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:
Relate the radius of nuclear fission fragments to their velocities using conservation laws and proportionality chains.

Step 2: Key Formula or Approach:
Nuclear radius scales as R ∝ m^(1/3) (from R = R₀A^(1/3)). Momentum conservation gives m ∝ 1/v. Chaining these: R ∝ (1/v)^(1/3).

Step 3: Detailed Explanation:
To find the radius ratio from the velocity ratio, flip the velocity fraction and take the cube root. If fragment velocities are in the ratio 2:1, then the radii scale as (1/2)^(1/3) relative to each other—the slower, more massive fragment has the larger radius. This chained proportionality condenses three separate physical laws into a single mental operation.

Step 4: Final Answer:
The radius ratio is the cube root of the inverse velocity ratio.