Need help with exam problem
Some hints:


Total charge of a homogeneously charged volume is charge densisty times volume
The electric field created by a homogeneously charged sphere is the same as that of a point charge of same magnitude, and that is \$\boldsymbol{E}(\boldsymbol{r})=\frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}\frac{\boldsymbol{r}}{r}\$
(assuming origin at point charge) 
You get the total electric field by superpostion of ("adding") fields of both spheres \$\boldsymbol{E}_{total} = \boldsymbol{E}_1 + \boldsymbol{E}_2\$
Electric force is electric field strength times charge (of probe) \$\boldsymbol{F}_{elec} = q\boldsymbol{E}\$
Kinetic energy is half mass times velocity squared \$W_{kin}=\frac{1}{2}mv^2\$
Energy/work is force integrated over space (path) \$W=\int_{path} \boldsymbol{F}_{elec}(\boldsymbol{s}) d\boldsymbol{s}\$


Putting all that together should solve the problem.