Consider relativity. If a stick accelerates from a stationary state, how will it look like during the acceleration? I mean how it will be shortened in the sight of an observer in a stationary coordinate system.
It seems I was not clear enough. If the stick becomes shorter, then while this is happening, the observer should see the two ends have different velocities. Then, how big are they?
 A: You're correct that different parts of the object have to accelerate differently. The details are pretty complicated—when you push one end of an object, transient pressure waves bounce around inside it, and it eventually settles down into a new equilibrium state, which happens to be moving and (in the original reference frame) shorter than before.
A simpler situation is a rocket ship that fires its rockets at a constant rate for a fixed amount of time. When the rockets are turned on and off there will be complicated transients, but in between it will settle into a steady state of constant acceleration, and if it accelerates for long enough you can ignore the transients. The interesting thing about this situation is that, in the steady state, the "top" of the ship (the part farthest from the rockets, which will feel like the top if you're standing in it) has a smaller acceleration than the bottom of the ship. You can measure the difference with any accelerometer, like the one in your smartphone. This wouldn't be true in the Newtonian case, and it seems internally inconsistent at first glance, but you can think of it as being related to the ship's increasing length contraction relative to the original rest frame.
A: If we assume
(1) the 'stick' (rod) does not contract according to an observer fixed at any point on the rod and
(2) any point on the rod has constant proper acceleration and
(3) the rod is momentarily at rest at $t=0$ in some inertial frame of reference
then every point on the rod is a Rindler observer which means that every point on the rod has a different proper acceleration!
The worldlines of the ends of the rod and points in between trace hyperbolic world lines as so



but does it still have a well defined proper length can be shown in
  the graph?

From the article "Accelerated Travels" at www.mathpages.com:

Consequently the line from the origin through any point on the
  hyperbolic path represents the space axis for the co-moving inertial
  coordinates of the accelerating worldline at that point. The same
  applies to any other hyperbolic path asymptotic to the same
  lightlines, so a line from the origin intersects any two such
  hyperbolas at points that are mutually simultaneous and separated by a
  constant proper distance (since they are both a fixed proper distance
  from the origin along their mutual space axis). It follows that in
  order for a slender "rigid" rod accelerating along its axis to
  maintain a constant proper length (with respect to its co-moving
  inertial frames), the parts of the rod must accelerate along a family
  of hyperbolas asymptotic to the same lightlines, as illustrated below.



The x',t' axes represent the mutual co-moving inertial frame of the
  hyperbolic worldlines where they intersect with the x' axis. All the
  worldlines have constant proper distances from each other along this
  axis, and all have the same speed.

