Question
The velocity of particle $$A$$ is $$0.1\,m{s^{ - 1}}$$ and that of particle $$B$$ is $$0.05\,m{s^{ - 1}}.$$ If the mass of particle $$B$$ is five times that of particle $$A,$$ then the ratio of de-Broglie wavelengths associated with the particles $$A$$ and $$B$$ is
A.
2 : 5
B.
3 : 4
C.
6 : 4
D.
5 : 2
Answer :
5 : 2
Solution :
$${\text{Given,}}\,{v_A} = 0.1\,m{s^{ - 1}}\,{\text{and}}$$ $${v_B} = 0.05\,m{s^{ - 1}}{\text{also,}}$$ $${m_B} = 5{m_A}$$
$$\eqalign{
& {\text{de - Broglie wavelength,}}\,\lambda = \frac{h}{{mv}} \cr
& \therefore \,\,\frac{{{\lambda _A}}}{{{\lambda _B}}} = \frac{{\frac{h}{{{m_A}{v_A}}}}}{{\frac{h}{{{m_B}{v_B}}}}} = \frac{{{m_B}{v_B}}}{{{m_A}{v_A}}} \cr
& = \frac{{5{m_A} \times 0.05}}{{{m_A} \times 0.1}} = 5 \times 0.5 = 2.5 = \frac{5}{2} \cr
& \therefore \,\,{\lambda _A}:{\lambda _B} = 5:2 \cr} $$