Question

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

• A. $2$
• B. $\frac{1}{2}$
• C. $\left(\frac{1}{2}\right)^{1/3}$
• D. $\frac{1}{\sqrt{2}}$

Hint:

Remember the chaining of proportions for nuclear fission: Radius depends on the cube root of mass ($R \propto m^{1/3}$), and mass is inversely proportional to velocity ($m \propto 1/v$) due to momentum conservation. Combining these facts gives $R \propto \left(\frac{1}{v}\right)^{1/3}$. Simply flip the velocity ratio ($2/1 \rightarrow 1/2$) and take the cube root to find the answer instantly!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
An unstable atomic nucleus at rest undergoes fission, splitting into two separate fragments. Given the ratio of their ejection velocities, we need to determine the ratio of their physical nuclear radii.

Step 2: Key Formula or Approach:
1. Law of Conservation of Linear Momentum: Since no external forces act on the system, the initial momentum (zero) equals the final momentum: $$m_1 v_1 + m_2 v_2 = 0 \implies m_1 v_1 = m_2 v_2 \implies \frac{m_1}{m_2} = \frac{v_2}{v_1}$$ 2. Nuclear Mass-Radius Relation: The mass number ($A$) of a nucleus is directly proportional to its mass ($m$). The radius ($R$) of a nucleus relates to its mass number via: $$R = R_0 A^{1/3} \implies R \propto m^{1/3}$$

Step 3: Detailed Explanation:
We are given the velocity ratio of the two splitting pieces: $$\frac{v_1}{v_2} = \frac{2}{1}$$ Using the conservation of momentum relation, the mass ratio is the inverse of the velocity ratio: $$\frac{m_1}{m_2} = \frac{v_2}{v_1} = \frac{1}{2}$$ Now, substitute this mass ratio into the nuclear radius scaling relationship: $$\frac{R_1}{R_2} = \left(\frac{m_1}{m_2}\right)^{1/3}$$ $$\frac{R_1}{R_2} = \left(\frac{1}{2}\right)^{1/3}$$

Step 4: Final Answer:
The ratio of their nuclear radii is $\left(\frac{1}{2}\right)^{1/3}$, which corresponds to option (C).