The de-Broglie wavelength of a proton and α-particle are equal. The ratio of their velocities is :

Solution

Physics sir · 3 steps

Same wavelength means same momentum, so the lighter particle has to move faster.

1. Equal wavelength. The de Broglie relation is λ = h/mv. If the two wavelengths are equal, the momenta mv must be equal too.

   mₚ vₚ = m_α v_α ⇒ vₚ/v_α = m_α/mₚ

2. Mass of each. An α-particle is a helium nucleus: 2 protons and 2 neutrons. So m_α ≈ 4mₚ.
3. So. vₚ/v_α = 4mₚ/mₚ = 4, so the ratio is 4 : 1, option (c).

The idea to keep. Equal de Broglie wavelength means equal momentum, nothing more. Velocity then goes inversely with mass, so compare masses and flip the ratio.

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