\(v = \left(ax\right)^{3 / 2} \\ m = 0 .5~\text{kg} , a = 5 ~\text{m}^{- 1 / 2} \text{s}^{- 1} \)

We know that the acceleration is given by

\(a_{0} = \frac{dv}{dt} = v \frac{dv}{dx} = \left(ax\right)^{3 / 2} \frac{d}{dx} \left(ax\right)^{3 / 2} \\ = \left(ax\right)^{3 / 2} \times a \times \frac{3}{2} \times x^{1 / 2} = \frac{3}{2} a^{2} x^{2}\)

Step 2: Find the work done.

\(\text{Work done}= \int_{x = 0}^{x = 2} Fdx = \int_{0}^{2} \frac{3 \left(ma^{2} \right)x^{2} dx}{2} \\ = \frac{3}{2} \left(ma^{2} \right)\times \left[\right. x^{3} / 3 \left.\right]_{0}^{2} \\ = \frac{1}{2} \left(ma^{2}\right) \times 8 = \frac{1}{2} \times \left(\right. 0 .5 \left.\right) \times \left(\right. 25 \left.\right) \times 8 = 50 ~\text J\)

Hence, option (2) is the correct answer.