Question

The nucleus at rest disintegrate into two nuclear parts which have their velocities in the ratio \(2: 1\) The ratio of their nuclear sizes will be (A) \(2^{(1 / 3)}: 1\) (B) \(1: 2^{(1 / 3)}\) (C) \(3^{(1 / 2)}: 1\) (D) \(1: 3^{(1 / 2)}\)

Short answer

The ratio of their nuclear sizes will be (A) \(2^{(1/3)}:1\).

Step-by-step solution

Understand the problem and conservation of linear momentum

The problem presents that a nucleus at rest disintegrated into two nuclear parts. We know the velocity ratio of these two parts after disintegration. To find the ratio of their nuclear sizes, we will use the conservation of linear momentum, which states that the total linear momentum before disintegration is equal to the total linear momentum after disintegration.

Set up the linear momentum equation

Let the mass of the first part be \(m_1\) with velocity \(v_1\), and the mass of the second part be \(m_2\) with velocity \(v_2\). Since the nucleus was initially at rest, the initial momentum is zero. After disintegration, the momentum of each part must be equal and opposite so that the total momentum remains zero. Now we know the ratio of their velocities, \(v_1 : v_2 = 2 : 1\), so we can write \(v_1 = 2v_2\). We can set up the equation for the conservation of linear momentum as follows: \[m_1v_1 = m_2v_2\]

Substitute the velocity ratio

Now, we can substitute the given velocity ratio into the linear momentum equation: \[m_1(2v_2) = m_2v_2\] We can simplify and solve for \(m_1\): \[m_1 = \frac{m_2v_2}{2v_2}\] \[m_1 = \frac{1}{2}m_2\] The ratio of the masses of the two nuclear parts will be: \[\frac{m_1}{m_2} = \frac{1}{2}\]

Calculate the ratio of nuclear sizes

Since the nuclear parts are presumed to be spheres, the volume ratio would be the cube root of the mass ratio. To do this, we can write: \[\frac{V_1}{V_2} = \left(\frac{m_1}{m_2}\right)^{(1/3)}\] Substituting the mass ratio, we have: \[\frac{V_1}{V_2} = \left(\frac{1}{2}\right)^{(1/3)}\] \[\frac{V_1}{V_2} = 2^{-1/3} : 1\] Hence, the correct answer is: (A) \(2^{(1 / 3)}: 1\)