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Minimum time to stationkeeping, constant Gs

  1. Jan 22, 2009 #1
    A mechanics problem which has to have been studied, but I can't find any references on the web...

    Assume that a body in the plane is in motion at coordinates (x,0) with velocity vector (a,b). The body can be accelerated at 1 unit/sec^2 in any direction (constant G force).

    The question is how to apply the acceleration in order to bring the body to rest at (0,0) in minimum time.

    A naive solution which is easily tractable would involve applying all acceleration perpendicular to the line between the body and the origin in order to reduce the problem to 1 dimension -- first cancel all "angular momentum" and then solve the 1-D problem.

    However, there are some obvious cases where this is clearly suboptimal. For instance, the body starts at (100,0) with velocity vector (-20,1). No matter what we do, the body is going to overshoot the target by a fair bit, and the majority of the acceleration at the beginning should be directed to slowing the body down. Better to miss the target a little bit while traveling slower than to hit the target dead on at a higher speed.

    Is there a closed form solution for the desired angle at which the acceleration should be applied? The body has no "memory", so the desired acceleration at any point in time is simply a function of the position and the velocity vector.
  2. jcsd
  3. Jan 23, 2009 #2


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    Welcome to PF!

    Hi jbrennen ! Welcome to PF! :smile:

    Hint: As you say, we can define an angle θ which depends only on r and v … θ(r,v).

    So what is the differential equation for θ? :wink:
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