Let,
m be the mass of the man,
m′ be the mass of the boy,
v be the velocity of the man,
v′ be the velocity of the boy.
As per the question,
12mv2=12(12m′v′2)
12mv2=14m′v′2
12mv2=14m2v′2
v=v′2
v′=2v
In later case
12m(v+1)2=12m′v′2
12m(v+1)2=12(m2)(2v)2
12m(v+1)2=mv2
v2+2v+1=2v2
v2−2v−1
v2−2v=1
v2−2v+1=2
(v−1)2=2
v−1=√2
v=√2+1
and v′=2v
v′=2(√2+1)
Therefore, the speed of man is √2+1 and the speed of boy is 2(√2+1).