In a nuclear fission reaction of an isotope of mass "M" ,three similar daughter nuclei of same mass are formed.The speed of a daughter nuclei in terms of mass defect `Delta M` will be :

Options

A. `c sqrt((2Delta M)/(M))`
B. `c sqrt((3Delta M)/(M))`
C. `(Delta Mc^(2))/(3)`
D. `sqrt((2c Delta M)/(M))`

Answer

To solve the problem of finding the speed of daughter nuclei formed in a nuclear fission reaction in terms of the mass defect \( \Delta M \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Reaction**:
In a nuclear fission reaction, a nucleus of mass \( M \) splits into three daughter nuclei of equal mass. Therefore, the mass of each daughter nucleus is \( \frac{M}{3} \).

2. **Mass Defect**:
The mass defect \( \Delta M \) is the difference between the mass of the original nucleus and the total mass of the daughter nuclei formed. This mass defect is converted into energy according to Einstein's equation \( E = \Delta M c^2 \).

3. **Kinetic Energy of Daughter Nuclei**:
The energy released during the fission will be converted into the kinetic energy of the daughter nuclei. Since there are three daughter nuclei, the total kinetic energy (KE) can be expressed as:
\[ KE = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \frac{1}{2} m_3 v_3^2 \]
Given that all daughter nuclei have the same mass \( \frac{M}{3} \) and the same speed \( v \), we can simplify this to:
\[ KE = 3 \times \frac{1}{2} \left(\frac{M}{3}\right) v^2 = \frac{1}{2} M v^2 \]

4. **Equating Energy**:
The total energy released in the fission reaction is equal to the kinetic energy of the daughter nuclei:
\[ \Delta M c^2 = \frac{1}{2} M v^2 \]

5. **Solving for Speed \( v \)**:
Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{2 \Delta M c^2}{M} \]
Taking the square root of both sides gives:
\[ v = \sqrt{\frac{2 \Delta M c^2}{M}} \]

### Final Answer:
The speed of a daughter nucleus in terms of mass defect \( \Delta M \) is:
\[ v = \sqrt{\frac{2 \Delta M c^2}{M}} \]