Question

In a particle accelerator, a neutron moving with velocity v ≈ 2 × 103 m/s collides elastically with a C-12 nucleus at rest. Assuming head-on collision, the fraction a of neutron's kinetic energy transferred to the carbon nucleus is (Ignore relativistic effects)

Answer 1

48/169

Answer 2

In a head-on elastic collision between a neutron (mass $m_n$) moving with velocity $v_n$ and a C-12 nucleus (mass $m_C$) at rest ($v_C = 0$), both linear momentum and kinetic energy are conserved.

Conservation of linear momentum:

$m_n v_n + m_C v_C = m_n v_n' + m_C v_C'$

$m_n v_n = m_n v_n' + m_C v_C'$ (since $v_C = 0$)

Conservation of kinetic energy:

$\frac{1}{2} m_n v_n^2 + \frac{1}{2} m_C v_C^2 = \frac{1}{2} m_n v_n'^2 + \frac{1}{2} m_C v_C'^2$

$\frac{1}{2} m_n v_n^2 = \frac{1}{2} m_n v_n'^2 + \frac{1}{2} m_C v_C'^2$ (since $v_C = 0$)

Solving these two equations for the final velocities $v_n'$ and $v_C'$ gives:

$v_C' = \frac{2 m_n}{m_n + m_C} v_n$

The initial kinetic energy of the neutron is $K_n = \frac{1}{2} m_n v_n^2$.

The kinetic energy transferred to the carbon nucleus is its final kinetic energy, $K_C' = \frac{1}{2} m_C v_C'^2$.

The fraction $\alpha$ of the neutron's kinetic energy transferred to the carbon nucleus is:

$\alpha = \frac{K_C'}{K_n} = \frac{\frac{1}{2} m_C v_C'^2}{\frac{1}{2} m_n v_n^2} = \frac{m_C}{m_n} \left(\frac{v_C'}{v_n}\right)^2$

Substitute the expression for $v_C'/v_n$:

$\alpha = \frac{m_C}{m_n} \left(\frac{\frac{2 m_n}{m_n + m_C} v_n}{v_n}\right)^2 = \frac{m_C}{m_n} \left(\frac{2 m_n}{m_n + m_C}\right)^2 = \frac{4 m_n^2 m_C}{m_n (m_n + m_C)^2} = \frac{4 m_n m_C}{(m_n + m_C)^2}$

The mass of a C-12 nucleus is approximately 12 times the mass of a neutron ($m_C \approx 12 m_n$). Let $m_n = m$ and $m_C = 12m$.

$\alpha = \frac{4 m (12m)}{(m + 12m)^2} = \frac{48 m^2}{(13m)^2} = \frac{48 m^2}{169 m^2} = \frac{48}{169}$

Therefore, the fraction of neutron's kinetic energy transferred to the carbon nucleus is $\frac{48}{169}$. The initial velocity value is not needed for this calculation.