Explanation

(c) For first part,
$u = 0,\ t = T$ and acceleration $= a$
$\therefore v = 0 + aT = aT$ and $S_{1} = 0 + \frac{1}{2}aT^{2} = \frac{1}{2}aT^{2}$
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 $v^{2} = u^{2} - 2aS_{2} \Rightarrow S_{2} = \frac{u^{2}}{2a_{1}} = \frac{1}{2}\frac{a^{2}T^{2}}{a_{1}}$
$S_{2} = \frac{1}{2}aT \times T_{1}\ \ \ \ \left( As\ a_{1} = \frac{aT}{T_{1}} \right)$
$\therefore v_{av} = \frac{S_{1} + S_{2}}{T + T_{1}} = \frac{\frac{1}{2}aT^{2} + \frac{1}{2}aT \times T_{1}}{T + T_{1}}$
$= \frac{\frac{1}{2}aT(T + T_{1})}{T + T_{1}} = \frac{1}{2}aT$