Analysis of the Problem:
To find the minimum velocity required for the charge -q to reach the surface of the solid sphere of charge KQ, we need to consider the electrostatic potential energy and kinetic energy of the system. The charge -q will experience both attractive and repulsive forces due to the charges Q and KQ.

Key Steps:
- Calculate the electrostatic potential energy between the charge -q and the solid sphere of charge Q at a distance R.
- Calculate the electrostatic potential energy between the charge -q and the solid sphere of charge KQ at a distance 5R.
- Equate the sum of the potential energies with the initial kinetic energy of the charge -q.
- Use the conservation of energy principle to find the minimum velocity required for the charge -q to reach the surface of the solid sphere of charge KQ.

Calculation:
1. The electrostatic potential energy between the charge -q and the solid sphere of charge Q at a distance R is given by:
\[ U = \frac{k|Qq|}{R} \]
2. The electrostatic potential energy between the charge -q and the solid sphere of charge KQ at a distance 5R is given by:
\[ U = \frac{k|KQq|}{5R} \]
3. The initial kinetic energy of the charge -q is given by:
\[ KE = \frac{1}{2}mv^2 \]
4. According to the conservation of energy principle, the sum of potential energies equals the initial kinetic energy:
\[ \frac{k|Qq|}{R} + \frac{k|KQq|}{5R} = \frac{1}{2}mv^2 \]
5. Substituting the given values of Q, K, and R, we can solve for the minimum velocity v required for the charge -q to reach the surface of the solid sphere of charge KQ.

Conclusion:
By following the steps outlined above and performing the necessary calculations, you can determine the minimum velocity required to project the charge -q along the line joining the centres of the two fixed solid spheres of charges Q and KQ. This analysis demonstrates the application of electrostatic principles in determining the motion of charged particles in an electric field.