Answer
$v =1.04m/s$
Work Step by Step
The energy from the speed and friction of the block is converted to the energy
used in compressing the spring
$\Delta K + E_{th} =E_s $
$(K_{_2}-K_{_1}) + E_{th} =E_s $
$\frac{1}{2}m(v_{_2}-v_{_1}) = E_s- E_{th} $
$\frac{1}{2}m(v_{_2}^2-v_{_1}^2) = E_s- E_{th} $
$v_{_2}^2-v_{_1}^2 = \frac{2(E_s - E_{th})}{m}$
$-v_{_1}^2 = \frac{2(E_s - E_{th})}{m} -v_{_2}^2$
$-v_{_1}^2 = \frac{2(E_s - E_{th})}{m} -v_{_2}^2$
$v_{_1}^2 = -({\frac{2(E_s - E_{th})}{m} -v_{_2}^2})$
$v =\sqrt {-({\frac{2(E_s - E_{th})}{m} -v_{_2}^2})}$
$v =\sqrt {-({\frac{2(-0.9 - 0.46)}{2.5} -0^2})}$
$v =1.04m/s$
