Problem $1$: Real mechanical systems may involve the deflection of nonlinear springs. In the figure, a block of mass $m$ is released a distance $h$ above a nonlinear spring. The resistance force $F$ of the spring is given by:

$F=-(k_{1}d+k_2{d}^{\frac{3}{2}})$

Conservation of energy can be used to show that

$0=\frac{2k_2{d}^{\frac{5}{2}}}{5}+\frac{k_1{d}^2}{2}-mgd-mgh$

Using Newton's method in the $2^{nd}$ equation, solve for the contraction distance $d,$ given the following parameter values: $k_1=40,000\frac{g}{s^2},k_2=40\frac{g}{s^2{m}^5},m=95g,g=9\mathrm{.}81\frac{m}{s^2},$ and $h=0\mathrm{.}43m.$