Head-on Collision of Neutron with Nucleus

In a head-on collision between a neutron and a nucleus, we need to determine the fraction of energy retained by the nucleus after the collision. To solve this problem, we can use the concept of conservation of momentum and conservation of energy.

Conservation of Momentum:
In a collision, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces are acting on the system.

Conservation of Energy:
In an ideal elastic collision, both momentum and kinetic energy are conserved. However, in a real-world scenario, there may be some energy losses due to various factors like deformation, heat, and sound generation.

Given Data:
- Speed of neutron (v) = 10^6 m/s
- Mass number of the nucleus (A) = 80
- Assume the initial velocity of the nucleus (V) is zero.

Calculating the Fraction of Energy Retained:

1. Calculate the initial momentum of the system.
- Momentum (p) = mass (m) × velocity (v)
- Momentum of neutron (p_neutron) = m_neutron × v_neutron
- Momentum of nucleus (p_nucleus) = m_nucleus × v_nucleus (initially zero)
- Total initial momentum (p_initial) = p_neutron + p_nucleus

2. Apply the conservation of momentum to find the final velocity of the nucleus.
- Momentum of the neutron after collision = -p_neutron (opposite direction)
- Momentum of the nucleus after collision = p_nucleus
- Total final momentum (p_final) = -p_neutron + p_nucleus

3. Calculate the final kinetic energy of the system.
- Kinetic energy (KE) = 0.5 × mass × velocity^2
- Initial kinetic energy (KE_initial) = 0.5 × m_neutron × v_neutron^2
- Final kinetic energy (KE_final) = 0.5 × m_neutron × (-v_neutron)^2 + 0.5 × m_nucleus × v_nucleus^2

4. Calculate the fraction of energy retained by the nucleus.
- Fraction of energy retained = (KE_final - KE_initial) / KE_initial

Explanation:
In a head-on collision, the neutron transfers some of its momentum and energy to the nucleus. The fraction of energy retained by the nucleus depends on the masses and velocities of both the neutron and the nucleus.

By applying the conservation of momentum, we can find the final velocity of the nucleus after the collision. Then, using the conservation of energy, we can calculate the final kinetic energy of the system. The difference between the initial and final kinetic energies gives us the energy lost during the collision.

The fraction of energy retained by the nucleus is obtained by dividing the energy lost by the initial kinetic energy. This fraction represents the efficiency of the collision in terms of energy transfer.

It's important to note that in real-world scenarios, collisions are not perfectly elastic, and some energy is always lost due to various factors such as deformation, heat generation, and sound production.