Question

A nucleus of uranium decays at rest into nuclei of thorium and helium. Then:
A. The helium nucleus has less kinetic energy than the thorium nucleus.
B. The helium nucleus has more kinetic energy than the thorium nucleus.
C. The helium nucleus has less momentum than the thorium nucleus.
D. The helium nucleus has more momentum than the thorium nucleus.

Answer 1

Hint: Assume that the net force on the given system is zero. Then we can apply the law of conservation of momentum. Then we can discuss the momentums of the nuclei after the fission. To compare the kinetic energies of the nuclei, we can use the relation between the kinetic energy and the momentum of a particle.

Formula used:
$K=\dfrac{{{p}^{2}}}{2m}$

Complete answer:
It is given that a nucleus of uranium is at rest. It decays into two nuclei of thorium and helium. Meaning the nucleus of thorium breaks into nuclei of thorium and helium. This process is called fission When this happens the nuclei of thorium and helium will move apart with some kinetic energy.
Let us assume that there is no external force affecting the system of these nuclei. The nucleus of uranium breaks apart due the internal forces. And we know that the net internal force of a system is always zero. Therefore, the net force on the system is zero.
This means the momentum of the system will be conserved. Meaning the momentum of the system before the fission and have the fission will be the same.
Since the uranium nucleus is at rest initially, the momentum before fission is zero. Therefore, the net momentum of the after the fission will be zero.

Therefore, the momentums of the nuclei of thorium and helium will be of equal magnitudes and in opposite directions so that the net momentum of the system is zero.
Now, let us discuss their kinetic energy. For this, we will use the relation between momentum (p) and the kinetic energy (K) of a particle.
i.e. $K=\dfrac{{{p}^{2}}}{2m}$ …… (i), where m is the mass of the particle.

Here, the magnitudes of the momentums of both the nuclei are the same. Therefore, the kinetic energy of the nucleus will depend only on the mass of the nucleus.
From equation (i), we get that the nuclei have more mass and will have less kinetic energy. Whereas the nucleus with less mass will have less kinetic energy.
The helium nucleus has less mass than a thorium nucleus. Therefore, the helium nucleus has more kinetic energy than the thorium nucleus.

So, the correct answer is “Option B”.

Note:
We can also say that if the momentum of a particle is constant, then the kinetic energy of the particle is inversely proportional to the mass of the particle.
Therefore, less the mass, more will be the kinetic energy of the particle.