Question

A nucleus moving with a velocity $\overrightarrow{v}$ emits an $\alpha$-p article. Let the velocities of the $\alpha$-particle and the remaining nucleus be $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ and their masses be $m_1$, and $m_2$ respectively. Choose the correct option.

• A. $\overrightarrow{v}$,$\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ must be parallel to each other.
• B. None of the two of $\overrightarrow{v}$,$\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ should be parallel to each other.
• C. $\overrightarrow{v_1}$ + $\overrightarrow{v_2}$ must be parallel to $\overrightarrow{v}$ .
• D. $m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}$ must be parallel to $\overrightarrow{v}$ .

Answer 1

The correct option is D

$m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}$ must be parallel to $\overrightarrow{v}$ .

If initially the nucleus is moving with a velocity $\overrightarrow{v}$, then considering the nucleus as a point mass,
its centre of mass is moving with $\overrightarrow{v}$.
Now after the disintegration of nucleus into $\alpha$-particle and the residual, the COM should be moving with the same velocity as there is no net external force acting upon it.
And we also know,
${\overrightarrow{v}}_{CM}=\frac{m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}}{m_1+m_2}$
Which clearly shows that ($m_1\overrightarrow{v_1}+m_2\overrightarrow{v_2}$) must be parallel to $\overrightarrow{v}$.