Find Trajectory from A to B: Approaches & Solutions

[VladZH]

Hello

Given:
Point A
Body B with angular velocity ω
C body with radius r
Spacecraft with constant velocity v.

We neglect the gravity of the bodies B, C

The problem:
Find the shortest trajectory for spacecraft from A to B

What approaches might be here?

How might the solution be changed if we consider the gravity of C and v would be initial velocity?

 

[berkeman]

Thread closed temporarily for Moderation...

 

[berkeman]

Thread reopened. @VladZH -- is this problem for schoolwork? Can you show us your ideas for approaches to use on this type of problem?

 

[VladZH]

This is problem for my video game

I tried to solve a simpler problem when we don't have the body C.

Let P(r, φ) is a point on the circle. Let s between A and P. Hence, the time for spacecraft from A to B equals Δt=s/v
The time for body B to get P is Δt=Δφ/ω. We get d/v=Δφ/ω where φ=sω/v
Now we can find φ=φ_B + Δφ and direct the spacecraft towards P by a straight line.

But the problem here is to express s in terms of φ. If we substitute s by this formula we get φ with cosine and φ without cosine on different sides of equation
s = √r_s²+r_B²-2r_sr_Bcos(φ_s-φ)
And I have no idea how to solve it

Then if we consider body C I thought about kind of force that pushes the spacecraft out of straight line trajectory. Moreover there are two possible trajectories near opposite sides of body C. The optimal one depends on speeds of body B and speed of the spacecraft

Thank you