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, m1, and m2 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 v1 and v2 be the respective velocities of the daughter nuclei having masses m1and m2.
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 v1,v2 be their respective velocities. Therefore, total linear momentum after disintegration = m1v1 +m2 v2. 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 v1 and v2 are in opposite directions
