Polar Coordinates for Velocity and Acceleration of a Particle

Date: 03/11/97 at 16:59:12
From: Nick Manville
Subject: Use of polar coordinates for velocity and acceleration of a 
particle. 

In England this is a new topic on our A-level syllabus (our final-year 
exams) and our teacher really doesn't have a clue. Do you know of any 
internet resources that might be able to enlighten me on the subject?
 
I know that velocity components are dr/dt and rdo/dt and that 
components of acceleration are d^2r/dt^2 - r(do/dt)^2, etc.
I would appreciate any help at all in understanding how these are 
generated and/or how to use them.

Many thanks,
Nick Manville

Date: 03/12/97 at 19:38:13
From: Doctor Charles
Subject: Re: Use of polar coordinates for velocity and acceleration of 
a particle. 

I am one of the few brit doctors here and consider polar coordinates 
to be one of my more familiar areas. I had a look at a couple of web 
pages we have links to here and couldn't find much that was 
enlightening as such - basically just the definitions and a few 
formulae. I presume that you are looking for some understanding of the 
problem and here is just a small start which might help. Unfortunately 
- having studied some tensor calculus I know more about them than was 
certainly in my A-level syllabus so I have a little difficulty in 
reconciling the slightly easier A-level version with the more 
complete tensor version - but anyway...

The thing which makes Cartesian co-ordinates so easy is that the basis 
vectors always point in the same direction (the i vector - in the 
direction of the x axis always points straight 'right'. This means 
that when you differentiate a vector you just have to differentiate 
its components.

i.e. d/dt([x(t),y(t)]) = [ dx/dt, dy/dt ]

or using i,j notation which is more helpful for understanding in this 
case:

 d/dt ( xi + yj ) = (dx/dt)i + (dy/dt)j    because i, j are constant 

However with polar co-ordinates the basis vector in the 'r' direction 
always points away from the origin so at x=0, y=1 it points up but at 
x=1, y=0 it points right.

Normally we use 'e sub r' for the basis vector in the radial direction 
(away from the origin and 'e sub theta' for the basis vector an the 
tangential (anticlockwise) direction. I'll denote these e_r, e_0 for 
ease of typing.

So if we have a vector: r(t) e_r + th(t) e_0 and we want to 
differentiate it (using r' for dr/dt for speedy typing 
again) we get:

 r'(t) e_r + th'(r) e_0 + r(t) d/dt(e_r) + th(t) d/dt(e_0)

So we need to know how  e_r and e_0 differentiate.

Well if we move in the r direction both vectors stay in the same 
direction, so 

 d/dr(e_r) = 0 = d/dr(e_0)

But if we move in the theta direction - around the origin - the basis 
vectors rotate. Because the vectors don't change side this is just 
like adding a very small displacement vector perpendicular to the 
original basis vector for each very small displacement of theta. (It's 
best to draw a diagram but I can't think of how best to do this right 
know.)

Because the basis vectors stay as unit vectors the size of this 
displacement is d0 for a small displacement d0. Again my reasoning 
isn't sound but if you draw a diagram showing arrows for typical 
r basis vector at two points (r,th) and (r,th+dth) and then look at a 
vector parallel to the second at the same point of the first and look 
at the triangle the form you should be able to draw the vector 
d(th) * e_0 joining the tips of these two vectors. This is difficult 
to explain but perhaps you get my gist?

Then we have d/dr(e_r) = 0, d/dth(e_r) = e_0, d/dr(e_0) = 0,
d/dth(e_0) = -e_r (Can you see why the minus sign?)

Any now we have done the hard part, the rest is easy(ish!). Using the 
chain rule (and partial derivatives) we have:

      d/dt(e_r) = dr/dt * d/dr(e_r)  +  dth/dt * d/dth(e_r)
                =  r'   *    0       +   th'   * e_0
                =  th' * e_0

and similarly  d/dt(e_0) =  -th' * e_r

So if we have a particle at position (r,th)  then its position vector 
is  r(t) * e_r so we can find its velocity by differentiating:

v = r'(t) * e_r + r(t) * d/dt(e_r)
  = r'(t) * e_r + r(t) * th'(t) * e_0

and acceleration:

a = r''(t) * e_r + r'(t) * th'(t) * e_0
   + r'(t)*th'(t)*e_0 + r(t)*th''(t)*e_0 - r(t)*th'(t)*-th'(t)*e_r

  = [r''(t) - r(t)*th'(t)^2] * e_r  +  [2*r'(t)*th'(t) + r(t)*th''(t)] 
* e_0

And this works with any vector - not just position vectors.

-Doctor Charles,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/