For First part,

u = 0, t = T and acceleration = a

\(\therefore \mathbf{V} = 0 + a \mathbf{T} = a \mathbf{T}\) and \(S_1 = 0 + \frac{1}{2} a t^2 = \frac{1}{2} a t^2\)

For Second part,

u = aT, retardation= a 1 , \(\mathbf{V} = 0\) and time taken = T 1 (let)

\Theta = u - a_{1} T_{1} \(\Rightarrow \vec{a} T = \vec{a}_1 T_1\)

and from v^{2} = u^{2} + 2aS_{2} \(\Rightarrow S_{2} = \frac{u^{2}}{2a_{1}} = \frac{1}{2} a^{2} T^{2} \frac{1}{a_{1}}\)

\(S_2 = \frac{1}{2} a T \times T_1\) \(\left( \text{As } a_{1} = \frac{a^{T}}{T_{1}} \right)\)

\(v_{av} = \frac{S_1 + S_2}{T + T_1} = \frac{\frac{1}{2} a T^2 + \frac{1}{2} a T \times T_1}{T + T_1}\)

\(= \frac{\frac{1}{2} a T \left(T + T_1 \right)}{T + T_1}\) \(= \frac{1}{2} a T^{2}\)