Kinetic energy is defined as the half of the product of mass and square times the velocity. K.E=‌
1
2
Mv2,K.E where is the kinetic energy, M is the mass and v is the velocity of the object whose kinetic energy We are calculating.
First Case: The kinetic energy of the man is half the kinetic energy of the boy with half the mass. The man has a mass M and velocity v1 whereas the boy has mass ‌
M
2
and velocity v2 . ‌
1
2
Mv12=‌
1
2
(‌
1
2
‌
M
2
v22) v12=‌
1
4
(v22)
Second Case: The kinetic energy of the man is equal to the kinetic energy of the boy with half the mass. The man has a mass M and velocity v1+1 whereas the boy has mass ‌
M
2
and velocity v2. ‌
1
2
M(v1+1)2=‌
1
2
‌
M
2
v22 (v1+1)2=‌
1
2
v22 By taking the ratio both these relations we get ‌