Re: Problem: acceleration perpendicular to velocity

["John S. Denker" <jsd@MONMOUTH.COM>]

At 07:28 AM 9/18/01 -0400, David Abineri wrote:

> ... velocity vector pointing to the top of
> the page and an acceleration vector pointing to the right side of the
> page at a particular instant in time. It asks if the speed is increasing
> or decreasing?
...
> it could be an object on the left side of a circular
> motion motion in which case the acceleration could the the centripetal
> acceleration and the speed constant.

Yes, that's a good way to look at it.

> It seems that one would have such a situation on a projectile at the
> peak of its vertical motion (turning the page by 90 degrees) and then
> its speed would be increasing.

That's fallacious.

In the projectile motion scenario, on the way up the speed is decreasing,
and on the way down the speed is increasing.  The peak of the trajectory is
right where we cross over from decreasing to increasing.  So the best
answer is that it is neither decreasing nor increasing at this special point.

It may help to draw the graph of speed versus time.

In general:  Consider a coordinate system where the current velocity vector
defines what we mean by "forward".
 -- Any acceleration vector in the forward hemisphere means the speed is
increasing.
 -- Any acceleration vector in the rear hemisphere means the speed is
decreasing.
 -- Any acceleration vector that is exactly perpendicular to the current
velocity means there is a change in direction with no change in speed.  Can
you prove this, rigorously?  Hint: start from the definition of speed and
turn the crank.

> They say the speed is not changing.

Yup.

> Is the question in fact ambiguous?

I don't think so.