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

Explanation

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

### Step 1: Understand the System
We have a nucleus of mass \( M + \Delta m \) that is at rest. It decays into two daughter nuclei, each of mass \( \frac{M}{2} \).

### Step 2: Apply Conservation of Momentum
Since the initial nucleus is at rest, the total initial momentum is zero. After the decay, the momentum of the two daughter nuclei must also sum to zero.

Let the speed of each daughter nucleus be \( v \). Therefore, the momentum of the first nucleus is \( \frac{M}{2} v \) and the momentum of the second nucleus is \( \frac{M}{2} (-v) \) (since they move in opposite directions).

Setting the total momentum to zero:
\[ \frac{M}{2} v + \frac{M}{2} (-v) = 0 \]
This confirms that momentum is conserved.

### Step 3: Calculate the Mass Defect
The mass defect \( \Delta m \) is the difference between the initial mass and the total mass of the products:
\[ \text{Mass defect} = (M + \Delta m) - \left(\frac{M}{2} + \frac{M}{2}\right) = (M + \Delta m) - M = \Delta m \]

### Step 4: Apply Conservation of Energy
The energy released during the decay can be expressed using the mass defect:
\[ E = \Delta m c^2 \]
This energy is converted into the kinetic energy of the two daughter nuclei.

### Step 5: Write the Kinetic Energy Expression
The total kinetic energy (KE) of the two daughter nuclei is:
\[ \text{Total KE} = 2 \times \left(\frac{1}{2} \times \frac{M}{2} v^2\right) = \frac{M}{2} v^2 \]

### Step 6: Set the Energies Equal
Setting the kinetic energy equal to the energy released:
\[ \frac{M}{2} v^2 = \Delta m c^2 \]

### Step 7: Solve for \( v^2 \)
Rearranging the equation gives:
\[ v^2 = \frac{2 \Delta m c^2}{M} \]

### Step 8: Solve for \( v \)
Taking the square root of both sides:
\[ v = \sqrt{\frac{2 \Delta m c^2}{M}} \]

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

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