b
(b)

\(\tan \theta=\frac{1}{2}=\frac{F_0}{t_0-4}\)

\(\Rightarrow F_0=\frac{t_0-4}{2}\)

Total change in momentum should be zero, then only it will retain its initial momentum.

So, positive area of \(F-t\) curve should be equal to negative area of \(F-t\) curve till time \(t_0\).

\(\frac{1}{2}(4)(1)=\frac{1}{2}\left(t_0-4\right) F_0\)

\(8=\frac{\left(t_0-4\right)}{2} \cdot \frac{\left(t_0-4\right)}{2}\)

\(\left(t_0-4\right)^2=32\)

\(t_0=4+2 \sqrt{2}\)