Head-on Collision of a Neutron with an Atom

When a neutron collides head-on and elastically with an atom, it transfers some of its kinetic energy to the atom. To determine the amount of kinetic energy retained by the neutron after the collision, we need to consider the conservation of momentum and kinetic energy.

Conservation of Momentum
In an elastic collision, both momentum and kinetic energy are conserved. The momentum before the collision is given by the equation:

Initial momentum = Final momentum

Since the neutron is initially moving with velocity u and the atom is at rest, the initial momentum is given by:

Initial momentum = mu

After the collision, the neutron will move in the opposite direction with a final velocity v and the atom will gain some velocity in the same direction. The final momentum is given by:

Final momentum = mv + Mv'

where m is the mass of the neutron, M is the mass of the atom, v is the final velocity of the neutron, and v' is the final velocity of the atom.

Since the atom is much heavier than the neutron, we can assume that the final velocity of the atom is negligible compared to the final velocity of the neutron:

Final momentum ≈ mv

Therefore, we can write the conservation of momentum equation as:

mu = mv

Conservation of Kinetic Energy
The initial kinetic energy of the neutron is given by the equation:

Initial kinetic energy = 1/2 mu^2

After the collision, the kinetic energy is given by:

Final kinetic energy = 1/2 mv^2

In an elastic collision, the kinetic energy is conserved. Therefore, the final kinetic energy will be equal to the initial kinetic energy:

1/2 mv^2 = 1/2 mu^2

Determining the Retained Kinetic Energy
To find the retained kinetic energy, we need to solve the equation above for v. Dividing both sides of the equation by m and simplifying, we get:

v^2 = u^2

Taking the square root of both sides, we obtain:

v = u

Therefore, the final velocity of the neutron after the collision is equal to its initial velocity. Substituting this value into the equation for the final kinetic energy, we find:

Final kinetic energy = 1/2 mu^2

Hence, the neutron retains all of its initial kinetic energy after the head-on collision with the atom.

(A-B) 2/(A+B)2E