by PrepWiser.