The speed of daughter nuclei is

Options

A. `c (Delta m)/(M+Delta m)`
B. `c sqrt((2Delta m)/(M))`
C. `c sqrt((Delta m)/(M))`
D. `c sqrt((Delta m)/(M+Delta m))`

Explanation

To find the speed of the daughter nuclei after the decay of a parent nucleus, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a parent nucleus (let's denote it as nucleus A) that decays into two daughter nuclei (let's denote them as nuclei B and C).
- We need to find the speed of the daughter nuclei after the decay.

2. **Assumptions**:
- Let the mass of the parent nucleus A be \( m \).
- When it decays, we assume that the masses of the daughter nuclei B and C are equal, thus each has a mass of \( \frac{m}{2} \).

3. **Conservation of Energy**:
- According to the conservation of energy, the energy released during the decay (which is related to the change in mass) will be converted into kinetic energy of the daughter nuclei.
- The change in mass (\( \Delta m \)) can be related to the energy released using Einstein's equation: \( E = \Delta m c^2 \).

4. **Kinetic Energy of Daughter Nuclei**:
- The kinetic energy (KE) of each daughter nucleus can be expressed as:
\[ KE_B = \frac{1}{2} \left(\frac{m}{2}\right) v^2 \]
\[ KE_C = \frac{1}{2} \left(\frac{m}{2}\right) v^2 \]
- Therefore, the total kinetic energy of both daughter nuclei is:
\[ KE_{total} = KE_B + KE_C = \frac{1}{2} \left(\frac{m}{2}\right) v^2 + \frac{1}{2} \left(\frac{m}{2}\right) v^2 = \frac{m}{2} v^2 \]

5. **Equating Energy**:
- The total kinetic energy of the daughter nuclei should equal the energy released during the decay:
\[ \Delta m c^2 = \frac{m}{2} v^2 \]

6. **Solving for Speed \( v \)**:
- Rearranging the equation gives:
\[ v^2 = \frac{2 \Delta m c^2}{m} \]
- Taking the square root of both sides, we find:
\[ v = c \sqrt{\frac{2 \Delta m}{m}} \]

7. **Conclusion**:
- The speed of the daughter nuclei is given by the formula:
\[ v = c \sqrt{\frac{2 \Delta m}{m}} \]
- This indicates that the speed of the daughter nuclei is dependent on the change in mass and the mass of the parent nucleus.