A body of mass 500g moves along x -axis such that it's velocity...

It is given that

$v = 10\sqrt{x}m{s}^{- 1}$

Force is defined by $F = m\dfrac{dv}{dt} . . . ( i )$

Multiplying and dividing equation (i) by $dx$

$F = m\dfrac{dv}{dt}\dfrac{dx}{dx} ⇒ F = m\dfrac{dx}{dt}\dfrac{dv}{dx} = mv\dfrac{dv}{dx} . . . ( ii )$

Finding the values of the terms in equation (ii)
$\dfrac{dv}{dx} = \dfrac{10}{2\sqrt{x}} = \dfrac{5}{\sqrt{x}}$
$v\dfrac{dv}{dx} = a = 10\sqrt{x} \times \dfrac{5}{\sqrt{x}} = 50m{s}^{- 2}$

Hence, the force is
$F = \dfrac{1}{2} \times 50 = 25N$.