Answer
$F=43.9N$
Work Step by Step
According to the principle of work and energy
$T_A+\Sigma U_{A-B}=T_B$
$\implies \frac{1}{2}mv_A^2-U_w+U_F-U_S=\frac{1}{2}mv_B^2$
$\implies 0+F\Delta S-\frac{1}{2}kS^2-mgS=\frac{1}{2}mv_B^2$
We plug in the known values to obtain:
$F(0.5-0.391)-\frac{1}{2}(100)(0.15)^2-(0.8)(9.81)(0.15)=\frac{1}{2}(0.8)(2.5)^2$
This simplifies to:
$F=43.9N$
