Answer
a) $7.71\frac{m}{s}$
b) $x=0.26m$
Work Step by Step
$E_K=\frac{mv^2}{2}$
$E_P=mgh$
a) $\frac{m(v_i)^2}{2}=\frac{mv^2}{2}-mgh$
$v=\sqrt{(v_i)^2+2gh}=\sqrt{(4.5\frac{m}{s})^2+2(9.8\frac{m}{s^2})(2.0m)}=7.71\frac{m}{s}$
b) $\frac{mv^2}{2}=\frac{kx^2}{2}-mgx$
$\frac{(62kg)(7.71)^2}{2}=\frac{(5.8\times10^4\frac{N}{m})x^2}{2}-(62kg)(9.8\frac{m}{s^2})x$
$0=29000x^2-608x-1840$
$x=0.26m$