I want to move a particle in a spiral at a constant speed. Note that that is not a constant angular speed. This is proving rather difficult, and I'll go through my method so far below.
The spiral in question is a classic Archimedean spiral with the polar equation
r = ϑ, and the parametric equations
x = t*cos(t), y = t*sin(t). This looks like this:
I want to move a particle around the spiral, so naively, I can just give the particle position as the value of t, and the speed as the increase in t. That way the particle moves round the spiral at a constant angular speed. However, this means that the further out from the centre it gets, the faster its (non angular) speed becomes.
So, instead of having my speed in the increase in t, I want my speed as the increase in arc length. Getting the arc length of a spiral is the first challenge, but due to the fact that the Archimedean spiral I am using isn't too insane, the arc length function is , where
a = 1. This allows me to convert theta values to the arc length, but that is the precise opposite of what I need. So I need to find the inverse of the arc length function, and at that hurdle, Wolfram-Alpha has failed me.
So is it possible to find the inverse of the arc length function? The function is a one to one mapping, if you exclude negative values of theta.