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

\(\int \limits_u^v d v=\int \limits_0^t a d t\)

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

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

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

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

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