Use the method of characteristics to solve $u_t+uu_x+\frac{1}{2}u=0$. 
Use the method of characteristics to solve 
  $$u_t+uu_x+\frac{1}{2}u=0, \quad t>0, \quad {-\infty}<x<\infty$$
  $$u(x,0)=\sin(x)\quad {-\infty}<x<\infty$$
  (solution may be expressed in implicit form). Show that a shock solution is possible if $t=t_c=2\ln(2)$.

Notes: Although I have used the method of characteristics to solve the wave equation and have cover basic theory in class. But, I am bit lost in application. Any hints would be appreciated. 

The shock between states $u_L,u_R$ moves with speed $(u_L+u_R)/2$. This follows from applying conservation to an infinitesimal rectangular control volume that has part of the shock path as a diagonal. The source term has no effect on this. Past the shock formation time, the solution is double-valued, and in this region the shock is governed by the ODE
$$\frac{dx_s}{dt}=\frac{1}{2}(u_1+u_2)$$
There will not be an explicit solution to this because $u_{1,2}$ are only given implicitly (but probably that was not what you meant}
