How Musk will probably succeed at getting a fully reusable rocket system built

Third in a series
Part 1  :  showed that by using a hydrogen/lox engine on the second stage, Musk could still get to orbit with mass to spare for making his rocket system fully reusable.
Part 2  :  a speculative post about how this might be accomplished in terms of potential flight trajectories.

With this third part of the series, the trajectories are refined to a potential method using only fuel and hardly any new hardware.  The same basic rocket, with the same basic amount of fuel can do the job.  How?

In part 1, the spreadsheet calcs showed that the rocket system will be less in mass than what the current system uses.  Just add the mass back in as fuel on the first stage and re-work the spreadsheet to find what the potential delta v can be achieved on a second burn.

The second burn is proposed to be at the highest point in the trajectory.  Since the rocket is still rising even after 1st stage separation, we simply wait until it stops rising.  Gravity will help slow down the forward velocity as well.  At the highest point in the trajectory, the first stage will be going at its slowest speed before speeding up again on its way down.

Instead of letting it speed up again, a second burn is initiated.  It is assumed that the velocity is sufficient slow now that a second burn in the opposite direction of movement will stop the forward motion away from the launch site and send it back toward the launch site with fuel to spare.

Thus, a new trajectory is formed that can be vectored back to the launch site.  Provided that there is enough fuel left, it can be used for a soft landing.  Parachutes can assist to slowing the first stage on its way down.

Frankly, I don't know for sure that there will be enough fuel for this manuever.  However, it is without doubt that the vehicle will slow down as it continues to rise after the first stage separates from the second stage.  It is also without doubt that the delta v in the opposite direction will not be insignificant.  I am guessing about a 2 engine ( out of 9 ) burn for about 60 seconds will be available.

The rocket must remain above the atmosphere so that drag will not become an issue.  There should be  as much forward speed as possible in order for the first stage to make it back to the launch site.  As before, the rocket will reach its highest altitude at its slowest forward speed.  I'm thinking that it is desirable for the forward speed to be as close to zero and as close to the launch site as possible.

By doing this, the attempt will be to hit the launch site like a ballistic missile would.  But instead of crashing into it, it would be slowed down by parachute and/or, if possible, by another rocket burn.

Compare original configuration in column C with the proposed configuration in column J (highlighted in green)