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

Dual Nature of Radiation PYQ — JEE Main 2023 · 1 Feb · Shift 1 · Q54 with Solution

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Correct answer: C

For a particle,

\lambda = \frac{h}{p}

If the proton and the alpha particle have the same de Broglie wavelength, then their momenta are equal:

p_p = p_\alpha

For non-relativistic motion,

K = \frac{p^2}{2m}

Since both particles have the same momentum $p$ ,

K \propto \frac{1}{m}

Therefore,

\frac{K_p}{K_\alpha} = \frac{m_\alpha}{m_p}

An alpha particle has mass approximately equal to $4$ times the mass of a proton:

m_\alpha \approx 4m_p

Thus,

\frac{K_p}{K_\alpha} = \frac{4m_p}{m_p} = 4

So the ratio is

K_p : K_\alpha = 4 : 1

Hence, the correct option is C.