As we know that,
$$
\text { W.D. }=\int_{x_1}^{x_2} \vec{F} \cdot \overline{d x}=\int_{x_1}^{x_2} m \vec{a}_0 \cdot \overrightarrow{d x}
$$
As given that, $m=0.5 \mathrm{~kg}, a=5 \mathrm{~m}^{-1 / 2} \mathrm{~s}^{-1}$, work done $(W)=$ ?
$$
v=a x^{3 / 2}
$$
We also know that,
Acceleration,
$$
\begin{aligned}
&a_0=\frac{d v}{d t}=v \cdot \frac{d v}{d x}=a x^{3 / 2} \frac{d}{d x}\left(a x^{3 / 2}\right) \\
&=a x^{3 / 2} \times a \times \frac{3}{2} \times x^{1 / 2}=\frac{3}{2} a^2 x^2 \\
&\text { Now, Force }=m a_0=m \frac{3}{2} a^2 x^2 \\
&\text { From(i), } \\
&\text { Work done }=\int_{x=0}^{x=2} F d x \\
&=\int_0^2\left[\frac{3}{2} m a^2 x^2\right] d x \\
&=\frac{3}{2} m a^2 \times\left(\frac{x^3}{3}\right)_0^2 \\
&=\frac{1}{2} m a^2 \times 8 \\
&=\frac{1}{2} \times(0.5) \times(25) \times 8=50 \mathrm{~J} \\
&
\end{aligned}
$$