Question 507: Atoms & Nuclei

Question

A nucleus breaks into two nuclear parts, which have their velocity ratio $$2: 1$$. The ratio of their nuclear radii will be

Accepted Answer

When a nucleus breaks into two parts, conservation of momentum applies. Let's denote the masses of the two nuclear parts as $$m_1$$ and $$m_2$$, and their velocities as $$v_1$$ and $$v_2$$ respectively. According to the conservation of momentum:
$$m_1 v_1 = m_2 v_2$$
Given that the velocity ratio is $$v_1 : v_2 = 2 : 1$$, we can write:
$$\frac{v_1}{v_2} = 2 \implies v_1 = 2v_2$$
Substituting this into the momentum equation:
$$m_1 (2v_2) = m_2 v_2 \implies 2m_1 = m_2 \implies \frac{m_1}{m_2} = \frac{1}{2}$$
Now, the nuclear radius $ R $ is related to the mass number $ A $ (which corresponds to the mass) by the relation:
$$R \propto A^{1/3}$$
Thus, the ratio of the radii $R_1$ and $R_2$ of the two nuclear parts will be:
$$\frac{R_1}{R_2} = \left( \frac{A_1}{A_2} \right)^{1/3} = \left( \frac{m_1}{m_2} \right)^{1/3} = \left( \frac{1}{2} \right)^{1/3} = \frac{1}{2^{1/3}}$$
Therefore, the correct answer is:
Option C
$$\frac{1}{2^{1 / 3}}$$

Answer

$$\sqrt{2}$$

Answer

$$\frac{1}{2}$$

Answer

$$\frac{1}{2^{1 / 3}}$$

Answer

$$\frac{1}{\sqrt{2}}$$

Atoms & Nuclei

A nucleus breaks into two nuclear parts, which have their velocity ratio $2: 1$. The ratio of their nuclear radii will be

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