Question

A particle of mass $M$ at rest decays into two particle of masses $m_1$ and $m_2$ having non zero velocities. The ratio of the de Broglie wavelength of particles ${\lambda }_1/{\lambda }_2$ :

• 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 velocity of particle of mass $m_1$ is $v_1$ and velocity of particle of mass $m_2$ is $v_2$.

There is no external force is acting so momentum will be conserved.

Before Decay, Linear momentum$=0$,

After decay, Linear momentum$=m_1{v}_1+m_2{v}_2$

$\Rightarrow m_1{v}_1+m_2{v}_2=0$ $\Rightarrow m_1{v}_1=-m_2{v}_2$

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

de Broglie wavelength, $\lambda =\frac{h}{mv}$

$\Rightarrow \frac{{\lambda }_1}{{\lambda }_2}=\frac{m_2{v}_2}{m_1{v}_1}=1$

Hence, ${\lambda }_1:{\lambda }_2=1:1$