Answer
Velocity of the father $=2.4$ $m/s$
Work Step by Step
Let the mass and the velocity of the father be $m$ and $v_{f}$ respectively,
and the mass and the velocity of the son be $\frac{m}{2}$ and $v_{s}$ respectively.
The expression of kinetic energy is:
$Kinetic$ $Energy= \frac{1}{2}mv^{2}$
Therefore "father's Kinetic energy is half of the son's Kinetic Energy implies :
$\frac{1}{2}m(v_{f})^{2} = \frac{1}{2} \times [ \frac{1}{2}\frac{m}{2}(v_{s})^{2} ]$
$mv_{f}^{2} = \frac{m}{4}v_{s}^{2}$
$4mv_{f}^{2}=v_{s}^{2}$
$2v_{f} = v_{s}$ . . . . . . . . . . . . . .(1)
If father's velocity is increased by $1 m/s$, it becomes $v_{f}+1$, and their kinetic energies become equal.
Therefore we can write:
$\frac{1}{2}m (v_{f}+1)^{2} = \frac{1}{2}(\frac{m}{2})v_{s}^{2}$
Substituting the value of $v_{s}$ as $2v_{f}$ (from equation 1 ) in the above equation and solving gives:
$\frac{1}{2}m (v_{f}+1)^{2} = \frac{1}{2}(\frac{m}{2}) 2v_{f}^{2}$
Solving this quadratic equation gives:
$v_{f}= \frac {2+ \sqrt 8}{2} = 1+\sqrt 2 = 1+1.4$
$v_{f}= 2.4$ $m/s$
So, velocity of father is $2.4$ $m/s$.
