A heavy nucleus at rest breaks into two fragments which fly off with velocities in the ratio `8 : 1`. The ratio of radii of the fragments is.

Options

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

Answer

To solve the problem step by step, we will apply the principles of conservation of momentum and the relationship between mass, volume, and radius of the fragments.

### Step 1: Understand the conservation of momentum
When a heavy nucleus breaks into two fragments, the total momentum before the break must equal the total momentum after the break. Since the nucleus is initially at rest, the total initial momentum is zero.

### Step 2: Set up the momentum equation
Let the masses of the two fragments be \( m_1 \) and \( m_2 \), and their velocities be \( v_1 \) and \( v_2 \). According to the conservation of momentum:
\[ m_1 v_1 + m_2 v_2 = 0 \]
This implies:
\[ m_1 v_1 = -m_2 v_2 \]

### Step 3: Relate mass to volume and radius
The mass of a nucleus can be expressed in terms of its volume and density. For spherical fragments, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass \( m \) is proportional to the volume:
\[ m \propto r^3 \]
We can express the masses of the fragments as:
\[ m_1 = k r_1^3 \quad \text{and} \quad m_2 = k r_2^3 \]
where \( k \) is a constant that includes the density.

### Step 4: Substitute masses into the momentum equation
Substituting the expressions for \( m_1 \) and \( m_2 \) into the momentum equation gives:
\[ (k r_1^3) v_1 = (k r_2^3) v_2 \]
The constant \( k \) cancels out:
\[ r_1^3 v_1 = r_2^3 v_2 \]

### Step 5: Express the ratio of radii
Rearranging the equation gives:
\[ \frac{r_1^3}{r_2^3} = \frac{v_2}{v_1} \]
Taking the cube root of both sides results in:
\[ \frac{r_1}{r_2} = \left(\frac{v_2}{v_1}\right)^{1/3} \]

### Step 6: Use the given velocity ratio
We are given that the velocities are in the ratio \( v_1 : v_2 = 8 : 1 \). Therefore:
\[ \frac{v_1}{v_2} = 8 \implies \frac{v_2}{v_1} = \frac{1}{8} \]
Substituting this into the radius ratio equation gives:
\[ \frac{r_1}{r_2} = \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2} \]

### Step 7: State the final ratio of the radii
Thus, the ratio of the radii of the fragments \( r_1 : r_2 \) is:
\[ r_1 : r_2 = 1 : 2 \]

### Final Answer
The ratio of the radii of the fragments is \( 1 : 2 \).

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