Answer
$3.71cm$
Work Step by Step
We can find the required compression done by putty-block system as follows:
$P_i=m_1v_{1i}+m_2v_{2i}=0+m_2v_{2i}=m_2v_{2i}$
and $P_f=(m_1+m_2)v_f$
According to law of conservation of momentum
$m_2v_{2i}=(m_1+m_2)v_f$
This can be rearranged as:
$v_f=\frac{m_2v_{2i}}{m_1+m_2}$
We plug in the known values to obtain:
$v_f=\frac{(0.05Kg)(2.3m/s)}{0.43Kg+0.05Kg}=0.2396m/s$
At equilibrium
$\frac{1}{2}Kx^2=\frac{1}{2}mv_f^2$
This can be rearranged as:
$x^2=\frac{(m_1+m_2)v_f^2}{K}$
$\implies x=\sqrt{\frac{(m_1+m_2)v_f^2}{K}}$
We plug in the known values to obtain:
$x=\sqrt{\frac{(0.43Kg+0.05Kg)(0.2396m/s)^2}{20.0N/m}}$
$x=0.03712m$
$x=3.71cm$