The velocity of a particle $A$ is 3 times the velocity of proton. If the ratio of the de Broglie wavelengths…

The velocity of a particle $A$ is 3 times the velocity of proton. If the ratio of the de Broglie wavelengths of the particle $\mathrm{A}$ and the proton is 3:2, the mass of the particle $\mathrm{A}$ is ( $\mathrm{m}_{\mathrm{p}}$ - mass of proton)
  1. $\frac{2}{9} \mathrm{~m}_{\mathrm{p}}$
  2. $\frac{2}{3} \mathrm{~m}_{\mathrm{p}}$
  3. $\frac{2}{5} \mathrm{~m}_{\mathrm{p}}$
  4. $\frac{2}{7} \mathrm{~m}_{\mathrm{p}}$

Solution

De-Broglie wavelength $\lambda=\frac{\mathrm{h}}{\mathrm{p}}$ or $\lambda \propto \frac{1}{\mathrm{mv}}$ $ \begin{aligned} & \therefore \frac{\lambda_{\mathrm{A}}}{\lambda_{\mathrm{P}}}=\frac{\mathrm{m}_{\mathrm{p}} \mathrm{v}_{\mathrm{p}}}{\mathrm{m}_{\mathrm{A}} \mathrm{v}_{\mathrm{A}}} \text { or, } \frac{3}{2}=\frac{\mathrm{m}_{\mathrm{p}} \mathrm{v}_{\mathrm{p}}}{\mathrm{m}_{\mathrm{A}} 3 \mathrm{v}_{\mathrm{p}}} \Rightarrow \frac{3}{2}=\frac{\mathrm{m}_{\mathrm{p}}}{3 \mathrm{~m}_{\mathrm{A}}} \\ & \therefore \mathrm{m}_{\mathrm{A}}=\frac{2}{9} \mathrm{~m}_{\mathrm{p}} \end{aligned} $

Asked in: AP EAMCET 2023 (15 May Shift 1)

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