A neutron travelling with some velocity, collides ellastically head on, with the nucleus of an atom of mass number A at rest. The fraction fo Total energy retained by neutron is??

To determine the fraction of total energy retained by a neutron after it collides elastically with a nucleus at rest, we need to consider the principles of conservation of momentum and kinetic energy. This scenario involves a neutron colliding with a nucleus, which can be modeled using the equations of elastic collisions in one dimension.

Understanding the Collision Dynamics

In an elastic collision, both momentum and kinetic energy are conserved. Let's denote:

• m_n: mass of the neutron
• M: mass of the nucleus (which can be approximated as A times the mass of a proton, m_p)
• v_n: initial velocity of the neutron
• u_n: final velocity of the neutron after the collision
• u_M: final velocity of the nucleus after the collision

Applying Conservation Laws

From the conservation of momentum, we have:

m_nv_n = m_nu_n + Mu_M

From the conservation of kinetic energy, we have:

0.5m_nv_n² = 0.5m_nu_n² + 0.5Mu_M²

Solving for Final Velocities

To find the final velocities, we can use the equations derived from these conservation laws. For an elastic collision, the final velocities can be expressed as:

• u_n = (m_n - M)/(m_n + M) * v_n
• u_M = (2m_n)/(m_n + M) * v_n

Calculating the Fraction of Energy Retained

The kinetic energy of the neutron before the collision is:

KE_initial = 0.5m_nv_n²

After the collision, the kinetic energy of the neutron is:

KE_final = 0.5m_nu_n² = 0.5m_n[(m_n - M)/(m_n + M)]²v_n²

The fraction of the total energy retained by the neutron is then given by:

Fraction = KE_final / KE_initial

Substituting the expressions we derived:

Fraction = [(m_n - M)² / (m_n + M)²]

Final Expression

Thus, the fraction of total energy retained by the neutron after the elastic collision with the nucleus is:

Fraction = [(m_n - M)² / (m_n + M)²]

In practical terms, if we consider the mass of the neutron and the nucleus, we can substitute these values to find the specific fraction for any given mass number A. This approach allows us to understand how the neutron's energy is affected by the mass of the nucleus it collides with.