Minimum time to stationkeeping, constant Gs

[jbrennen]

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.

 

[tiny-tim]

Welcome to PF!

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.
…
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.

Hi jbrennen ! Welcome to PF!

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?

 

[charng]

I would approach this problem by first defining the problem clearly and breaking it down into smaller components. It seems like this is a problem in classical mechanics, specifically in the field of dynamics. I would start by identifying the equations of motion for the body in motion and the acceleration vector. Then, I would use mathematical techniques such as calculus and optimization to determine the optimal angle at which the acceleration should be applied in order to minimize the time to reach the target point (0,0). This may involve solving differential equations and using techniques such as the Euler-Lagrange equations to find the optimal path for the body.

Next, I would research any existing literature on similar problems, particularly in the field of spacecraft dynamics and stationkeeping. I would also consult with other experts in the field to see if they have encountered a similar problem and how they approached it.

If there is no existing literature or closed form solution for this specific problem, I would conduct experiments and simulations to test different scenarios and gather data on the optimal angle of acceleration. This data could then be used to develop a mathematical model or algorithm for solving the problem in the future.

Overall, while this may be a challenging problem, it is certainly one that can be studied and solved using scientific methods and techniques. With proper research and analysis, I am confident that a solution can be found for this mechanics problem.