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 ( ) v m M m U + = 2 (b) Hence show that the neutron loses a fraction f of its energy where ( ) 2 4 m M mM f + = (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​