Question

A nucleus of mass $M+\vartriangle m$ is at rest and decays into two daughter nuclei of equal mass of $\dfrac{M}{2}$ each. The speed of light is c, then speed of daughter nuclei is ?
$\begin{aligned} & (A)c\dfrac{\vartriangle m}{M+\vartriangle m} \\\ & (B)c\sqrt{\dfrac{2\vartriangle m}{M}} \\\ & (C)c\sqrt{\dfrac{\vartriangle m}{M}} \\\ & (D)c\sqrt{\dfrac{\vartriangle m}{M+\vartriangle m}} \\\ \end{aligned}$

Answer 1

The total energy of products equal to the total energy releases. By substituting the values and equating them we will get a new equation. Substituting for momentum as the product of mass and velocity and then by rearranging then we will get the value of the speed of daughter nuclei formed by decay.

Complete answer:
$M+\vartriangle m\to \dfrac{M}{2}+\dfrac{M}{2}$
Total kinetic energy of products= Total energy released.
$\Rightarrow \dfrac{{{P}^{2}}}{2m}+\dfrac{{{P}^{2}}}{2m}$=(mass defect ) ${{c}^{2}}$
where,$m=\dfrac{M}{2}$
$\Rightarrow 2\dfrac{{{P}^{2}}}{2m}=\left[ (M+\vartriangle m)-(\dfrac{M}{2}+\dfrac{M}{2}) \right]\times {{c}^{2}}$
$\Rightarrow \dfrac{{{P}^{2}}}{m}=\left[ \vartriangle m{{c}^{2}} \right]$
$\Rightarrow \dfrac{{{m}^{^{2}}}{{v}^{2}}}{m}=\vartriangle m{{c}^{2}}$
$\Rightarrow \dfrac{M}{2}{{v}^{2}}=\vartriangle m{{c}^{2}}$
$\Rightarrow {{v}^{2}}=\vartriangle m{{c}^{2}}\times \dfrac{2}{M}$
$\Rightarrow v=\sqrt{\dfrac{2\vartriangle m}{M}}c$

Hence, option (B) is correct.

Additional information:
- Radioactive decay results in a reduction of summed rest mass, once the released energy (the disintegration energy) has escaped in some way. Although decay energy is sometimes defined as associated with the difference between the mass of the parent nuclide products and the mass of the decay products, this is true only of rest mass measurements, where some energy has been removed from the product system. This is true because the decay energy must always carry mass with it, wherever it appears (see mass in special relativity) according to the formula $E=m{{c}^{2}}$ .
- The decay energy is initially released as the energy of emitted photons plus the kinetic energy of massive emitted particles (that is, particles that have rest mass). If these particles come to equilibrium with their surroundings and photons are absorbed, then the decay energy is transformed to thermal energy, which retains its mass.
- Decay energy therefore remains related to a particular measure of mass of the decay system, called invariant mass, which doesn't change during the decay, albeit the energy of decay is distributed among decay particles. The energy of photons, the kinetic energy of emitted particles, and, later, the thermal energy of the surrounding matter, all contribute to the invariant mass of the system. Thus, while the sum of the rest masses of the particles is not conserved in radioactive decay, the system mass and system invariant mass (and also the system total energy) is conserved throughout any decay process.

Note:
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is considered radioactive. Three of the foremost common sorts of decay are alpha decay, beta decay and gamma decay, all of which involve emitting one or more particles or photons.