A neutron in a nuclear reactor moves at 4.80 x 106 m/s

Question

A neutron in a nuclear reactor moves at 4.80 x 106 m/s and makes an elastic head-on collision with a stationary carbon atom. The mass of the carbon atom is 12 times the mass of the neutron. 

a) Determine the velocity of each after the collision. 

b) Determine the fraction (percentage) of the neutron’s kinetic energy transferred to the carbon atom.

Answer 1

This can be viewed from the standpoint of conservation of momentum and energy.

The total momentum before collision is equal to the total momentum after collision. In equation form, this is

M_n x V_ni = M_n x V_nf + M_carbon x V_carbon where M_n = mass of neutron, V_ni = initial velocity of neutron, V_nf = final velocity of neutron. Since the M_carbon = 12 x M_n. This can now be written

M_n x V_ni = M_n x V_nf + 12 x M_n x V_carbon which simplifies to

1) V_ni = V_nf + 12 x V_carbon There are two unknowns V_nf and V_carbon.

The total kinetic energy after collision is equal to the initial kinetic energy of the neutron.

1/2 M_n x V_ni² = 1/2 M_n x V_nf² + 1/2 x 12 M_n x V²_carbon This simplifies to

2) V_ni² = V_nf² + 12 x V²_carbon This has the same unknowns as Equation 1).

These can be solved simultaneously with a lot of algebra, but it can be done.

Answer 2

So E M.,

The fastest way to a solution in such problems is by mentally (or otherwise) transforming to Center-of-Mass coordinates. Imagine that YOU are at the CM for the system, that is, you are moving towards the carbon at 1/13 of the neutron speed (remember, CM uses net momentum, total mass = 13). Then after the collision, everything just rebounds IN THAT CM frame-of reference. So the C atom is moving away from you at 1/13 initial neutron speed, (=2/13 initial neutron speed in stationary frame=="S-frame") and the neutron is heading back FROM YOU at 12/13 of initial S-frame speed, so that is 11/13 backwards in S-frame. You work out the kinetic energies -- but remember that you ONLY need the new E(C) and the original E(n), the new E(n) is guaranteed to be the remainer, since it is an elastic collision.

-- Cheers, --Mr. d.