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On the third flight of SpaceX's Falcon 1, the following failure occurred according to one narrative.

We turned off the first-stage engine and then proceeded to separate the vehicle stages; however, when the stages uncoupled, there was still a little leftover "kick," or thrust, in the first-stage engine. Our first stage literally rear-ended our second stage immediately after we had tried to separate the two sections of the vehicle. It was a devastating, emotional experience.

Sounds logical. You wouldn't want to wait until your prior stage fully putters out completely before separating because (among other things) you'll lose altitude. Ideally, you'd want to be delivering as much thrust as possible as constantly as possible, and that might entail having some kick left in the prior stage.

In the general sense (not specific to SpaceX), how is this avoided in vertically staged rockets?

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This type of failure is known as separation failure which is caused due to timing error in separating the stages

Musk said SpaceX would be releasing video of the staging event that clearly shows that the first and second stages separated as planned about 2 minutes and 20 seconds into the flight, but that unanticipated residual thrust from the redesigned Merlin engine caused the first stage to bump the second stage just as it began to fire

This is normally overcome in the rockets by

  • shutting down the rocket engine properly (cutting down the fuel source(liquid propellent) in case of solid propellent it is impossible to shut down so timing is very important the fuel is entirely burnt before shut down)

Source:

The separation of large boosters uses direct staging. Each booster is connected to the rocket’s core through a bar mechanism in the front joint side and the round head mechanism in the back joint side. When the engine of the booster stops working, the connection bolt and separation nut will be unlocked separately. The small separation rockets that are bound to the boosters are ignited and separate the boosters from the core. The boosters fall down and separation is complete.

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  • $\begingroup$ Also, if you engage the second stage engine fast enough, it will simply push the other stage off-course. $\endgroup$
    – SF.
    Aug 26, 2013 at 12:25
  • $\begingroup$ If the first stage still have fuel it could accelerate since the second stage is still yet to kick start(small time difference in igniting second stage ) $\endgroup$
    – Hash
    Aug 26, 2013 at 12:31
  • $\begingroup$ If the first stage still keeps thrust then the stages won't separate as the first stage still pushes against the second. Only in case of a hiccup, when the first stage pretty much cuts off, and then reignites after separation this would be possible - or in case the disengaging system pushes them apart briefly but doesn't provide continuous thrust. $\endgroup$
    – SF.
    Aug 26, 2013 at 12:54
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    $\begingroup$ For the record, Solid Propellent isn't impossible to shut down, one simply has to eject the fuel, which is actually fairly commonly and easily done. $\endgroup$
    – PearsonArtPhoto ♦
    Sep 24, 2013 at 18:11
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    $\begingroup$ @PearsonArtPhoto Solid propellent typically only burns while a certain combustion pressure is maintained. Hence, while solid propellent motors cannot be actively throttled, they can be "shut off" by venting adequate pressure (even violently) from the combustion chamber. $\endgroup$
    – quasinormalized
    Apr 24, 2018 at 18:06

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