In this problem, we consider a particle moving on a helix:
a) Suppose the position vector of a particle as a function of time $t$ is given by $r(t) = ( \cos t, \sin t, 3t )$. Find the speed of the particle.
b) The position vector of another particle moving on the same helix at a different speed is given by $r(t) = ( \cos 2t, \sin 2t, 6t)$. Find the magnitude of the acceleration.
c) A particle moves on the same helix with constant speed equal to $v$. Find the magnitude of its acceleration.
The part that I am struggling with is part (c). This is my understanding thus far. We are given the constant speed, which suggests that there is still an acceleration (due to changes in direction). I understand that speed is a magnitude of first derivative of the position vector. From that, how do we go about finding the magnitude of acceleration? Is there some sort of a formula that I am missing. I am just a little confused about how they can be related.
If you want the particle to follow the same helix at a constant speed, it must have a parametrisation of the form $$ r(t) = (\cos at, \sin at, 3at) $$ for some constant $a$ (you have seen this with $a = 1$ and $a = 2$ already in parts (a) and (b)). You want the speed to be $v$, so you need to figure out which value of $a$ makes it so.
Once you've found the value of $a$ (expressed in terms of $v$) insert that into the parametrisation above, then find the magnitude of the accelration.