Question #278383

A neutron in a reactor makes an elastic headon collision with the nucleus of an atom initially at rest. Assume: The mass of the atomic nucleus is about 11.3 the mass of the neutron. What fraction of the neutron’s kinetic energy is transferred to the atomic nucleus?


1
Expert's answer
2021-12-12T16:41:29-0500

m = mass of neutron

M = mass of nucleus = 11.3 m

vi = initial velocity of neutron before collision = ?

Vi = initial velocity of nucleus before collision = 0 m/s

vf = final velocity of neutron after collision

Vf = final velocity of nucleus after collision

using conservation of momentum

mvi+MVi=mvf+MVf(m)vi+(11.3m)(0)=mvf+(11.3m)Vfvi=vf+(11.3)Vfvivf=(11.3)Vf      eq1mv_i + MV_i = mv_f + MV_f \\ (m) v_i + (11.3 m) (0) = m v_f + (11.3 m) V_f \\ v_i = v_f + (11.3) V_f \\ v_i - v_f = (11.3) V_f \;\;\; eq-1

using conservation of kinetic energy

mvi2+MVi2=mvf2+MVf2(m)vi2+(11.3m)(0)2=mvf2+(11.3m)Vf2vi2=vf2+(11.3)Vf2vi2vf2=(11.3)Vf2(vivf)(vi+vf)=(11.3)Vf2(13.7)Vf(vi+vf)=(11.3)Vf2vi+vf=Vf        eq2m v^2_i + M V^2_i = m v^2_f + M V^2_f \\ (m) v^2_i + (11.3 m) (0)^2 = m v^2_f + (11.3 m) V^2_f \\ v^2_i = v^2_f + (11.3) V^2_f \\ v^2_i - v^2_f = (11.3) V^2_f \\ ( v_i - v_f )( v_i + v_f ) = (11.3) V^2_f \\ (13.7) V_f ( v_i + v_f ) = (11.3) V^2_f \\ v_i + v_f = V_f \;\;\;\; eq-2

adding eq-1 and eq-2

(vivf)+(vivf)=(11.3)Vf+Vf2vi=12.3Vfvi=6.15Vf( v_i - v_f ) + ( v_i - v_f ) = (11.3) V_f + V_f \\ 2 v_i = 12.3 V_f \\ v_i = 6.15 V_f

fraction of energy transferred is given as

fraction=(0.5)MVf2(0.5)mvi2fraction=MVf2mvi2fraction=(11.3m)Vf2m(6.15Vf)2fraction=11.3(6.15)2=0.2980.3fraction = \frac{(0.5) MV^2_f }{(0.5) m v^2_i } \\ fraction = \frac{M V^2_f }{m v^2_i } \\ fraction = \frac{(11.3 m) V^2_f }{m (6.15 V_f )^2 } \\ fraction = \frac{11.3}{(6.15)^2} = 0.298 ≈ 0.3


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS