Collision between a Neutron and Nucleus

When a neutron traveling with a velocity v and kinetic energy E collides perfectly elastically head-on with a nucleus at rest, we can determine the fraction of the total kinetic energy retained by the neutron.

Conservation of Momentum

In an elastic collision, both momentum and kinetic energy are conserved. Let's consider the collision between the neutron and the nucleus.

- Momentum conservation: The total momentum before the collision is equal to the total momentum after the collision. Since the nucleus is at rest initially, its momentum is zero. Therefore, the initial momentum is simply the momentum of the neutron, given by p = mv.

- Kinetic energy conservation: The total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy of the neutron is given by E = (1/2)mv^2.

Calculating the Fraction of Retained Kinetic Energy

To find the fraction of the total kinetic energy retained by the neutron, we need to determine the final velocities of both the neutron and the nucleus after the collision.

Let's assume the final velocity of the neutron is v' and the final velocity of the nucleus is V. Since the collision is perfectly elastic, kinetic energy is conserved.

After the collision, we can write the following equations:

- Momentum conservation: mv = mv' + mV

- Kinetic energy conservation: E = (1/2)mv^2 = (1/2)mv'^2 + (1/2)mV^2

Calculating the Final Velocities

From the momentum conservation equation, we can solve for V:

V = (mv - mv') / m

Substituting this value of V into the kinetic energy conservation equation:

E = (1/2)mv'^2 + (1/2)m((mv - mv') / m)^2

Simplifying and solving for v':

E = (1/2)mv'^2 + (1/2)m((v - v')^2)

Expanding the squared term:

E = (1/2)mv'^2 + (1/2)m(v^2 - 2vv' + (v')^2)

Rearranging the equation:

2E = mv'^2 + m(v^2 - 2vv' + (v')^2)

Now, we can use the fact that the total kinetic energy is conserved to simplify the equation:

2E = mv^2 - 2mvv' + mv'^2 + mv'^2

Simplifying further:

2E = mv^2 + 2mv'^2 - 2mvv'

Dividing both sides by 2m:

E = v^2 + v'^2 - vv'

Now, we need to solve for v' in terms of v and E. Rearranging the equation:

v' = (E + vv') / (v + v')

Fraction of Retained Kinetic Energy

We can now calculate the fraction of the total kinetic energy retained by the neutron by dividing the final kinetic energy of the neutron by the initial kinetic energy.

The final kinetic energy of the neutron is given by:

E' = (1/2)