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-Broglie wavelengths of the particles, ${\lambda }_1/{\lambda }_2$ is

• A. $m_1/m_2$
• B. $m_2/m_1$
• C. $1.0$
• D. $\sqrt{m_2}/\sqrt{m_1}$

Answer 1

The correct option is C $1.0$

By the law of conservation of momentum
$0=m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}\Rightarrow m_1\overrightarrow{v_1}=-m_2\overrightarrow{v_2}$
Negative sign indicates that both the particles are moving in the opposite direction.
And we say that the magnitudes of individual momentums are same. i.e,
$m_1{v}_1=m_2{v}_2$
Now de-Broglie wavelengths
${\lambda }_1=\frac{h}{m_1{v}_1}$ and ${\lambda }_2=\frac{h}{m_2{v}_2}$;
$\therefore \frac{{\lambda }_1}{{\lambda }_2}=\frac{m_2{v}_2}{m_1{v}_1}=1$