How Can I Solve Non-Linear Equations for Motion with Multiple Variables?

[Jim_Shah]

I know this might be kind of an easy question but I'm just getting back into physics for school. I'm looking to take an object moving through space with an initial velocity vector, V0, and an initial position, X0, in 3D space. I'm looking at a way to get to another position, X1, and at time t when I get there, my velocity is 0. I've come up with the following equations but am a little stuck on the best way to solve...(Some sort of estimation since they are non-linear equations with more variables than equations). It seems that there might need to be some point in between t0 and t where the acceleration vector would need to be changed in order to achieve zero velocity at X1 so I've included that in these equations

X1 = X0 + V0t + 1/2(A1*t1^2) + 1/2(A2 * (t - t1)^2)
0 = V0 + 2*A1*t1 + 2*A2*(t-t1)

X1,X0,V0 are known
|A1| = |A2| and are known

I'm just looking for some advice as to a next step in coming up w/ t1, t, and the direction for A1 & A2 in order to make these equations work.

Thanks for the help.

 

[whozum]

Lets take the easy cases where $x_0 = v_0 = 0$, then you know that $|a_1t|= |a_2t|$ in order for the velocity to return to zero. There will be two periods of acceleration where the accelerations are in opposite directions.

If you want to start off with an initial velocity, then $v_0 + a_1t = a_2t$ (again, magnitudes). [itex] a_1 \ and \ a_2 [/tex] will need opposite linear directions for these equations to hold, they can be expanded for motion in two and three dimensions.

These will be your velocity functions of time. To figure out your position functions, you can integrate each wrt t.

$$x(t) = \int_{t=0}^{t} (v_0 + a_1t - a_2t)dt$$

Solving the integral gives

$$x(t) = v_0t + \frac{1}{2}a_1t^2 - \frac{1}{2}a_2t^2$$ which is what you would expect. For a given $x_0, x_f, x_i, v_0 + a_1 = a_2$, the end result is just a quadratic which can be solved for t with the quadratic formula.

Is this what you are looking for?

 

[Jim_Shah]

I still think there are things missing here. Since v0, a1, a2 are vectors, the directions are important. Therefore some of the equations you've shown do not work. I'll try to explain the problem w/ an example perhaps.

I'm at point X0 = (3,4,5) at time t = 0. At some time, t, I want to reach point X1 = (-7,-2,-1) and when I reach that point I want my velocity to equal zero. I have an initial velocity vector of (2,2,2). I am also applying an acceleration with a constant magnitude but the direction can be changed. Is there a way to solve for the directions of the acceleration and when they need to be applied in order to satisfy these conditions?

I'm not expecting a solution to the example...just using it to identify the problem. I've tried to encapsulate those requirements in my equations above...unfortunately I'm a LaTex noob so you'll have to bear with my equations for now. The problem I am running into is that I have 4 unknowns (t - the time when the destination is reached and the velocity is zero, t1 - the time when the second acceleration is applied, a1 - the direction of acceleration for the first time period, a2 - the direction of the acceleration for the second time period) and 2 equations. The idea of two acceleration periods is intiuitive at this point since if you think of an object moving towards a destination you will need to apply a force in the opposite direction to slow it down. It is also a requirement that t be minimal ( or a close approximation to) since a simple way would be to slow to a stop and then move directy towards the destination after stopping.

If I need to do more explaining I will...just let me know. Thanks for the help.

 

[whozum]

Recall that I stated that the accelerations must be exactly anti parallel for the equations to work, but with that premise, they will work in any linear path, as any linear path can be condensed to a single dimensional problem.

I think your trying to say your path is non-linear, in which case I'm not in a position to help you right now.

What are your two equations?

 

[Jim_Shah]

Yes...I am looking for a non-linear path. The two equations are just the ones listed in the first post with X1, X0, A1, and A2 being vectors.

Thanks