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Thread: Intercept of two accelerating objects

  1. #1
    Jun 2011

    Intercept of two accelerating objects

    My background is in computer graphics programming, so please forgive me if any terminology or notation seems off.

    This problem would take place in two dimensional space. There are two objects each with known positions (x,y). Each object has a known initial velocity (Vx, Vy). Each object can accelerate at a known rate, but cannot accelerate faster or slower than its given rate, unless its acceleration is 0. Each object can accelerate in any direction, but each object has a known speed limit that it cannot accelerate past.

    If we are given the direction of acceleration for one object, assuming that each object will simply coast in a straight line once its speed limit is reached, how do we determine what direction the second object must accelerate in order to collide with the first object (assuming each only occupies a single point given by their position).

    Any suggested resources or references to read up on to better understand the problem are greatly appreciated.
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  2. #2
    Senior Member
    May 2010
    Los Angeles, California

    Re: Intercept of two accelerating objects

    The problem as stated is too vague. We need to know the exact position $\displaystyle \mathbf{r}_1(t)$ of the first object as a function of time before we can say anything about the second object. To do this, you need to give specifics (i.e. force, intial postion and initial velocity) of the first object in order to determine $\displaystyle \mathbf{r}_1(t)$.
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  3. #3
    Jun 2011

    Re: Intercept of two accelerating objects

    Sorry, for such a delayed response. I shouldn't have posted so soon before leaving town...

    If I may, let me change the problem a bit to simplify it:
    I know if you are given initial velocity and acceleration, you can easily plot change in position over time. What I would like to figure out is essentially how to do the reverse, finding the necessary direction of acceleration to reach a given displacement.

    Let's assume that instead of a moving body trying to reach another moving body, like I originally posted, we simply have one moving body at the origin and a stationary destination point at (x, y). If we know the initial velocity of Vx and Vy of the object and the magnitude of acceleration but not the individual vectors (we don't know what direction force is being applied but we know how much) is it possible to figure out what direction the object needs to accelerate to reach its destination in the shortest amount of time? Lets also forget about the whole speed limit restriction I mentioned earlier, so the object can accelerate at a constant rate indefinitely.

    I guess what I would need to end up with is a formula that plots the direction of acceleration as the dependent variable with the time it takes the object to reach its destination as the independent variable. I believe that most values for the direction of acceleration would never be able to reach the point in any amount time, so there would be a limited number of solutions.

    Here's what I've been working with so far -
    The equations for distance traveled with constant acceleration:
    dx = Vx * T + (Ax * T^2 ) / 2 | dy = Vy * T + (Ay * T^2 ) / 2

    where T = time, Vx and Vy = initial velocity, Ax and Ay = acceleration vectors, and dx and dy would essentially be the point (x, y) since we are starting at the origin.

    Both Ax and Ay are unknown, but Ax^2 + Ay^2 = A^2 where A is the magnitude of acceleration and A is known, so if we designate 'u' as the angle between Ax and A such that tan(u) = Ay/Ax, I assumed it would be easier to use cos(u)*A in place of Ax and sin(u)*A for Ay so that we are solving for the single angle 'u'.

    So now:
    dx = Vx*T + cos(u)*(A*T^2)/2 | dy = Vx*T + cos(u)*(A*T^2)/2

    which I rearranged into:
    cos(u) = ( (dx - Vx*T) * 2 ) / (A * T^2) | sin(u) = ( (dy - Vy*T) * 2 ) / (A * T^2)

    if I remember correctly, tan(u) = sin(u) / cos(u) so:

    tan(u) = (dy - Vy*T) / (dx - Vx*T)

    u = tan^-1( (dy - Vy*T) / (dx - Vx*T) )

    I'm not sure if this equation is helpful to me though, because it seems like the magnitude of acceleration would have a big impact on the necessary direction but that has been reduced out of the end result.

    I've been reluctant to plug in actual numbers for this problem, but I've tried plotting several different variations on the computer, and can't really derive any conclusions from the result. If having arbitrary numbers is necessary to explain the problem better then we could use the following:

    Vx = +25 m/s
    Vy = +100 m/s
    x = +450m
    y = + 850 m
    A = 5 m/s^2

    Again, sorry if it's silly to respond after such a long hiatus, but this problem and many variations of it have been driving me nuts for months, and I was wondering if anyone might find it interesting.
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