This work is concerned with the motion of a rigid sphere which is accelerated by an accelerating fluid inside a cylindrical tube where both the sphere and the fluid are initially at rest. The investigation is valid for laminar flow and slow sphere motion relative to the fluid. In Part I an exact solution of the Navier-Stokes equation is obtained for the flow of an incompressible Newtonian fluid in a cylindrical tube under the influence of an imposed pressure gradient which increases linearly with time. Then a one-dimensional momentum balance is used to solve the same problem. Also, since the exact solution does not converge well for very small times, a small time solution is derived using Laplace transform techniques. The solutions are compared graphically and approximate regions of validity for each are obtained. In Part II the motion of a sphere is studied where the sphere has been placed in the fluid of Part I before the fluid motion begins, so that both the sphere and the fluid are initially at rest. A lower bound is found for the velocity of the sphere with respect to the tube for each of the solutions for the fluid velocity of Part I. The results are applied to the problem of the mechanism of seating of prosthetic aortic heart valves.