In this problem, we are required to apply conservation of energy.

Electric potential energy:

$\overline{){\mathbf{U}}{\mathbf{=}}\frac{\mathbf{k}{\mathbf{q}}_{\mathbf{1}}{\mathbf{q}}_{\mathbf{2}}}{\mathbf{r}}}$

Law of conservation of energy:

$\overline{){{\mathbf{K}}}_{{\mathbf{i}}}{\mathbf{+}}{{\mathbf{U}}}_{{\mathbf{i}}}{\mathbf{+}}{{\mathbf{W}}}_{{\mathbf{nc}}}{\mathbf{=}}{{\mathbf{K}}}_{{\mathbf{f}}}{\mathbf{+}}{{\mathbf{U}}}_{{\mathbf{f}}}}$, where W_{nc} is the work done by non-conservative forces such as friction.

A proton and an alpha particle are momentarily at rest at a distance r from each other. They then begin to move apart. Find the speed of the proton by the time the distance between the proton and the alpha particle doubles.

Both particles are positively charged. The charge and the mass of the proton are, respectively, e and m. The charge and the mass of the alpha particle are, respectively, 2e and4m.

Find the speed of the proton V_{f}(p) by the time the distance between the particles doubles.

Express your answer in terms of some or all of the quantities e, m, r, and epsilon 0.

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