Various theories have been posited for example that the instantaneous impact point just disappears over Africa, another that it goes once around the world before going poof.Neither seems satisfactory. They may be close enough for government work, but I don't work for the government.First on a perfectly spherical airless world.As the launch approaches orbit the efficient trajectory is horizontal, which means that it is either at the apogee or perigee of the instantaneous orbit. For the IIP we only need to consider the apogee case. The perigee is halfway round the orbit, and the IIP must be between the apogee and perigee. The IIP will vanish when the perigee is at the planet's surface exactly halfway round. Which is neither over Africa (for a launch from the US) nor all the way round the planet.For a planet with atmosphere.An object in low orbit will decay and has a theoretical IIP. Although good luck figuring out where it is until the last few orbits. So for a launch to a low orbit the IIP should never really disappear. It just whips around the planet many times, probably approaching infinite speed. Eventually (for very small values of eventually) it still exists but becomes incalculable (or fades into a sea of probability).For a non-spherical planet.You need to take into account things like mass cons and precession that affect orbits and possibly topography (The IIP may tend to the highest mountain)Since the problem is so ill conditioned relativistic and quantum may eventually rear their ugly heads.There are probably also pragmatic or legal answers, I'd be interested in them too, particularly if they can explain why anyone thinks an IIP should vanish over Africa.