FIGURE 25.8a shows the potential-energy curve—a hyperbola—for two like charges as a function of the distance r between them. Also shown is the total energy line for two charged particles approaching each other with equal but opposite momenta.

FIGURE 25.8 The potential-energy diagrams for two like charges and two opposite charges.

You can see that the total energy line crosses the potential-energy curve at $r_{\text{min}}$. This is a turning point. The two charges gradually slow down, because of the repulsive force between them, until the distance separating them is $r_{\text{min}}$. At this point, the kinetic energy is zero and both particles are instantaneously at rest. Both then reverse direction and move apart, speeding up as they go. $r_{\text{min}}$ is the distance of closest approach.

Two opposite charges are a little trickier because of the negative energies. Negative total energies seem troubling at first, but they characterize bound systems. FIGURE 25.8b shows two oppositely charged particles moving apart from each other with equal but opposite momenta. If $E_{\text{mech}}<0$, as shown, then their total energy line crosses the potential-energy curve at $r_{\text{max}}$. That is, the particles slow down, lose kinetic energy, reverse directions at maximum separation $r_{\text{max}}$, and then “fall” back together. They cannot escape from each other. Although moving in three dimensions rather than one, the electron and proton of a hydrogen atom are a bound system, and their mechanical energy is negative.

Two oppositely charged particles can escape from each other if $E_{\text{mech}}>0$. They’ll slow down, but eventually the potential energy vanishes and the particles still have kinetic energy. The threshold condition for escape is $E_{\text{mech}}=0$, which will allow the particles to reach infinite separation $(U\to 0)$ at infinitesimally slow speed $(K\to 0)$. The initial speed that gives $E_{\text{mech}}=0$ is called the escape speed.