The paper treats the problem of objects in an incompressible fluid of the second grade. The second grade fluid is capable of exhibiting both the normal stress differences, and thus this model has been studied in great detail during last years. First, the authors study the problem of a thin, but heavy, lamina falling under the influence of gravity between two vertical plates. This is followed by the study of a heavy cylinder falling under the influence of gravity inside a vertical tube filled with a second grade fluid. The velocities of the objects and of the fluid are determined by solving an initial-boundary value problem which is treated by using Laplace transform. Explicit exact solutions are obtained. These results can be considered as the first logical step to understand the drag reduction due to the addition of polymers to water.