What happens to the IIP as a launch approaches orbital velocity?

[Barley]

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.

[edkyle99]

Picture the IIP as a spot traveling downrange along the projected launch track at the Earth's surface.  At first it is not far from the initially slow moving vehicle, but as downrange velocity increases the spot moves faster and faster, and moves further and further away from the vehicle location.  During the final moments just before orbital velocity is reached the spot really moves.  It passes across Africa in seconds during an eastward launch from Florida, for example.  Then, when orbital velocity is achieve, the spot disappears as the projected track raises above the Earth's surface.

 - Ed Kyle

[Jim]

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.

Once an object reaches orbit, the IIP is no longer relevant and not used.   The calculations used for determining IIP are only for ascent and not for reentry or mascon effects.   Different calculations are needed once on orbit to determine reentry and that is only a prediction.  The reentry calculations have to take into account the shape and mass of the object in orbit. 
 The ascent IIP assumes a vacuum. 

The IIP "disappears" over Africa because is moves so fast and there is no point in worrying about it anymore.