Conference rusure::math

    Hmm. The problem is not very well specified. As stated, the bottom
    of the ladder experiences an infinite acceleration in order to attain
    an instantaneous velocity of V. This will tend to separate the top
    of the ladder from the bottom. Also vaporize the ladder. In these
    conditions it is not clear that the top will come down at all.
    

Okay:
   	A ladder of length L is nearly upright, leaning against a wall.
	The bottom of the ladder is being pulled away from the wall at
	velocity V.

As before:
	What is the velocity of the *top* of the ladder when it's
	at height L/2?  What is it's velocity as it hits the ground?

    How close is the velocity V to the speed of light?  Must we include
    relativistic effects here?
    

    Aside from relativistic effects, this is really a nasty problem.
    Long objects which fall over tend to bend; this is why smokestacks
    usually break when pushed over. You also need to know how large
    V is w/r/t the length of the ladder in order to determine if the
    ladder will remain in contact with the wall as it descends. So there
    are still a lot of undefined assumptions in the problem statement.

Well, I guess nobody's willing to do any real work on this.

    I've looked at the problem.  Some partial results follow.
    
    When the ladder is at an angle O from the ground, the bottom of the
    ladder is traveling at a velocity of V cos O in a direction parallel to
    the ladder and V sin O in a direction normal to the ladder.  Assuming
    the ladder to be a rigid body, the top of the ladder must also be
    traveling at a velocity of V cos O in the direction parallel to the
    ladder.  However, its normal component is unknown. 
    
    The above includes the assumption that the bottom of the ladder remains
    on the ground.  I have not yet figured out under what circumstances
    this is true.
    
    If we knew how much energy the ladder had, it would be simple to
    determine the last component of the ladder's velocity.  However, energy
    may be added or removed by three things, the force which causes the
    bottom of the ladder to move at velocity V, the force on the ladder
    from the ground, and the force on the ladder from the wall.  One might
    assume these last two forces to do no work on the ladder, since the
    ladder does not move because of them, but that is not necessarily true
    -- even though the end of the ladder touching the wall/ground does not
    move, the rest of the ladder is affected by these forces. 
    
    
    				-- edp

    Be very careful if you try to solve this problem using energy. 
    You must be certain to include rotational energy the ladder acquires
    when it is falling.

Re .4:

Prove:

	A smokestack always breaks apart at the point 1/3rd of the way
up from the bottom when it is dynamited, and starts tipping over.

Sounds under-specified?  Someone noticed this from inspecting photos
and films of falling smokestacks, and it was proved in some place like
the Amateur Scientist or Mathematical Games section of Scientific American.
(Does this ring a bell with someone?).
				/AHM

Let the mass of the ladder be M, assume this is uniformly distributed
along its length, and let the acceleration of gravity be G.  The ladder
is, of course, assumed to be rigid.

The potential enery available as the ladder stands is:

	   L     MGl      MGL
	Integral --- dl = ---
	   0      L        2
	
(i.e., recall that force=mass*acceleration, and energy=force*distance,
 and we're integrating over the length of the ladder).

Now if the top of the ladder were constrained to remain against the
wall as the bottom is moved, it's height at time t would be:

	sqrt (L� - t�V�)

and it's velocity would be:

	- t / sqrt (L� - t�V�)

(negative because it's moving downward; this is the derivative of the
 height with respect to time).


Well, it's a start.

    How about considering the rotational position, velocity, and
    acceleration of the ladder around the bottom of the ladder?  With
    that focus, there would be only one force to consider:  gravity.
    
    
    				-- edp