A running man has half the kinetic energy of that of a boy of half of his mass. The man speeds up by 1 m/s. So as to have same kinetic energy as that of the boy. The original speed of the man is :

Let mass and spend of man be M and v respectively.

Let speed of the boy be v, them

$\frac{1}{2}M{v}^2=\frac{1}{2}\left[\begin{array}{c}\frac{1}{2}\left(\frac{M}{2}\right)v^2\end{array}\right]$ ...(i)

$\frac{1}{2}M(v+1{)}^2=\frac{1}{2}\left[\begin{array}{c}\frac{M}{2}\end{array}\right]v^2$ ...(ii)

Dividing Eq. (i) by (ii) we obtain 

or $\frac{v^2}{(v+1{)}^2}=\frac{1}{2}$

or $\frac{v^2}{v+1}=\frac{1}{\sqrt{2}}$

or $\sqrt{2}v=v+1$ 

or $v(\sqrt{2}-1)=v+1$

or $v=\frac{1}{\sqrt{2}-1}m/s$