A neutron of mass m moving with velocity v collides head-on and elastically with a stationary nucleus of mass M. (a) Show that the velocity of the nucleus after the collision, U, is given by U 2m (m+M) (b) Hence show that the neutron loses a fraction f of its energy where

4mM (m+M)²

(c) A fast neutron enters a target of carbon nuclei which may be assumed to have masses 12 times that of the neutron. How many head-on collisions will it take before the neutron loses 95% of its energy?

(d) Suggest one reason why in a real reactor a neutron is likely to make more collisions with the moderator nuclei before losing this much energy​