neutron in a reactor makes an elastic head-on collision with the nucleus of an atom initially at rest.
Assume: The mass of the atomic nucleus is about 12.5 the mass of the neutron.
What fraction of the neutron's kinetic energy is transferred to the atomic nucleus?
To solve this problem, we can use the concept of conservation of momentum and kinetic energy.
Let's denote the mass of the neutron as mn and the mass of the atomic nucleus as mnucleus. We are given that mnucleus is about 12.5 times the mass of the neutron, so mnucleus = 12.5 * mn.
Before the collision, the neutron is moving and the atomic nucleus is at rest. So the initial momentum of the system is only due to the neutron: p_initial = mn * v_neutron, where v_neutron is the velocity of the neutron.
After the collision, the neutron and the atomic nucleus will move with different velocities. Let's denote the velocities of the neutron and the atomic nucleus after the collision as v_neutron' and v_nucleus', respectively.
According to the conservation of momentum, we have:
p_initial = p_final
mn * v_neutron = mn * v_neutron' + mnucleus * v_nucleus'
In an elastic collision, the kinetic energy is conserved. So the initial kinetic energy of the system is given by:
KE_initial = (1/2) * mn * v_neutron^2
The final kinetic energy of the system is given by the sum of the kinetic energy of the neutron and the kinetic energy of the atomic nucleus:
KE_final = (1/2) * mn * v_neutron'^2 + (1/2) * mnucleus * v_nucleus'^2
The fraction of the neutron's kinetic energy transferred to the atomic nucleus can be calculated as:
Fraction transferred = (KE_initial - KE_final) / KE_initial
Now let's calculate this fraction. First, let's solve the momentum conservation equation for v_neutron':
mn * v_neutron = mn * v_neutron' + mnucleus * v_nucleus'
v_neutron' = (mn * v_neutron - mnucleus * v_nucleus') / mn
Substituting this expression for v_neutron' in the equation for KE_final, we have:
KE_final = (1/2) * mn * [(mn * v_neutron - mnucleus * v_nucleus') / mn]^2 + (1/2) * mnucleus * v_nucleus'^2
Simplifying this equation gives us:
KE_final = (1/2) * mn * [(mn * v_neutron - mnucleus * v_nucleus')^2 / mn^2] + (1/2) * mnucleus * v_nucleus'^2
= [(mn * v_neutron - mnucleus * v_nucleus')^2 / 2mn] + (mnucleus * v_nucleus'^2 / 2)
Now we can substitute this expression for KE_final into the fraction transferred equation:
Fraction transferred = (KE_initial - KE_final) / KE_initial
= [((1/2) * mn * v_neutron^2) - {[(mn * v_neutron - mnucleus * v_nucleus')^2 / 2mn] + (mnucleus * v_nucleus'^2 / 2)}] / ((1/2) * mn * v_neutron^2)
Now we just need to plug in the given mass ratio mnucleus = 12.5 * mn, cancel out terms, and simplify the expression to get the final fraction transferred value. The specific numerical result will depend on the values of the neutron's mass and velocity.