A fly flying towards an oncoming steam train hits its front and is carried along with it.

physics/fly.and.train New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A fly flying towards an oncoming steam train hits its front and is
carried along with it.  The fly's velocity changes from one direction
to the other, so it must pass through zero; at that instant the fly is
touching the train, so must the train also be stopped?

[ Editor: problem statement reconstructed from the postings, a March
1995 rec.puzzles thread "The Fly And The Steam Train", of which only the
reply below was collected. ]

Show solution

In article <3k72rh$k5r@decaxp.harvard.edu> dsfergem@fas.harvard.edu
(David Fergemann) writes:
>The fly's velocity _must_ and _does_ pass through v=0.  Velocities are
>continuous.

Who says so?  And why?

>There is such a concept as "instantaneous velocity."  It's impossible
>to _measure_ velocity for an instant, but that doesn't mean velocity
>is undefined.

Instantaneous velocity is found by averaging velocity over shorter and
shorter intervals around your target time.  You can't just ignore the
intervals.

>The "paradox" is that the fly can be approximated by a point mass, and
>its motion is described almost perfectly by this approximation.  But,
>if you continue to treat the fly as a point mass when you consider the
>train, you come to ridiculous conclusions about the train's motion.

Yes, this is a good insight into the problem.  The "paradox" is that we
use a certain point-mass approximation for the fly, but then apply it
using a set of assumptions under which that approximation is no good.
The question is, what's wrong with the approximation?  That's how to
resolve the paradox.

The paradox arises when we assume that the fly flies at a constant
velocity right up to a certain instant in time (the collision with the
train), after which it again moves at a constant velocity (with the
train).  The graph of the fly's velocity vs. time then looks something
like this:

  ^
v |
  |      .___________________
  |
  |
  |
  |
  |
  |
  +--------------------------> t
  |
  _______.

That's the graph of a discontinuous function.  Another way to look at
it is that the fly's velocity is the derivative of its position in
space as a function of time:

  ^
x |
  |        /
  |       /
  |      /
  |\    /
  | \  /
  |  \/
  |
  +------------------> t

This function has a sharp corner at the time of collision, and the
velocity is undefined there.

Now this model isn't altogether bad; we use it for calculating the
results of simple collisions between bodies in physics.  It's only a
bad model if we try to ask when the velocity is zero, because the model
doesn't define a time when the velocity is zero.

If we want to find the zero velocity, we have to go to a more realistic
model, where the train is allowed to accelerate the fly over a period
of time rather than having to do it instantaneously.  Any of several
simple models will do, any one of which will show that the fly achieves
velocity zero while it is still in motion relative to the train, and
the train never achieves zero velocity.

This of course is just one way of analyzing one formalization of the
paradox.  There have been several explanations of this paradox posted,
and as far as I've seen, *all* have hit in one way or another on the
inconsistencies in the implict assumptions of the paradox.

-- David A. Karr <karr@cs.cornell.edu>  (March 1995)