The momentum $p$ of a particle is given by the product of its mass $m$ and its velocity $v$, that is, $p = m \cdot v$. For a given momentum, the relationship between mass and velocity can be understood as inversely proportional. This means that as the mass increases, the velocity decreases to maintain the same momentum, and vice versa.

The kinetic energy ($K.E.$) of a particle is given by the formula $K.E. = \frac{1}{2} m v^2$. This equation shows that the kinetic energy depends on both the mass of the particle and the square of its velocity.

Given that four particles $A, B, C, D$ have masses $\frac{m}{2}, m, 2 m, 4 m$, respectively, and all have the same momentum, we can assume the momentum of each particle to be $p$. This common value of momentum allows us to express the velocity of each particle in terms of its mass and the common momentum $p$. The velocity $v$ of each particle will be $v = \frac{p}{m}$.

Thus, for each particle, we can determine the velocity as follows:

• For $A$: $v_A = \frac{p}{\frac{m}{2}} = \frac{2p}{m}$


• For $B$: $v_B = \frac{p}{m}$


• For $C$: $v_C = \frac{p}{2m} = \frac{p}{2m}$


• For $D$: $v_D = \frac{p}{4m}$

Now, substituting these velocities into the kinetic energy formula yields the kinetic energies for each particle:

• $K.E._A = \frac{1}{2} \cdot \frac{m}{2} \cdot \left(\frac{2p}{m}\right)^2 = \frac{1}{2} \cdot \frac{m}{2} \cdot \frac{4p^2}{m^2} = \frac{2p^2}{m}$


• $K.E._B = \frac{1}{2} \cdot m \cdot \left(\frac{p}{m}\right)^2 = \frac{1}{2} \cdot m \cdot \frac{p^2}{m^2} = \frac{p^2}{2m}$


• $K.E._C = \frac{1}{2} \cdot 2m \cdot \left(\frac{p}{2m}\right)^2 = \frac{1}{2} \cdot 2m \cdot \frac{p^2}{4m^2} = \frac{p^2}{4m}$


• $K.E._D = \frac{1}{2} \cdot 4m \cdot \left(\frac{p}{4m}\right)^2 = \frac{1}{2} \cdot 4m \cdot \frac{p^2}{16m^2} = \frac{p^2}{8m}$

Comparing these kinetic energies, we see that the particle $A$ has the maximum kinetic energy, as it is inversely related to mass in this scenario, and $A$ has the least mass but the highest velocity squared component, thus maximizing its kinetic energy. Therefore, the correct answer is:

Option D: A