c
\(v _{0}=\alpha t _{1}\) and \(0= v _{0}-\beta t _{2} \Rightarrow v _{0}=\beta t _{2}\)

\(t _{1}+ t _{2}= t\)

\(v _{0}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)= t\)

\(\Rightarrow v _{0}=\frac{\alpha \beta t }{\alpha+\beta}\)

Distance \(=\) area of \(v - t\) graph

\(=\frac{1}{2} \times t \times v _{0}=\frac{1}{2} \times t \times \frac{\alpha \beta t }{\alpha+\beta}=\frac{\alpha \beta t ^{2}}{2(\alpha+\beta)}\)