A heavy nucleus at rest breaks into two fragments which fly off with velocities in the ratio 27:1. The ratio of the radii of the fragments (assumed spherical) is

Options

A. `1:3`
B. `1:4`
C. `4:1`
D. `2:1`

Answer

To solve the problem, we need to find the ratio of the radii of two fragments after a heavy nucleus breaks into them, given that their velocities are in the ratio of 27:1.

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a heavy nucleus that breaks into two fragments. The velocities of these fragments are given in the ratio \( v_1 : v_2 = 27 : 1 \).

2. **Apply Conservation of Momentum**:
Since the nucleus is initially at rest, the total momentum before and after the break must be conserved. Therefore, we have:
\[ m_1 v_1 = m_2 v_2 \]
where \( m_1 \) and \( m_2 \) are the masses of the two fragments.

3. **Express Mass Ratio in Terms of Velocity Ratio**:
From the momentum conservation equation, we can express the mass ratio as:
\[ \frac{m_1}{m_2} = \frac{v_2}{v_1} \]
Substituting the given velocity ratio:
\[ \frac{m_1}{m_2} = \frac{1}{27} \]

4. **Relate Mass to Volume and Radius**:
For spherical fragments, the mass is related to the volume and density. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass can be expressed as:
\[ m = \rho V = \rho \left( \frac{4}{3} \pi r^3 \right) \]
Therefore, mass is proportional to the cube of the radius:
\[ m \propto r^3 \]

5. **Set Up the Mass Ratio in Terms of Radius**:
Using the mass proportionality, we can write:
\[ \frac{m_1}{m_2} = \frac{r_1^3}{r_2^3} \]
Substituting the mass ratio we found earlier:
\[ \frac{r_1^3}{r_2^3} = \frac{1}{27} \]

6. **Take the Cube Root**:
To find the ratio of the radii, we take the cube root of both sides:
\[ \frac{r_1}{r_2} = \frac{1}{3} \]

7. **Final Result**:
Therefore, the ratio of the radii \( r_1 : r_2 \) is:
\[ r_1 : r_2 = 1 : 3 \]

### Conclusion:
The ratio of the radii of the fragments is \( 1 : 3 \).

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