(c)

$a=\alpha t^{3 / 2}$ (acceleration is a function of time)

$\int \limits_u^v d v=\int \limits_0^t a d t$

$\Rightarrow \int \limits_u^v d v=\int \limits_0^t \alpha t^{3 / 2} d t$

$\left.\Rightarrow v\right|_u ^v=\left.\alpha \frac{t^{3 / 2}+1}{\frac{3}{2}+1}\right|_0 ^t$

$\Rightarrow (v-u)=\alpha \times \frac{2}{5} \quad\left(t^{5 / 2}-0\right)$

$\Rightarrow v-u=\frac{2}{5} \alpha t^{5 / 2}$

$\Rightarrow v=u+\frac{2}{5} \alpha t^{5 / 2}$