There has been much discussion recently about whether piano hammers can
accelerate after they have been released to fly freely at the string.

First the definition of acceleration.  Acceleration is defined as the
rate of change of velocity.  Both velocity and acceleration are vector
quantities;  this means that the direction of movement matters too.
A piano hammer is constrained to move in only one direction, though
strictly speaking it is moving in an arc rather than a straight line.
The effect of this deviation from a straight line is negligible.

Ordinary Newtonian mechanics require that to accelerate a mass you must
apply a force to it.  Force = mass * acceleration.  Once you have
stopped applying the force there will be _no_ acceleration.  In the
case of a piano action there will be small frictional forces that will
cause slight negative acceleration (i.e., slow down the hammer) after it
has been released.

There can be no real argument about this. If the hammer is flying free
then it is _not_ accelerating.  Newtonian mechanics are an extremely
good approximation to reality for the speeds and dimensions of a piano
hammer;  we do not need to drag in relativistic, gravitational or any
other forms of exotic mechanics.

Jeffrey Borinsky