  We establish the equivalence of free piston and delta shock, for the
one-space-dimensional pressureless compressible Euler equations. The delta
shock appearing in the singular Riemann problem is exactly the piston that may
move freely forward or backward in a straight tube, driven by the pressureless
Euler flows on two sides of it in the tube. This result not only helps to
understand the physics of the somewhat mysterious delta shocks, but also
provides a way to reduce the fluid-solid interaction problem, which consists of
several initial-boundary value problems coupled with moving boundaries, to a
simpler Cauchy problem. We show the equivalence from three different
perspectives. The first one is from the sticky particles, and derives the
ordinary differential equation (ODE) of the trajectory of the piston by a
straightforward application of conservation law of momentum, which is
physically simple and clear. The second one is to study a coupled
initial-boundary value problem of pressureless Euler equations, with the piston
as a moving boundary following the Newton's second law. It depends on a concept
of Radon measure solutions of initial-boundary value problems of the
compressible Euler equations which enables us to calculate the force on the
piston given by the flow. The last one is to solve directly the singular
Riemann problem and obtain the ODE of delta shock by the generalized
Rankine-Hugoniot conditions. All the three methods lead to the same ODE.
