An object with mass 500 g moves along x-axis with speed $$v = 4\sqrt{x}$$ m/s. The force acting on the object is :

The speed of the object varies with its position as $$v = 4\sqrt{x}$$, where $$x$$ is the coordinate (in metres) measured along the +x-axis.

First convert the mass to SI units: given mass $$m = 500\,$$g $$= 0.5\,$$kg.

To find the force we need the acceleration. For one-dimensional motion, acceleration can be expressed in terms of position as
$$a = v\,\frac{dv}{dx}$$

STEP 1 - Differentiate the speed with respect to $$x$$:
$$v = 4x^{1/2} \; \Rightarrow \; \frac{dv}{dx} = 4 \cdot \frac{1}{2}x^{-1/2} = 2x^{-1/2}$$

STEP 2 - Use $$a = v\,\dfrac{dv}{dx}$$:
Substitute $$v = 4x^{1/2}$$ and $$\dfrac{dv}{dx} = 2x^{-1/2}$$:
$$a = \left(4x^{1/2}\right)\left(2x^{-1/2}\right) = 8 \text{ m\,s}^{-2}$$

The acceleration turns out to be a constant $$8 \text{ m\,s}^{-2}$$ (independent of $$x$$).

STEP 3 - Apply Newton’s second law $$F = ma$$:
$$F = 0.5 \times 8 = 4 \text{ N}$$

Hence the force acting on the object is $$4 \text{ N}$$.

Therefore, Option D is correct.