Motion of a material whose particle velocity is a linear function of the spatial position

The equations of motion are derived for plane one-dimensional motion of a material whose particle velocity is a linear function of the spatial position. It is proved the mass density is uniform at all times if it is initially uniform. A closed form solution to the equations of motion is obtained for a special case. Sensitivity of the solution to various parameters is investigated. An equation for the efficiency of a power history is derived and discussed.