## More conservation of energy

Now that we know about electric potential energy, we can revisit our old friend, the law of conservation of energy. It’s basically the same as before: start out with

Delta E_"k" = -Delta E_"el",

then rearrange for the unknown quantity and substitute everything else.

### Example

An alpha particle with a charge of 3.2 × 10−19 C moving at 1.0 × 106 m/s from infinity approaches a gold nucleus whose charge is 1.3 × 10−17 C. How close does the alpha particle get?

We by substituting the energy formulae into the equation given above:

1/2m(v_2^2-v_1^2) = -kq_1q_2(1/r_2-1/r_1).

The particle will stop moving when it gets to the closest point, so we make v_2 zero. It starts at infinity, and one over infinity, for our purposes, is zero, so that gets rid of another term. We are left with

1/2mv_1^2 = (kq_1q_2)/r_2.

Solving for final radius, we have

r_2 = (2kq_1q_2)/(mv_1^2),

and working that out gives us the answer, 1.1 × 10−11 m.