A nucleus travelling with a velocity v emits an α -particle. Let...

According to the law of conservation of linear momentum, the total momentum of the system remains conserved since no external force acts on it.

Initial momentum of the nucleus = $(m_1 + m_2) \vec{v}$

Final momentum of the system after emission = $m_1 \vec{v}_1 + m_2 \vec{v}_2$

Equating the initial and final momentum:

$(m_1 + m_2) \vec{v} = m_1 \vec{v}_1 + m_2 \vec{v}_2$

Since $(m_1 + m_2)$ is a positive scalar quantity, the vector $m_1 \vec{v}_1 + m_2 \vec{v}_2$ is a scalar multiple of $\vec{v}$ and lies in the same direction.

Thus, $m_1 \vec{v}_1 + m_2 \vec{v}_2$ must be parallel to $\vec{v}$.