1. Given data

• Initial nucleus is at rest.
• Initial mass of nucleus $= M + \Delta m$.
• It decays into two identical daughter nuclei, each of mass $\dfrac{M}{2}$.

We need the speed of each daughter nucleus.

2. Use conservation of momentum

Since the parent nucleus is initially at rest, total initial momentum is zero.

After decay, the two daughter nuclei have equal masses and must move in opposite directions with equal speeds $v$.

So momentum is automatically conserved.

3. Use conservation of energy

Initial energy: $E_i = (M+\Delta m)c^2$

Final energy consists of:

• Rest energy of two daughters: $2\left(\frac{M}{2}c^2\right)=Mc^2$
• Plus their kinetic energies.

Hence total kinetic energy released is $K = E_i - Mc^2 = (M+\Delta m)c^2 - Mc^2 = \Delta m\,c^2$

4. Distribute kinetic energy equally

Since the two daughter nuclei are identical, each gets equal kinetic energy: $K_1 = K_2 = \frac{\Delta m\,c^2}{2}$

5. Apply non-relativistic kinetic energy

For each daughter nucleus, mass is $\dfrac{M}{2}$, so $K_1 = \frac{1}{2}\left(\frac{M}{2}\right)v^2 = \frac{Mv^2}{4}$

Set this equal to $\dfrac{\Delta m c^2}{2}$: $\frac{Mv^2}{4} = \frac{\Delta m c^2}{2}$

Multiply by $4$: $Mv^2 = 2\Delta m c^2$

So, $v^2 = \frac{2\Delta m}{M}c^2$

Therefore, $v = c\sqrt{\frac{2\Delta m}{M}}$

6. Match with options

This corresponds to:

$\boxed{\text{B: } c\sqrt{\frac{2\Delta m}{M}}}$

7. Comparison with stored correct answer

Stored correct answer: B

Our derived answer: B

So they agree.