\(u = 0, t = T \) and acceleration \(= a\)

\(\therefore v = 0 + aT = aT\)and \({S_1} = 0 + \frac{1}{2}a{T^2} = \frac{1}{2}a{T^2}\)

For Second part,

\(u = aT,\) retardation \(=a_1\), \(v = 0\) and time taken \(= T_1\) (let)

\(\therefore \)\(0 = u - {a_1}{T_1}\)\( \Rightarrow aT = {a_1}{T_1}\)

and from \({v^2} = {u^2} - 2a{S_2}\) \( \Rightarrow {S_2} = \frac{{{u^2}}}{{2{a_1}}} = \frac{1}{2}\frac{{{a^2}{T^2}}}{{{a_1}}}\)

\({S_2} = \frac{1}{2}aT \times {T_1}\) \(\left( {As\,\,{a_1} = \frac{{aT}}{{{T_1}}}} \right)\)

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

\( = \frac{{\frac{1}{2}aT\;(T + {T_1})}}{{T + {T_1}}}\) \( = \frac{1}{2}aT\)