Retardation \(\rightarrow a\)

Initial velocity \(\rightarrow u\)

\((I)\) For total journey

\(v=u+a t\)

\(0=u-a T\)

\(\Rightarrow u=a T \ldots (i)\)

\(d=u T-\frac{1}{2} a T^2\)

Dividing by \(2\) on both sides

\(\frac{d}{2}=\frac{u T}{2}-\frac{1}{2} \frac{a T^2}{2} \ldots (ii)\)

On comparing equation \((i)\) and \((iii)\)

\(\frac{u T}{2}-\frac{1}{2} \frac{a T^2}{2}=u t-\frac{1}{2} a t^2\)

Put \(u=a T\)

\(\Rightarrow \frac{a T^2}{2}-\frac{a T^2}{4}=a T t-\frac{1}{2} a t^2\)

\(\Rightarrow \frac{T^2}{4}=T t-\frac{t^2}{2}\)

Multiplying by 4 on both sides

\(T^2=4 T t-2 t^2 \Rightarrow 2 t^2-4 T t+T^2=0\)

On solving this quadratic equation,

\(t=T-\frac{T}{\sqrt{2}} \Rightarrow t=T\left(1-\frac{1}{\sqrt{2}}\right)\)

\((II)\) For half journey

\(\frac{d}{2}=u t-\frac{1}{2} a t^2 \ldots (iii)\)