Minimum speed of the bullet to penetrate a sphere

1. The problem statement, all variables and given/known data

A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge + q. If it strikes the surface of sphere with speed u, find the minimum speed u so that it can penetrate through the sphere. (Neglect all resistance forces or friction acting on bullet except electrostatic forces)

2. Relevant equations

3. The attempt at a solution

I think if the bullet can just reach the center ,then it can penetrate through the sphere because the electric field inside a sphere is radially outwards .As it reaches the center and moves just a little bit outwards radially ,the electric field will push the bullet outwards .

The problem can be approached either by conservation of energy or by work kinetic energy theorem.

1) By conservation of energy

kq2/R+(1/2)mu2 = (3/2)kq2/R

2) By work energy theorem

## \int_{R}^{0}\frac{kq^{2}r}{R^3} \hat{r} \cdot dr\hat{r} = \frac{1}{2}mu^2 ##

Have I approached the problem correctly or is there something more in it ?

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