A proton moving with one tenth of velocity of light has a certain de Broglie wavelength of λ. An alpha particle having certain kinetic energy has the same de-Brogle wavelength λ. The ratio of kinetic energy of proton and that of alpha particle is:

Solution

Physics sir · 4 steps

Same de Broglie wavelength means same momentum, so the lighter particle carries more kinetic energy.

1. Equal wavelength. From λ = h/p, two particles with the same λ must have the same momentum p. The speed c/10 only fixes what that common λ is, and we do not need its value.
2. Energy from momentum. Write kinetic energy in terms of momentum. With p the same for both, K depends only on mass, and it goes as 1/m.

   K = p²/2m ⇒ Kₚ/K_α = m_α/mₚ

3. Put in masses. An alpha particle has 2 protons and 2 neutrons, so m_α ≈ 4mₚ. That gives Kₚ/K_α = 4mₚ/mₚ = 4.
4. So. The ratio of kinetic energies of the proton and the alpha particle is 4 : 1, option (c).

The idea to keep. Equal de Broglie wavelengths always mean equal momenta. Then use K = p²/2m: kinetic energy goes as 1/m, so the lighter particle has more energy.

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