Back On Track, Part I

Sorry about that. Back to normal, scheduled updates.

Let me introduce you to one of my favorite subjects, math. I've been working on a problem interesting to me these past few days:

Say you have a vehicle traveling forward at a speed given by v(t), that can only change its direction by turning with an angular velocity ω(t). What is the equation describing the position of the object, , at any given time?

The reason this is so interesting a question is that it involves combining both linear and angular motion; a complex and non-trivial task. First, let us set some constraints on the problem to give us an idea of where we have to go: let us set a constant speed of V and constant angular velocity of T, to simplify our equations vastly. We can easily see that the total distance traveled is given by the following equation:

By similar math, the vehicle will have traveled a total arc of θ(t) = Tt. But that is not enough to fully describe the motions of the vehicle. One more piece is needed, the turning radius.

Consider for a moment the formula s=θr, the formula for the arc length of a circle. But what does this have to do with velocity? Assume with me for a moment that the vehicle is moving along a circle (a fact we will prove in a little bit). Using the formulas we have defined earlier, we recognize that s(t) = θ(t)r. We have left the radius in boldface just to illustrate how the parts fit together. Defining some equation r(t) as the turning radius, and solving for r, we find:

It is a reasonable to assume the vehicle is moving in a circle, but assumptions are not good enough for mathematics. We will show that the vehicle is actually traveling in a circle. More rigorously we define x(t) and y(t).

Why? The actual velocity of the vehicle at any time t is a vector with an angle of θ(t) and a magnitude of v(t). Notice an important distinction; speed is a scalar quantity, and hence there is no arrow over our speed function. Velocity is separate from speed. The same goes for the position  and the x-coordinate x(t). The appropriate x and y is obtained by integrating the respective components of the velocity vector, namely the formulas you see in the integrals. Once again, we will assume a constant speed of V and a constant angular velocity of T. Integrating these formulas out using previously discussed, we find that:

This is actually a tricky answer. The naive interpretation of "moving in a circle" would be to assume that the x component is found by multiplying the radius by the cosine, and the y component by the sine, but one of the harshest lessons mathematicians learn is to never place too much trust intuition. It can point you in the right direction, but it is no substitute for proof. I actually fell victim to this. There is one point of interest in these two equations, the two constants of integration. The point (X, Y) is the center of the turning circle. To discuss proof of this is another non-trivial problem which will lead us into tomorrow's discussion.

PS: Images are kind of mangled in this post. I had to tinker around with blogger for a little bit before fixing them. Next post will have functional images.