Question

Electron, proton, neutron and alpha particle have the same momentum. Which particle has more kinetic energy?

Answer 1

Hint: It is given in the question that the momentum of all the particles are the same. Thus, we have to find the relation between Kinetic energy with mass. Then by analysing the mass of every particle with the relation of kinetic energy to the mass, we will find the answer.

Complete step by step answer:
As per the given question the momentum of the given electron, proton, neutron and alpha particle are equal.
Let us consider the momentum of the particles to be $p$.
Momentum of a particle is defined as the product of its velocity and mass.
So, mathematically we get,
Momentum $p = m \times v$
The variables are,
$m = $ mass of the particle
$v = $ velocity of the particle
Let the kinetic energy of the particle be $K$.
The formula for Kinetic energy is,
$K = \dfrac{1}{2}m{v^2}$
The variables are,
$K = $ Kinetic energy
$m = $ mass of the particle
$v = $ velocity of the particle
Now, we can say, $K \propto m{v^2}$
Also, it can be arranged as, $K \propto {\left( {mv} \right)^2} \times \dfrac{1}{m}$
Substituting the value of $p = mv$ we get,
$K \propto {p^2} \times \dfrac{1}{m}$
In the given question it is stated that the momentum $p$ of the particle is constant.
Hence, we can say that, $K \propto \dfrac{1}{m}$
The mass of the given particles are,
Mass of electron is $9 \times {10^{ - 31}}{\text{ }}kg$
Mass of proton is $1.67 \times {10^{ - 27}}{\text{ }}kg$
Mass of neutron is $1.67 \times {10^{ - 27}}{\text{ }}kg$
Mass of alpha particle is $6.6 \times {10^{ - 27}}{\text{ }}kg$
Mass of electron < Mass of proton$ \approx $ Mass of neutron < Mass of alpha particle
Hence, according to the relation $K \propto \dfrac{1}{m}$ we get,
The mass of the particle which is the smallest has the most kinetic energy. So, here the mass of the electron is the least. So, its kinetic energy is most.
Kinetic energy of electron > Kinetic energy of proton$ \approx $ Kinetic energy of neutron > Kinetic energy of alpha particle.
Electrons have the most Kinetic energy.

Note: It must be noted that we have to formulate the relation between the kinetic energy and momentum in such a way that the variable which will be left must be able to answer the problem. If we had kept velocity as variable, then with the help of velocities of the particles we could not find any answer.