Question

A particle of mass $M$ at rest decays into two particles of masses $m_1$ and $m_2$, having non-zero velocities. The ratio of the de-Brogile wavelengths of the particles, $\frac{I_1}{I_2}$ is:

• A. $\frac{m_1}{m_2}$
• B. $\frac{m_2}{m_1}$
• C. $1:1$
• D. $\sqrt{\frac{m_2}{m_1}}$

Answer 1

The correct option is C $1:1$

Let particle 1 have mass $m_1$ and velocity $v_1$

Let particle 2 have mass $m_2$ and velocity $v_2$

conservation of momentum :

$m_1{v}_1+m_2{v}_2=0$

$m_1{v}_1=-m_2{v}_2$

$v_1/v_2=m_2/m_1$ .....(1)

De-broglie wave equation = $\lambda =h/mv$

So the ratio = $\frac{{\lambda }_1}{{\lambda }_2}=\frac{\lambda /m_1{v}_1}{\lambda /m_2{v}_2}=\frac{m_2{v}_2}{m_1{v}_1}$

$=\frac{m_2/m_1}{v_2/v_1}$

Using equation (1)

= 1