Answer
$v_A=26.8ft/s$
Work Step by Step
We can determine the required speed as follows:
$\Delta s_A=\frac{v_{A_1}+v_{A_2}}{2}(2)=\frac{6+v_{A_2}}{2}(2)=6+v_{A_2}$
$\implies \Delta s_B=2\Delta s_A=12+2v_{A_2}$
Now according to the principle of work and energy
$\frac{1}{2}m_Av_{A_1}^2+\frac{1}{2}m_Bv_{B_2^2}+W_A\Delta s_A-\mu_k N\Delta s_B=\frac{1}{2}m_Av_{A_2}^2+\frac{1}{2}m_Bv_{B}^2$
We plug in the known values to obtain:
$\frac{1}{2}(\frac{10}{32.2})+\frac{1}{2}(\frac{4}{32.2})(12)^2+10(6+v_{A_2})-(0.2\times+2v_{A_2})=\frac{1}{2}(\frac{10}{32.2})v_{A_2}^2+\frac{1}{2}(\frac{10}{32.2})4v_{A_2}^2$
This simplifies to:
$v_A=26.8ft/s$
