A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.

#### Solution 1

Let *m*, *m*_{1}, and *m*_{2} be the respective masses of the parent nucleus and the two daughter nuclei. The parent nucleus is at rest.

Initial momentum of the system (parent nucleus) = 0

Let *v*_{1} and *v*_{2} be the respective velocities of the daughter nuclei having masses *m*_{1}and *m*_{2}.

Total linear momentum of the system after disintegration = 'm_1v_1 + m_2v_2`

According to the law of conservation of momentum:

Total initial momentum = Total final momentum

`0 = m_1v_1 + m_2+v_2`

`v_1 = (-m_2v_2)/m_1`

Here, the negative sign indicates that the fragments of the parent nucleus move in directions opposite to each other.

#### Solution 2

Let m1, m2 be the masses of products and v_{1},v_{2} be their respective velocities. Therefore, total linear momentum after disintegration = m_{1}v_{1} +m_{2} v_{2}. Before disintegration, the nucleus is at rest.

Therefore, its linear momentum before disintegration is zero.

According to the principle of conservation of linear momentum

`m_1vecv_1 + m_2vecv_2 = 0 or vecv_2 = - (m_1vecv_1)/m_2`

Negative sign shows that v_{1} and v_{2} are in opposite directions