Sandford's Elevator

An elevator car has a constant upward acceleration of a = 3 ft/sec.  
If a bolt slips free from the bottom of the car and falls in 2 
seconds to the bottom of the shaft, how high is the car when the 
bolt lands?  (Neglect air resistance and assume the acceleration 
of gravity is g = 32 ft/sec^2.)

First Solution
Integrating the car's constant upward acceleration, from an arbitrary 
reference time t0 = 0, when the car's velocity is zero and its height 
is h0, to the time t1 when the bolt slips free, we have the car's 
velocity and height at the time t1:

          v1  =  a t1           h1  =  h0 + (a/2) t1^2

This is also the velocity and height of the bolt when it begins its 
free-fall.  Integrating the constant downward acceleration of gravity, 
g, over the free-fall time T, and taking v1 and h1 as the initial 
conditions, we have

        [1/2] T^2  +  [a t1] T  +  [(a/2)t1^2 + h0]  =  0

Solving this equation for t1, and adding the time to fall, T, gives 
tf, the time when the bolt hits the bottom of the shaft:
                    __________________________
         tf  =  +- / T^2 [1 - (g/a)] - 2[h0/a]

The elevator car has been accelerating upward at  a  since the initial 
instant t0 = 0, so its height at the time tf is

           hf  =  h0  +  (a/2) tf^2 

               =  (1/2) (a - g) T^2

Substituting the given values a = 3 ft/sec^2, g=-32 ft/sec^2, and 
T = 2 sec, we arrive at the result hf = 70 ft.


Second Solution
At the time t the bolt and the car have the same height and velocity.  
Subsequently they receed from each other with a constant relative 
acceleration of a-g (i.e., 35 ft/sec^2) for 2 seconds, at which point 
the bolt reaches the bottom of the shaft.  Therefore, the height of 
the car when the bolt lands is just the relative seperation between 
the bolt and the car after 2 seconds of a constant relative 
acceleration of 35 ft/sec^2, which immediately gives the result 
hf = (1/2)(a-g)T^2 = 70 ft.  


Supplemental Question
Since the statement of the problem gives us no other information, 
we assume the car experiences a constant upward acceleration of 
3 ft/sec^2 indefinitely.  Based on this assumption, and assuming 
the car does not crash, what is the maximum height the car could 
have had when the bolt slipped free?  The answer is 116.9878 ft,
as illustrated by the left-most dotted path in the figure below:

          

This occurs in the case when the elevator's path just skims the
ground at the lowest point on its path.  Of course, the elevator
could have a higher trajectory, and it might seem that the bolt
could be released from a higher point based on one of those higher
trajectories.  However, in order to reach the ground exactly 2
seconds following its release, the bolt would need to be released
nearer and nearer to the elevator's lowest position, so the initial
downward velocity of the bolt would be less, causing it to travel
a lesser distance in the 2 second fall to the ground.  The net
result is that it would need to fall from a lesser height.

On the other hand, if we remove the restriction that the elevator
not crash, we can consider lower trajectories for the elevator than
the one shown above.  This would move the bolt's release time further
to the left (for the left-most "possible path"), and allow the bolt
to be released from a greater height.  Given these allowances, there
is no upper limit (classically) to how far above the ground the bolt
might be released from an upwardly accelerating elevator, and still 
hit the ground in 2 seconds.


Discussion:  
Sandford's Elevator problem is interesting because it illustrates 
how a problem's statement can influence our approach to it's 
solution.  Suppose the problem had been stated as follows:

  An elevator has a constant upward acceleration of 3 ft/sec^2.  
  If a bolt slips free from the bottom of the elevator and falls 
  under the influence of gravity, how far is the elevator from 
  the bolt 2 seconds after they seperate?

In this form, we're compelled to take the relative rather than the 
absolute approach, and we arrive at the "Second Solution" immediately.
However, by simply rephrasing the problem to say that the bolt hits
the GROUND in two seconds, and asking how high *above the ground*
the elevator is when the bolt lands, the reader is encouraged to 
treat the problem in the absolute sense, i.e., relative to the ground, 
even though the ground contact is not relevant to the problem.  Thus, 
the reader is misled by a superfluous reference to the ground (i.e., 
the bolt striking the ground).  The artfulness of the problem is in 
the fact that the reference doesn't SEEM superfluous, because the
sylogistic loop is closed by asking, not how far the car is from 
the bolt, but how far the car is from the ground, even though they 
amount to the same thing.  This creates the conceptual illusion that 
the ground is a significant element of the problem.  Only when the 
arbitrary reference height h  algebraically drops out of the "First 
Solution" do we realize that it was irrelevant to begin with.

Of course, our "a priori" knowledge that the elevator shaft must have 
a bottom does impose certain conditions on the problem that can only 
be resolved via the absolute approach (i.e., the "First Solution"), 
which probably contributes to our psychological willingness to treat 
the ground as a significant element.  The allusion to a "crash" in 
the supplemental problem makes explicit the underlying anxiety that 
inclines us to prefer the fixed ground to an indeterminately moving 
elevator car as our frame of reference.  

For some reason, this reminds me of the following translation, due 
to Robert Bly, of a 15th-century Italian poem, which seems evocative
of the difficulty we face in thinking about Sandford's Elevator:

             "People possess four things
              that are no good at sea:
              anchors, rudders, oars,
              and the fear of going down."