Direction of Forces on an Euler Spiral Path | Intuitive Explanation

[sophiecentaur]

P√2 = F t₄₅. Suppose we try some other strategy for time t₄₅. Let the 45 degree direction be the y direction. If for any part of the time we direct some of the available F in the x direction then during that time F_y < F. Thus the integral of F_y over time t₄₅ will be less than P√2, and we will not have achieved the desired result.

I think haruspex's explanation qualifies as proof. If you need expansion on it, you should pinpoint exactly which point of it you want expanded.

As an alternative consider the reference frame where the astronaut was initially at rest. Here he has to accelerate from 0 to some speed along the 45° line as quickly as possible. Still not obvious?

Thanks for persevering, chaps. I think I have it now.

 

[thinkagain]

So is the 45 degree angle the optimum direction of thrust then? Thanks!

 

[A.T.]

So is the 45 degree angle the optimum direction of thrust then?

For the equal speed constraint, not for the fixed turn space constraint .

 

[thinkagain]

I see, so we still don't have an answer for that one then?

 

[haruspex]

I see, so we still don't have an answer for that one then?

As I posted, that would make quite a hard problem. In all likelihood, there is no analytical solution. Certainly out of the scope of a homework forum.

 

[jerromyjon]

I just figure after all that, if you can't make the turn quick enough you all stop thrust until you can make the turn by gradually redirecting to the y so you just barely have no x just in time and all y thrust at the boundary. If you have more than enough thrust you start all y direction until you have just enough time to all x up to the boundary and shift 90 degrees at the exact moment you cross, which in my simple mind... the latter resembles an inverse Euler spiral.