Explanation:

When a neutron collides head-on with an atom, it transfers some of its kinetic energy to the atom. The fractional kinetic energy retained by the neutron can be calculated using the concept of conservation of momentum and kinetic energy.

Conservation of Momentum:
In a head-on collision, the momentum before and after the collision should be conserved. The momentum of the neutron before the collision is given by its mass (m_n) multiplied by its velocity (v_n). The momentum of the atom before the collision is given by its mass (m_a) multiplied by its velocity (v_a).

Before collision: m_n * v_n + m_a * v_a

After collision: m_n * v'_n + m_a * v'_a

Since it is a head-on collision, the atom and neutron will move in opposite directions, so the velocities will have opposite signs.

Conservation of Kinetic Energy:
The kinetic energy before the collision is given by (1/2) * m_n * v_n^2. The kinetic energy after the collision is given by (1/2) * m_n * v'_n^2 + (1/2) * m_a * v'_a^2.

Since the atom is much heavier than the neutron, we can neglect its initial kinetic energy (m_a * v_a^2) as it is negligible compared to the neutron's kinetic energy.

Calculating the Fractional Kinetic Energy Retained:
To calculate the fractional kinetic energy retained by the neutron, we need to find the ratio of its kinetic energy after the collision to its initial kinetic energy.

Initial kinetic energy of the neutron = (1/2) * m_n * v_n^2
Final kinetic energy of the neutron = (1/2) * m_n * v'_n^2

Fractional kinetic energy retained = Final kinetic energy / Initial kinetic energy
= [(1/2) * m_n * v'_n^2] / [(1/2) * m_n * v_n^2]
= (v'_n^2) / (v_n^2)

Since the collision is head-on, the direction of the neutron's velocity changes, but its magnitude remains the same. So, v'_n = -v_n.

Fractional kinetic energy retained = [(-v_n)^2] / (v_n^2)
= 1

Therefore, the fractional kinetic energy retained by the neutron is 1, which can be represented as (A/A) or (A 1/A 1).