A nucleus of mass `M + Deltam` is at rest and decays into two daughter nuclei of equal mass `M/2` each. Speed of light is C.
The speed of deughter nuclei is :-

Options

A. `c(Deltam)/(M+Deltam)`
B. `csqrt((2Deltam)/(M))`
C. `csqrt((Deltam)/(M))`
D. `csqrt((Deltam)/(M+Deltam))`

Answer

To solve the problem, we will use the principles of conservation of energy and mass-energy equivalence. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a nucleus of mass \( M + \Delta m \) at rest that decays into two daughter nuclei, each of mass \( \frac{M}{2} \). We need to find the speed of the daughter nuclei after the decay.

### Step 2: Apply Conservation of Energy
According to the law of conservation of energy, the energy released during the decay (due to the mass defect) will be converted into the kinetic energy of the daughter nuclei.

The energy released can be expressed using Einstein's mass-energy equivalence:
\[ E = \Delta m c^2 \]

### Step 3: Calculate Kinetic Energy of Daughter Nuclei
Since the two daughter nuclei have equal mass \( \frac{M}{2} \) and are moving with the same speed \( v \), the total kinetic energy (KE) of the two nuclei can be expressed as:
\[ KE = 2 \times \left(\frac{1}{2} \times \frac{M}{2} \times v^2\right) = \frac{M}{2} v^2 \]

### Step 4: Set the Energy Released Equal to the Kinetic Energy
Now, we equate the energy released during the decay to the total kinetic energy of the daughter nuclei:
\[ \Delta m c^2 = \frac{M}{2} v^2 \]

### Step 5: Solve for \( v \)
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 us:
\[ v = c \sqrt{\frac{2 \Delta m}{M}} \]

### Final Answer
Thus, the speed of the daughter nuclei is:
\[ v = c \sqrt{\frac{2 \Delta m}{M}} \]