Answer
$k=773N/m$
Work Step by Step
The required stiffness of the spring can be determined as follows:
$T_1+V_1=T_2+V_2$
$\implies \frac{1}{2}mv_1^2+mgh_1+2\frac{1}{2}kx_1^2=\frac{1}{2}mv_2^2+mgh_2+2\frac{1}{2}kx_2^2$
We plug in the known values to obtain:
$0+2(\frac{1}{2}k(2-1)^2)=0+2(\frac{1}{2}k(\sqrt{(0.5)^2+(2)^2}-1)^2)-20(9.81)(0.5)$
This simplifies to:
$k=773N/m$
