$A$ stationary nucleus (mass number = $A$) emits an $\alpha$-particle with velocity $v$. Find the velocity of the daughter nucleus.

Consider the following statements:
$A.$ Atoms of each element emit a characteristic spectrum.
$B.$ According to Bohr's postulate, an electron in a hydrogen atom revolves in a certain stationary orbit.
$C.$ The density of nuclear matter depends on the size of the nucleus.
$D.$ A free neutron is stable, but a free proton decay is possible.
$E.$ Radioactivity is an indication of the instability of nuclei.
Choose the correct answer from the options given below:

Column $II$ gives certain systems undergoing a process. Column $I$ suggests changes in some of the parameters related to the system. Match the statements in Column $I$ to the appropriate process$(es)$ from Column $II$.

Column $I$	Column $II$
$(A)$ The energy of the system is increased	$(p)$ $System:$ $A$ capacitor, initially uncharged. $Process:$ It is connected to a battery.
$(B)$ Mechanical energy is provided to the system, which is converted into energy of random motion of its parts	$(q)$ $System:$ $A$ gas in an adiabatic container fitted with an adiabatic piston. $Process:$ The gas is compressed by pushing the piston.
$(C)$ Internal energy of the system is converted into its mechanical energy	$(r)$ $System:$ $A$ gas in a rigid container. $Process:$ The gas gets cooled due to colder atmosphere surrounding it.
$(D)$ Mass of the system is decreased	$(s)$ $System:$ $A$ heavy nucleus, initially at rest. $Process:$ The nucleus fissions into two fragments of nearly equal masses and some neutrons are emitted.
	$(t)$ $System:$ $A$ resistive wire loop. $Process:$ The loop is placed in a time-varying magnetic field perpendicular to its plane.

$A$ radioactive nucleus emits an $\alpha$-particle to become a stable nucleus. If the velocity of the $\alpha$-particle is $\upsilon$ and the mass number of the original radioactive nucleus is $A$,what will be the velocity of the daughter nucleus?

The graph between the number of decayed atoms $N'$ of a radioactive element and time $t$ is:

During the disintegration of a radioactive nucleus of mass number $208$ at rest, two alpha particles each with kinetic energy $E$ are emitted. The total kinetic energy of the emitted alpha particles and the daughter nucleus after the disintegration is