Calculating aim point to pass through other point

Question

We know for an object:

• Initial point O(x,y)
• Initial speed (s_x,s_y)
• Constant vector module aceleration (A)
• A point P(i,j) where we have to pass through in the future.

We need:

• A point W(x, y) so the angle alpha (or only the angle alpha), formed between O and W, can be used to calculate acceleration in both axis:

  Ax = A * cos(alpha)

  Ay = A * sin(alpha)

so that the object passes through W.

How do I define an elliptic motion system knowing two points of the orbit (and Vo)?

I'll appreciate alternatives to design a guided rocket too.

My attempt for solving this:

1. Calculate module of initial point O, initial speed So, destination point D. (Aceleration module A is given)

2. Use these terms to calculate the time we need to arrive to D with the equation:

   D = O + So*t + 1/2 A * t^2

3. Find out the value of alpha from this equation, knowing time:

   Dx = Ox + Sox * t + 1/2 A * cos(alpha) * t^2

I tried to implement this, but for some reason it doesn't work. Should this theorically work?

Answer 1

If I understand your problem correctly, you wish to determine, for a given initial location, initial velocity, desired target, and constant magnitude acceleration, what constant point the acceleration must be aimed at in order to pass through the desired target point at some future time.

This is the definition of circular motion around the aiming point, for a circle of radius r passing through both the initial location and the target, where by the well-known kinetic equations of circular motion the radius can be derived as:

r = v^2 / a

where v and a are respectively the magnitude of the starting velocity and acceleration.

Now it is sufficient to obtain, using some basic Euclidean geometry, the location of the point that is distance r from both the starting point and the target location, on the desired side so that the intersection occurs in the future rather than the past.